Taxas de Convergência para Métodos Iterativos Cíclicos em Problemas Mal Postos

Livre

0
0
77
1 year ago
Preview
Full text

  

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛

❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

P✉r❛ ❡ ❆♣❧✐❝❛❞❛

  

❚❛①❛s ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛

▼ét♦❞♦s ■t❡r❛t✐✈♦s ❈í❝❧✐❝♦s ❡♠

Pr♦❜❧❡♠❛s ▼❛❧ P♦st♦s

  

❘✉❜é♥ ❆❧❡① ▼❛rtí♥❡③ ▼✉ñ♦③

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳➸ ❉r✳ ❆♥tô♥✐♦ ▲❡✐tã♦

  ❋❧♦r✐❛♥ó♣♦❧✐s ❆❜r✐❧ ❞❡ ✷✵✶✺

  ❘✉❜é♥ ❆❧❡① ▼❛rtí♥❡③ ▼✉ñ♦③ ❚❛①❛s ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ▼ét♦❞♦s ■t❡r❛t✐✈♦s ❈í❝❧✐❝♦s ❡♠ Pr♦❜❧❡♠❛s ▼❛❧

P♦st♦s

  ❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐❝❛❞❛✱ ❞♦ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❋í✲ s✐❝❛s ❡ ▼❛t❡♠át✐❝❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛✱ ♣❛r❛ ❛ ♦❜✲ t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡✲ ♠át✐❝❛✱ ❝♦♠ ➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛✳ ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳➸ ❉r✳ ❆♥tô♥✐♦ ▲❡✐✲ tã♦✳

  ❋❧♦r✐❛♥ó♣♦❧✐s ❆❜r✐❧ ❞❡ ✷✵✶✺

  

❚❛①❛s ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ▼ét♦❞♦s

■t❡r❛t✐✈♦s ❈í❝❧✐❝♦s ❡♠ Pr♦❜❧❡♠❛s ▼❛❧

P♦st♦s

  ♣♦r

  ✶

  ❘✉❜é♥ ❆❧❡① ▼❛rtí♥❡③ ▼✉ñ♦③ ❊st❛ ❉✐ss❡rt❛çã♦ ❢♦✐ ❥✉❧❣❛❞❛ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ✏▼❡str❡✑✱ ➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛✱ ❡ ❛♣r♦✈❛❞❛ ❡♠ s✉❛

  ❢♦r♠❛ ✜♥❛❧ ♣❡❧♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐❝❛❞❛✳

  Pr♦❢✳ ❉r✳ ❉❛♥✐❡❧ ●♦♥ç❛❧✈❡s ❈♦♦r❞❡♥❛❞♦r ❞♦ ❈✉rs♦

  ❈♦♠✐ssã♦ ❊①❛♠✐♥❛❞♦r❛ Pr♦❢✳ ❉r✳ ▼❛r❝❡❧♦ ❙♦❜♦tt❦❛ ✲ ❯❋❙❈

  ✭Pr❡s✐❞❡♥t❡ ❞❛ ❇❛♥❝❛✮ Pr♦❢✳ ❉r✳ ❘❛♣❤❛❡❧ ❋❛❧❝ã♦ ❞❛ ❍♦r❛ ✲ ❯❋❙❈ Pr♦❢✳ ❉r✳ ❲❛❣♥❡r ❇❛r❜♦s❛ ▼✉♥✐③ ✲ ❯❋❙❈ Pr♦❢✳ ❉r✳ ❆❞r✐❛♥♦ ❉❡ ❈❡③❛r♦ ✲ ❋❯❘●

  ❋❧♦r✐❛♥ó♣♦❧✐s✱ ❆❜r✐❧ ❞❡ ✷✵✶✺✳

  ✶ ❇♦❧s✐st❛ ❞❛ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧ ❞❡ ◆í✈❡❧ ❙✉♣❡r✐♦r ✲ ❈❆P❊❙

  ✐✐ ✧■♣s❡ s❡ ♥✐❤✐❧ s❝✐r❡ ✐❞ ✉♥✉♠ s❝✐❛t✧✳

  Σωκρ´ ατ ης ✳

  ❆❣r❛❞❡❝✐♠❡♥t♦s

  ❆❣r❛❞❡ç♦✱ ❡♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❙❈✱ q✉❡ ❞❡✉ ✉♠❛ ♦♣♦rt✉♥✐❞❛❞❡ ♣❛r❛ ♠✐♠✱ ❡ t❛♠❜é♠ à ❈❆P❊❙✱ ♣♦r s❡✉ ✈❛❧✐♦s♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦✳

  ❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛ ❝❛❞❛ ✉♠ ❞♦s ♣r♦❢❡ss♦r❡s q✉❡ t✐✈❡ ♥❛s ❞✐✈❡rs❛s ❞✐s❝✐♣❧✐♥❛s✱ ❡♠ ♣❛rt✐❝✉❧❛r ❛ ♠❡✉ ♦r✐❡♥t❛❞♦r ❛❝❛❞ê♠✐❝♦ ■❣♦r ▼♦③♦❧❡✈s❦✐ ❡ ❛ ♠❡✉ ♦r✐❡♥t❛❞♦r ❞❡✜♥✐t✐✈♦ ❆♥tô♥✐♦ ▲❡✐tã♦✱ ✜♥❛❧♠❡♥t❡ q✉❡r♦ ❛❣r❛❞❡❝❡r ❛ ♦s ♣r♦❢❡ss♦r❡s q✉❡ ❛❝❡✐t❛r❛♠ s❡r ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛ ❞❛ ♠✐♥❤❛ ❞❡❢❡s❛✱ ♣❛r❛ t♦❞♦s✱ ♠✐s r❡s♣❡t♦s✳

  ▼❡♥çã♦ ❛♣❛rt❡ é ♣❛r❛ ❊❧✐s❛ ❆♠❛r❛❧✱ ♦❜r✐❣❛❞♦ ♣♦r t✉❞♦✳ ❆❣r❛❞❡ç♦✱ ♥ã♦ ♣♦❞✐❛ s❡r ❞❡ ♦✉tr❛ ♠❛♥❡✐r❛✱ ❛ ♠❡✉s ❛♠✐❣♦s ❡ ❝♦♥❤❡✲

  ❝✐❞♦s q✉❡ ❣❛♥❤❡✐ ❛q✉✐ ♥❡ss❛ ✐❧❤❛ ♠❛❧✉❝❛✿ ❑❛♠❛❧✱ ❘♦❜❡rt♦✱ ❆❧❡ss❛♥❞r♦✱ ▼❛r❝❡❧♦✱ ❲✐tr♦✱ ▼❛r✐♦✱ ❊❞✉❛r❞♦✱ ❘❛ú❧✱ ❡♥✜♠✳✳✳t✉❞❛ ❛ ❣❡♥t❡ q✉❡ ❝♦✲ ♥❤❡❝✐ ❣r❛ç❛s à ❯❋❙❈✱ ❡ ❡♠ ❡s♣❡❝✐❛❧✱ ❛❣r❛❞❡❝❡r à ♠✐♥❤❛ ❛♠✐❣❛ ❏ú✱ ❡❧❛ é ❞❡♠❛✐s✳

  ❆❣r❛❞❡ç♦✱ ❛ ♠✐♥❤❛ ❧✐♥❞❛ ❢❛♠í❧✐❛✱ ❛ ❢❛♠í❧✐❛ é ♦ ♣r✐♥❝✐♣❛❧ s✉♣♦rt❡ ♥❛ ✈✐❞❛✳

  ✐✈

  ❘❡s✉♠♦

  ◆❛ ❝❧❛ss❡ ❞♦s ♠ét♦❞♦s ❞❡ r❡❣✉❧❛r✐③❛çã♦ ✐t❡r❛t✐✈♦s✱ ♦s ♠ét♦❞♦s t✐♣♦ ❑❛❝③♠❛r③ sã♦ ✉♥s ❞♦s ♠ét♦❞♦s ♠❛✐s ✉t✐❧✐③❛❞♦s ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ♥❛ ♠❛t❡♠át✐❝❛ ❛♣❧✐❝❛❞❛✳ ◆♦ ❡♥t❛♥t♦✱ ♥❛ ❧✐t❡r❛t✉r❛ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ r❡✲ s✉❧t❛❞♦s s♦❜r❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡ ❛s r❡s♣❡❝t✐✈❛s t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♥ã♦ é ❛❜✉♥❞❛♥t❡✳ ❊st❡ tr❛❜❛❧❤♦ tr❛t❛ ❞❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ ❛❧❣✉♠❛s ✈❡rsõ❡s ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③✱ ♦❜t❡♥❞♦ ❝♦♥✈❡r❣ê♥❝✐❛ ❡ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ ♠ét♦❞♦ ♠♦❞✐✜❝❛❞♦ ❝♦♠ ✉♠ ♣❛râ♠❡tr♦ ❞❡ r❡❧❛①❛♠❡♥t♦✱ ❡ t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♥♦ ♠ét♦❞♦ ♣❛r❛ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❡♠ ❜❧♦❝♦ ♥❛s ✈❡rsõ❡s s✐♠étr✐❝❛ ❡ ♥ã♦ s✐♠étr✐❝❛✳ ❋✐♥❛❧♠❡♥t❡✱ ❝♦♠♣❛r❛✲s❡ ♠❡✲ ❞✐❛♥t❡ ❡①♣❡r✐♠❡♥t♦s ♥✉♠ér✐❝♦s ♦ ❞❡s❡♠♣❡♥❤♦ ❞♦s ♠ét♦❞♦s ❡st✉❞❛❞♦s ❝♦♠ ♦ ❞❡s❡♠♣❡♥❤♦ ❞❡ ♠ét♦❞♦s ❜❡♠ ❡st❛❜❡❧❡❝✐❞♦s✳

  P❛❧❛✈r❛s✲❈❤❛✈❡✿ Pr♦❜❧❡♠❛s ✐♥✈❡rs♦s✱ ♣r♦❜❧❡♠❛s ♠❛❧ ♣♦st♦s✱ ♠é✲ t♦❞♦s ❞❡ r❡❣✉❧❛r✐③❛çã♦ ✐t❡r❛t✐✈♦s✱ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③✱ t❛①❛s ❞❡ ❝♦♥✲ ✈❡r❣ê♥❝✐❛✳

  ✈✐

  ❆❜str❛❝t

  ■♥ t❤❡ ❝❧❛ss ♦❢ ✐t❡r❛t✐✈❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞s✱ ❑❛❝③♠❛r③ t②♣❡ ♠❡t❤♦❞s ❛r❡ s♦♠❡ ♦❢ t❤❛t ❛r❡ ♠♦r❡ ♦❢t❡♥ ✉s❡❞ t♦ s♦❧✈❡ ♣r♦❜❧❡♠s ✐♥ ❛♣♣❧✐❡❞ ♠❛t❤❡♠❛t✐❝s✳ ❍♦✇❡✈❡r✱ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ t❤❡ ❛♠♦✉♥t ♦❢ ❝♦♥✲ ✈❡r❣❡♥❝❡ r❡s✉❧ts ❛♥❞ t❤❡✐r ❝♦♥✈❡r❣❡♥❝❡ r❛t❡ ✐s ♥♦t ❛❜✉♥❞❛♥t✳ ❚❤✐s ✇♦r❦ ❞❡❛❧s ✇✐t❤ t❤❡ ❛♥❛❧②s✐s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ s♦♠❡ ✈❡rs✐♦♥s ♦❢ t❤❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ♠❡t❤♦❞✱ ♦❜t❛✐♥✐♥❣ ❝♦♥✈❡r❣❡♥❝❡ ❛♥❞ st❛❜✐❧✐t② ♦❢ t❤❡ ♠♦❞✐✜❡❞ ♠❡t❤♦❞ ✇✐t❤ ❛ r❡❧❛①❛t✐♦♥ ♣❛r❛♠❡t❡r✱ ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ r❛✲ t❡s ❢♦r ♠❡t❤♦❞ ❢♦r ❧✐♥❡❛r ❜❧♦❝❦ ♦♣❡r❛t♦rs ✐♥ ✈❡rs✐♦♥s s②♠♠❡tr✐❝❛❧ ❛♥❞ ♥♦♥✲s②♠♠❡tr✐❝❛❧✳ ❋✐♥❛❧❧②✱ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ♠❡t❤♦❞s ✐s ❝♦♠♣❛r❡❞ ✇✐t❤ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ✇❡❧❧✲❡st❛❜❧✐s❤❡❞ ♠❡t❤♦❞s✳

  ❑❡②✇♦r❞s✿ ■♥✈❡rs❡ ♣r♦❜❧❡♠s✱ ✐❧❧✲♣♦ss❡❞ ♣r♦❜❧❡♠s✱ ✐t❡r❛t✐✈❡ r❡❣✉✲ ❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞s✱ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③✱ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡s✳

  ✈✐✐✐

  ❈♦♥t❡ú❞♦

  ■♥tr♦❞✉çã♦ ✶

  ✶ ❘❡❣✉❧❛r✐③❛çã♦ ❞❡ Pr♦❜❧❡♠❛s ▼❛❧ P♦st♦s ♣♦r ▼ét♦❞♦s ■t❡r❛t✐✈♦s

  ✸ ✶✳✶ Pr♦❜❧❡♠❛s ▼❛❧ P♦st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❘❡❣✉❧❛r✐③❛çã♦ ♣♦r ▼ét♦❞♦s ■t❡r❛t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

  ✶✳✷✳✶ ▼ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳✷ ▼ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❈❧áss✐❝♦ ❡ ♦ ▼é✲ t♦❞♦ ▲♦♣✐♥❣✲▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③✭

  ▲

  ▲❑✮ ❡♠ ❊s✲ ♣❛ç♦s ❞❡ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

  ✶✳✷✳✸ ❘❡❧❛çã♦ ❍✐stór✐❝❛ ❞❡ ❘❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷ ❈♦♥✈❡r❣ê♥❝✐❛ ❞♦ ▼ét♦❞♦ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ✶✶ ✸ ❚❛①❛s ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ✷✼

  ✸✳✶ Pr♦❜❧❡♠❛s ■♥✈❡rs♦s ❡♠ ❆♥á❧✐s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✷ ▼ét♦❞♦s ❚✐♣♦ ❑❛❝③♠❛r③ ❡ ❖♣❡r❛❞♦r❡s ❡♠ ❇❧♦❝♦ ✳ ✳ ✳ ✳ ✷✾

  ✸✳✷✳✶ ▼ét♦❞♦ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ♥ã♦ ❙✐♠étr✐❝♦ ❝♦♠ Pré✲❈♦♥❞✐❝✐♦♥❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

  ✸✳✷✳✷ ▼ét♦❞♦ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❙✐♠étr✐❝♦ ❝♦♠ Pré✲ ❈♦♥❞✐❝✐♦♥❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

  ✸✳✸ ❚❛①❛s ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ▼ét♦❞♦ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ♥ã♦ ❙✐♠étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

  ✸✳✹ ❚❛①❛s ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ▼ét♦❞♦ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❙✐♠étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

  ✹ ❊①♣❡r✐ê♥❝✐❛s ◆✉♠ér✐❝❛s ✺✶ ✹✳✶ ▼ét♦❞♦s ■t❡r❛t✐✈♦s ❛ ❙❡r❡♠ ❈♦♠♣❛r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✷ ❉❡s❝r✐çã♦ ❞♦ Pr♦❜❧❡♠❛ ❚❡st❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

  ✹✳✷✳✶ ●❡r❛♥❞♦ ❉❛❞♦s ❊①❛t♦s ❡ ❝♦♠ ❘✉í❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

  ① ✹✳✸ ❘❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

  ✹✳✸✳✶ ❉❛❞♦s ❊①❛t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✸✳✷ ❉❛❞♦s ❝♦♠ ❘✉í❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

  ✹✳✹ ❈♦♥❝❧✉sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ❈♦♥❝❧✉sõ❡s ●❡r❛✐s

  ✺✾ ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✻✶

  ■♥tr♦❞✉çã♦

  ❯♠❛ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ ♣r♦❜❧❡♠❛s ❡♠ ❡♥❣❡♥❤❛r✐❛ ❡ ❝✐ê♥❝✐❛s ❛♣❧✐✲ ❝❛❞❛s ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ ❢ís✐❝❛ x

  ❛ ♣❛rt✐r ❞❡ ❞❛❞♦s ♦❜s❡r✈❛❞♦s ❞❡ ✉♠❛ ♦✉tr❛ q✉❛♥t✐❞❛❞❡ ❢ís✐❝❛ y✳ ❊ss❛s q✉❛♥t✐❞❛❞❡s ❡stã♦ ❝♦♥❡❝t❛❞❛s ♠❡❞✐❛♥t❡ ✉♠❛ r❡❧❛çã♦ ❢✉♥❝✐♦♥❛❧ ❞❛❞❛ ❞❡ ❢♦r♠❛ ❡①♣❧í❝✐t❛ ♦✉ ✐♠♣❧í❝✐t❛✳ ❊♠❜♦r❛ ❛ r❡❧❛çã♦ ❢✉♥❝✐♦♥❛❧ s❡❥❛ ❝♦♥❤❡✲ ❝✐❞❛ ❞❡ ❢♦r♠❛ ❡①♣❧í❝✐t❛✱ ♥❡♠ s❡♠♣r❡ é ♣♦ssí✈❡❧ ♦❜t❡r ❞❡ ❢♦r♠❛ ❛♥❛❧ít✐❝❛ ❛ q✉❛♥t✐❞❛❞❡ x ❡♠ ❢✉♥çã♦ ❞❡ y✳ P❛r❛ r❡s♦❧✈❡r ❡ss❡ ♣r♦❜❧❡♠❛✱ ✉s❛♠✲s❡ ❢❡rr❛♠❡♥t❛s ♥✉♠ér✐❝❛s✱ ♠❛s q✉❛♥❞♦ ❛ r❡❧❛çã♦ ❢✉♥❝✐♦♥❛❧ ✐♥✈❡rs❛ q✉❡ ❝♦✲ ♥❡❝t❛ y ❝♦♠ x ♥ã♦ t❡♠ ✏❜♦❛s ♣r♦♣r✐❡❞❛❞❡s✑ ❛s ❛♣r♦①✐♠❛çõ❡s ♥✉♠ér✐❝❛s t♦r♥❛♠✲s❡ ✐♥stá✈❡✐s✳ ❆q✉✐ ❡♥tr❛♠ ♥♦ ❥♦❣♦ ♦s ♠ét♦❞♦s ❞❡ r❡❣✉❧❛r✐③❛çã♦ ♣❛r❛ t♦r♥❛r ❡stá✈❡✐s ♣r♦❜❧❡♠❛s ❞❡ ❛♣r♦①✐♠❛çã♦ ✐♥stá✈❡✐s✳

  ◆♦ ❈❛♣ít✉❧♦ ✶✱ ❞❡✜♥❡♠✲s❡ ♦s ❝♦♥❝❡✐t♦s ❞❡ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ♠❛❧ ♣♦st♦✱ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ❡ ❛s ♥♦çõ❡s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡ t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ss❡s ♠ét♦❞♦s✳ ❉❡♣♦✐s✱ ❡①♣õ❡♠✲s❡ ♦s ♠ét♦❞♦s ♥♦s q✉❛✐s ❡stá ❜❛s❡❛❞♦ ♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✿ ♦s ♠ét♦❞♦s ❞❡ ▲❛♥❞✇❡❜❡r ❡ δ

  ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ♣❛r❛ ❞❛❞♦s ❡①❛t♦s✱ y✱ ❡ ❞❛❞♦s ❝♦♠ r✉í❞♦ y ✳ ❆q✉✐ ❛ t❡♦r✐❛ ♥ã♦ é tr❛t❛❞❛ ❞❡t❛❧❤❛❞❛♠❡♥t❡✱ ❥á q✉❡ ♦ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛✲ ❜❛❧❤♦ é ❡st❛❜❡❧❡❝❡r t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦s ♠ét♦❞♦s ❡st✉❞❛❞♦s✳ ◆♦ ❡♥t❛♥t♦✱ ✉♠❛ ❛❜♦r❞❛❣❡♠ r✐❣♦r♦s❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ♥❛s r❡❢❡rê♥✲ ❝✐❛s✳ ❋✐♥❛❧♠❡♥t❡✱ ❛♣r❡s❡♥t❛✲s❡ ✉♠ r❡s✉♠♦ ❞♦s r❡s✉❧t❛❞♦s ❞❡ ❝♦♥✈❡r❣ê♥✲ ❝✐❛✴t❛①❛s ♣❛r❛ ♦s ♠ét♦❞♦s t✐♣♦ ❑❛❝③♠❛r③ ♦❜t✐❞♦s ♥♦s ú❧t✐♠♦s ❛♥♦s✳

  ◆♦ ❈❛♣ít✉❧♦ ✷✱ ❛♣r❡s❡♥t❛✲s❡ ❡♠ ❞❡t❛❧❤❡ ❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ♣❛r❛ ♦♣❡r❛❞♦r❡s ♥ã♦ ❧✐♥❡❛r❡s✱ t❛♥t♦ ♣❛r❛ ❞❛❞♦s ❡①❛t♦s ❝♦♠♦ ♣❛r❛ ❞❛❞♦s ❝♦♠ r✉í❞♦✳

  ◆♦ ❈❛♣ít✉❧♦ ✸✱ ❛♣r❡s❡♥t❛♠✲s❡ ❛s ✈❡rsõ❡s ♥ã♦ s✐♠étr✐❝❛ ❡ s✐♠étr✐❝❛ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ♣❛r❛ ♦♣❡r❛❞♦r❡s ❡♠ ❜❧♦❝♦s ❧✐♥❡❛r❡s ❡ s❡ ❡st❛❜❡❧❡❝❡♠ t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ❛♠❜❛s ✈❛r✐❛♥t❡s ❞♦ ♠ét♦❞♦✳

  ◆♦ ❈❛♣ít✉❧♦ ✹✱ ❝♦♠♣❛r❛✲s❡ ♦ ❞❡s❡♠♣❡♥❤♦ ❞♦s ♠ét♦❞♦s ❡st✉❞❛❞♦s ♥♦s ❈❛♣ít✉❧♦s ✷ ❡ ✸ ❝♦♠ ♠ét♦❞♦s t✐♣♦ ▲❛♥❞✇❡❜❡r ❡ ♦ ♠ét♦❞♦ ❙t❡❡♣❡st ❉❡s❝❡♥t✳

  ✷

  ❈❛♣ít✉❧♦ ✶ ❘❡❣✉❧❛r✐③❛çã♦ ❞❡

Pr♦❜❧❡♠❛s ▼❛❧ P♦st♦s ♣♦r

▼ét♦❞♦s ■t❡r❛t✐✈♦s

  ◆❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ❞❡✜♥✐❞♦s ❝♦♥❝❡✐t♦s t❛✐s ❝♦♠♦ ✏Pr♦❜❧❡♠❛s ✐♥✲ ✈❡rs♦s✑✱ ✏Pr♦❜❧❡♠❛s ■♥✈❡rs♦s ▼❛❧ P♦st♦s✑ ❡ ♦ q✉❡ s❡ ❡♥t❡♥❞❡ ♣♦r ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ♣❛r❛ ❡ss❛ ❝❧❛ss❡ ❞❡ ♣r♦❜❧❡♠❛s ❥✉♥t♦ às ♥♦✲ çõ❡s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡ t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♠ét♦❞♦s ❞❡ r❡❣✉❧❛✲ r✐③❛çã♦✳ ❉❡✜♥✐❞♦ ✐ss♦✱ ❛♣r❡s❡♥t❛♠✲s❡ ❝♦♠♦ ❡str❛té❣✐❛s ❞❡ r❡❣✉❧❛r✐③❛✲ çã♦ ♣❛r❛ ♣r♦❜❧❡♠❛s ✐♥✈❡rs♦s ♠❛❧ ♣♦st♦s✱ ♦s ♠ét♦❞♦s ❞❡ ▲❛♥❞✇❡❜❡r ❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ♥♦ ❝♦♥t❡①t♦ ❞❡ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt ❣❡r❛✐s✳

  ✶✳✶ Pr♦❜❧❡♠❛s ▼❛❧ P♦st♦s

  ❖s ♣r♦❜❧❡♠❛s ✐♥✈❡rs♦s ♣❡rt❡♥❝❡♠ ❛ ✉♠❛ ❞❛s ár❡❛s ❞❛ ♠❛t❡♠át✐❝❛ ❛♣❧✐❝❛❞❛ ❝♦♠ ♠❛✐♦r ❝r❡s❝✐♠❡♥t♦ ♥❛s ú❧t✐♠❛s ❞é❝❛❞❛s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣♦rq✉❡ ♠✉✐t♦s ♣r♦❜❧❡♠❛s ♣ró♣r✐♦s ❞❛ ❡♥❣❡♥❤❛r✐❛ ❡ ❝✐ê♥❝✐❛s ♣♦❞❡♠ s❡ ❡♥q✉❛❞r❛r ♥❡ss❛ ❝❧❛ss❡ ❞❡ ♣r♦❜❧❡♠❛s✳

  ❙❡ ❡♥t❡♥❞❡ ♣♦r Pr♦❜❧❡♠❛ ■♥✈❡rs♦ ❛q✉❡❧❡ q✉❡ ❝♦♥s✐st❡ ❡♠ ❞❡t❡r♠✐♥❛r ❝❛✉s❛s ❡♠ ❢✉♥çã♦ ❞❡ ❡❢❡✐t♦s ♦❜s❡r✈❛❞♦s✳ ❙❡♥❞♦ ♠❛✐s ♣r❡❝✐s♦✱ ❛♦ ❢❛❧❛r ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ✐♠♣❧✐❝✐t❛♠❡♥t❡ s❡ ❡stá ❢❛❧❛♥❞♦ ❞❡ ✉♠ ♦✉tr♦ ♣r♦❜❧❡♠❛✱ q✉❡ s❡r✐❛ ♦ ♣r♦❜❧❡♠❛ ❞✐r❡t♦✳ ❆♠❜♦s ♣r♦❜❧❡♠❛s sã♦ ✐♥✈❡rs♦s ♠✉t✉❛♠❡♥t❡ ❡ ♣♦rt❛♥t♦ ❛ s♦❧✉çã♦ ❞❡ ✉♠ ❡♥✈♦❧✈❡ ❛ s♦❧✉çã♦ ❞♦ ♦✉tr♦✳

  ✹ ❋♦r♠❛❧♠❡♥t❡✱ ♥♦ ❝♦♥t❡①t♦ ❞♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✱ ❞❛❞♦ ✉♠ ♦♣❡r❛❞♦r

  F : X → Y ❡♥tr❡ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt X ❡ Y ✱ ❛ ❡q✉❛çã♦

  F (x) = y ✭✶✳✶✮

  é ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❣❡r❛❧ ❞♦ ♣r♦❜❧❡♠❛ ❞✐r❡t♦✲✐♥✈❡rs♦❀ ♦ ♣r♦❜❧❡♠❛ ❞✐✲ r❡t♦ é✿ ❞❛❞♦ x ∈ X ❛❝❤❛r y ∈ Y ✱ ❡ ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦✿ ❞❛❞♦ y ∈ Y ❛❝❤❛r x ∈ X✳ ●❡r❛❧♠❡♥t❡✱ ♦s ♣r♦❜❧❡♠❛s ✐♥✈❡rs♦s q✉❡ ❛♣❛r❡❝❡♠ ♥❛ ♣rát✐❝❛ ❝♦rr❡s♣♦♥❞❡♠ ❛ ♠♦❞❡❧♦s ♠❛t❡♠át✐❝♦s q✉❡ ♥ã♦ sã♦ ❜❡♠ ♣♦st♦s✱ ♦♥❞❡ s❡❣✉♥❞♦ ❍❛❞❛♠❛r❞✱ ✉♠ ♣r♦❜❧❡♠❛ ♠❛t❡♠át✐❝♦ ❞❡✜♥✐❞♦ ♠❡❞✐❛♥t❡ ❛ ❡q✉❛çã♦ ✭✶✳✶✮ é ❜❡♠ ♣♦st♦ s❡ ♣❛r❛ t♦❞♦s ♦s ❞❛❞♦s ❛❞♠✐ssí✈❡✐s ❡①✐st❡ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ x✱ ❛ q✉❛❧ ❞❡♣❡♥❞❡ ❝♦♥t✐♥✉❛♠❡♥t❡ ❞♦s ❞❛❞♦s✳ ❙❡ ✉♠❛ ❞❡ss❛s ❝♦♥❞✐çõ❡s é ✈✐♦❧❛❞❛✱ ♦ ♣r♦❜❧❡♠❛ é ❝❤❛♠❛❞♦ ❞❡ ♠❛❧ ♣♦st♦✱ ✐st♦ é✱ ♦ ❝r✐tér✐♦ ❞❡ ❍❛❞❛♠❛r❞ ❞✐③ q✉❡ ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ♥❛ ❡q✉❛çã♦ ✭✶✳✶✮

  −1

  é ❜❡♠ ♣♦st♦ s❡ ♦ ♦♣❡r❛❞♦r F é ✉♠❛ ❜✐❥❡çã♦ ❝♦♠ ✐♥✈❡rs❛ F ❝♦♥tí♥✉❛✳ ●❡r❛❧♠❡♥t❡✱ ♥❛ ♣rát✐❝❛✱ ♥ã♦ s❡ ❝♦♥❤❡❝❡ ❝♦♠ ♣r❡❝✐sã♦ ♦s ✈❛❧♦r❡s

  ❞♦s ❞❛❞♦s y ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦✱ ♣♦ré♠✱ só s❡ ❞✐s♣õ❡ ❞❡ ❞❛❞♦s δ ❛♣r♦①✐♠❛❞♦s y ✱ ♦s q✉❛✐s ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♣❡rt❡♥❝❡♠ à ✐♠❛❣❡♠ ❞♦ ♦♣❡r❛❞♦r F ✱ ❝✉❥♦ ❡rr♦ ♦✉ ♥í✈❡❧ ❞❡ r✉í❞♦ é ❝♦♠✉♠❡♥t❡ ❣❡r❛❞♦ ♣❡❧♦s ✐♥s✲ tr✉♠❡♥t♦s ❞❡ ♠❡❞✐çã♦ ♦s q✉❛✐s t❡♠ ✉♠❛ ♣r❡❝✐sã♦ ♦❜✈✐❛♠❡♥t❡ ❧✐♠✐t❛❞❛ ✭♥ã♦ ✐♥✜♥✐t❛✮✳ ❖s ❞❛❞♦s ❛♣r♦①✐♠❛❞♦s ✜❝❛♠ ❛ ✉♠❛ ❞✐stâ♥❝✐❛ q✉❡ ♥ã♦ s✉♣❡r❛ ♦ ✈❛❧♦r δ > 0✱ ✐st♦ é δ

  ✭✶✳✷✮ ky − y k ≤ δ ♦♥❞❡ δ é ❝❤❛♠❛❞♦ ♦ ♥í✈❡❧ ❞❡ ❡rr♦ ♦✉ ♥í✈❡❧ ❞❡ r✉í❞♦✳

  ❋r❡q✉❡♥t❡♠❡♥t❡ ♥❛ ♣rát✐❝❛✱ ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ♥❛ ❡q✉❛çã♦ ✭✶✳✶✮ é ♠❛❧ ♣♦st♦ ♦✉ ♠❛❧ ❝♦♥❞✐❝✐♦♥❛❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❛♥❞♦ ♦ ♦♣❡r❛❞♦r F é ❝♦♠♣❛❝t♦ ❡ ♥ã♦ ❞❡❣❡♥❡r❛❞♦✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ ❛ ✐♥✈❡rs❛ ♥ã♦ é ❝♦♥tí✲ ♥✉❛✳ ▼❛t❡♠❛t✐❝❛♠❡♥t❡✱ ❛ q✉❡stã♦ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣♦❞❡ s❡r ❡♥❢r❛q✉❡❝✐❞❛ ❛✉♠❡♥t❛♥❞♦ ♦ ❡s♣❛ç♦ s♦❧✉çã♦✳ ❊♠ r❡❧❛çã♦ à ✉♥✐❝✐❞❛❞❡ s✉❛ ❛✉sê♥❝✐❛ ✐♥❞✐❝❛ q✉❡ ❡stã♦ ❢❛❧t❛♥❞♦ ✐♥❢♦r♠❛çõ❡s ♥♦ ♠♦❞❡❧♦✳ P♦❞❡✲s❡ ❛❞✐❝✐♦♥❛r ♣r♦♣r✐❡❞❛❞❡s✱ ✐♠♣♦r ❝♦♥❞✐çõ❡s ❛♦ ♦♣❡r❛❞♦r✱ ♦✉ ❛✐♥❞❛✱ ❡s❝♦✲ ❧❤❡r ✉♠❛ ❡♥tr❡ ❛s ✈ár✐❛s s♦❧✉çõ❡s✱ ❛ q✉❛❧ ♠❡❧❤♦r ❝♦rr❡s♣♦♥❞❛ ❛♦ ♠♦❞❡❧♦✳ ❆❣♦r❛ ♥♦ ❝❛s♦ ❞❡ q✉❡ ❡①✐st❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ♥❛ ❡q✉❛çã♦ ✭✶✳✶✮ ❛ ✈✐♦❧❛çã♦ ❞♦ t❡r❝❡✐r♦ ❝r✐tér✐♦ ❞❡ ❍❛❞❛♠❛r❞✱ ✐st♦ é✱ ❛ ❞❡♣❡♥❞ê♥✲ ❝✐❛ ❝♦♥tí♥✉❛ ❞♦s ❞❛❞♦s✱ é ❞❡ ✈✐t❛❧ ✐♠♣♦rtâ♥❝✐❛✱ ❥á q✉❡ ❛ ❛✉sê♥❝✐❛ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❞❡♣❡♥❞ê♥❝✐❛ ❝♦♥tí♥✉❛ ❞♦s ❞❛❞♦s ♣♦❞❡ ♣r♦✈♦❝❛r sér✐❛s ❞✐✜❝✉❧❞❛❞❡s ♥✉♠ér✐❝❛s✳ P❛r❛ ♠❛♥t❡r ❡ss❛s ✐♥st❛❜✐❧✐❞❛❞❡s s♦❜ ❝♦♥tr♦❧❡✱ ♣r♦♣õ❡♠✲s❡ ♦s ❝❤❛♠❛❞♦s ▼ét♦❞♦s ❞❡ ❘❡❣✉❧❛r✐③❛çã♦✱ ♦s q✉❛✐s✱ ♣❡r♠✐✲ t❡♠ ♦❜t❡r ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❡stá✈❡❧ ❡ ❝♦♥✈❡r❣❡♥t❡ ✭❝♦♥❝❡✐t♦s ❛ s❡r❡♠ ❞❡✜♥✐❞♦s ♠❛✐s ♣❛r❛ ❢r❡♥t❡✮ ♣❛r❛ ✉♠❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❡♠ ❢✉♥çã♦ ❞♦ ♥í✈❡❧ ❞❡ r✉í❞♦ δ✳ ❊♠ t❡r♠♦s ❣❡r❛✐s✱ ♦ ♣r♦❝❡ss♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦

  ✺ é ❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ♠❛❧ ♣♦st♦ ♣♦r ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ♣r♦✲ ❜❧❡♠❛s ❜❡♠ ♣♦st♦s✳ ❆ ♠❛✐♦r ❞✐✜❝✉❧❞❛❞❡ ♥✉♠ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ♠❛❧

  ∗

  ♣♦st♦ ♥❛ ❢♦r♠❛ ❞❛ ❡q✉❛çã♦ ✭✶✳✶✮✱ é ♦ ❝á❧❝✉❧♦ ❞❡ ✉♠❛ s♦❧✉çã♦ x q✉❛♥❞♦ δ

  −1

  ❞✐s♣õ❡✲s❡ ❞❡ ❞❛❞♦s ❝♦♠ r✉í❞♦ y ✱ ❥á q✉❡ ❝♦♠♦ ♦ ♦♣❡r❛❞♦r ✐♥✈❡rs♦ F é ❞❡s❝♦♥tí♥✉♦✱ ♣♦❞❡ ❣❡r❛r ✭❡ ❣❡r❛❧♠❡♥t❡ ❣❡r❛✮ s♦❧✉çõ❡s ✐♥❛❞❡q✉❛❞❛s ♣❛r❛ δ ♦ ♣r♦❜❧❡♠❛✳ P♦rt❛♥t♦✱ é ♥❡❝❡ssár✐♦ ♣r♦❝✉r❛r ✉♠❛ ❛♣r♦①✐♠❛çã♦ x ❞❡ α

  ∗

  x q✉❡ s❡❥❛ ♣♦ssí✈❡❧ ❞❡ ❝❛❧❝✉❧❛r ❞❡ ♠❛♥❡✐r❛ ❡stá✈❡❧✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ δ ❞❡♣❡♥❞❛ ❝♦♥t✐♥✉❛♠❡♥t❡ ❞♦s ❞❛❞♦s ♦❜s❡r✈❛❞♦s y ✱ ❡ q✉❡ ❝♦♥✈✐r❥❛ à s♦✲

  ∗

  ❧✉çã♦ x q✉❛♥❞♦ ♦ ♥í✈❡❧ ❞❡ r✉í❞♦ t❡♥❞❡ ♣❛r❛ ③❡r♦✱ ❞❛❞❛ ✉♠❛ ❡s❝♦❧❤❛ ❛❞❡q✉❛❞❛ ❞♦ ♣❛râ♠❡tr♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ α✳

  P❛r❛ ✈❡r ❡♠ ❞❡t❛❧❤❡ ❛ t❡♦r✐❛ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ♣❛r❛ ♣r♦❜❧❡♠❛s ✐♥✈❡r✲ s♦s s✉❣❡r❡✲s❡ ❛♦ ❧❡✐t♦r ❝♦♥s✉❧t❛r ❬✽❪✱ ♦♥❞❡ sã♦ ❛♣r❡s❡♥t❛❞❛s ❛❧t❡r♥❛t✐✈❛s ❞❡ ❡s❝♦❧❤❛ ❞♦ ♣❛râ♠❡tr♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ α✳ ➱ ✐♠♣♦rt❛♥t❡ r❡ss❛❧t❛r q✉❡ ❛ ❡s❝♦❧❤❛ ❞♦ ♣❛râ♠❡tr♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ r❡q✉❡r ✉♠ ❜❛❧❛♥ç♦ ❡♥tr❡ ❡st❛✲ δ ❜✐❧✐❞❛❞❡ ❡ ♣r❡❝✐sã♦ ♣❛r❛ ❛ s♦❧✉çã♦ ❛♣r♦①✐♠❛❞❛ r❡❣✉❧❛r✐③❛❞❛ x α ✳ P❛r❛ t❛❧✱ ✉♠❛ ❡st✐♠❛t✐✈❛ ❞❡ δ ♥❛ r❡❧❛çã♦ ✭✶✳✷✮ é ❢✉♥❞❛♠❡♥t❛❧✳

  ◗✉❛♥❞♦ ✉♠ ♠ét♦❞♦ ❞❡ ❛♣r♦①✐♠❛çã♦ é ❞❡ ❢❛t♦ ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉✲ ❧❛r✐③❛çã♦✱ ✐st♦ é✱ é ❡stá✈❡❧✿ δ

δ→0

x α −→ x α

  ❡ é ❝♦♥✈❡r❣❡♥t❡✿ δ

α,δ→0

∗ x , α −→ x

  ❝❛❜❡ ❛ ♣❡r❣✉♥t❛ s♦❜r❡ ❛ r❛♣✐❞❡③ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛✳ ■♥❢❡❧✐③♠❡♥t❡✱ ❡♠ ❣❡r❛❧ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ♣❛r❛ r❡s♦❧✈❡r ✉♠ ♣r♦❜❧❡♠❛ ♠❛❧ ♣♦st♦ ♣♦❞❡ s❡r ❛r❜✐tr❛r✐❛♠❡♥t❡ ❧❡♥t❛✱ ♣♦r ✐ss♦ ❛s t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣♦❞❡♠ s❡r ♦❜t✐❞❛s s♦♠❡♥t❡ ❡♠ s✉❜❝♦♥❥✉♥t♦s ❞❡ X

  ∗

  ❡ ❞❡t❡r♠✐♥❛❞❛s ♣♦r ✐♥❢♦r♠❛çõ❡s ❛✲♣r✐♦r✐ s♦❜r❡ ❛ s♦❧✉çã♦ ❡①❛t❛ x ♦✉ s♦❜r❡ ♦s ❞❛❞♦s y✳ ❊st❛s ✐♥❢♦r♠❛çõ❡s ❛✲♣r✐♦r✐ sã♦ ❢♦r♠✉❧❛❞❛s ❡♠ t❡r♠♦s ❞❛s ❝❤❛♠❛❞❛s ❝♦♥❞✐çõ❡s ❞❡ ❢♦♥t❡ ♦✉ s♦✉r❝❡ ❝♦♥❞✐t✐♦♥s ✭♣❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s ✈❡❥❛ ❬✽❪✮✳ ❋✐♥❛❧♠❡♥t❡✱ ❡♥t❡♥❞❡✲s❡ ♣♦r t❛①❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ❛ t❛①❛ ♣❛r❛ ❛ q✉❛❧ ❛❝♦♥t❡❝❡ δ δ→0

kx α − x k −→ 0.

  ✻

  

✶✳✷ ❘❡❣✉❧❛r✐③❛çã♦ ♣♦r ▼ét♦❞♦s ■t❡r❛t✐✈♦s

  ◆❛ ♣rát✐❝❛✱ ♦ ♣r♦❜❧❡♠❛ ❞✐r❡t♦ é ♠✉✐t♦ ♠❡❧❤♦r ❡♥t❡♥❞✐❞♦ q✉❡ ♦ ♣r♦✲ ❜❧❡♠❛ ✐♥✈❡rs♦✱ ♣♦r ✐ss♦✱ ♣❛r❛ r❡s♦❧✈❡r ♦s ♣r✐♠❡✐r♦s ❡①✐st❡ ✉♠❛ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ ❜♦♥s ❛❧❣♦r✐t♠♦s ❡ s♦❢t✇❛r❡ ❝♦♠♣✉t❛❝✐♦♥❛❧✱ ✐st♦ é✱ s♦❢t✲ ✇❛r❡ ♣❛r❛ ❛✈❛❧✐❛r F (x) ❞❛❞♦ ✉♠ ❞❡t❡r♠✐♥❛❞♦ x ∈ X✳ ❊♥tã♦✱ é ✉♠❛ ❜♦❛ ✐❞❡✐❛ t❡♥t❛r r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ♠❡❞✐❛♥t❡ ✐t❡r❛çõ❡s s✉❝❡ss✐✈❛s✱ ✉s❛♥❞♦ ❡ss❛ ❜❛t❡r✐❛ ❞❡ s♦❢t✇❛r❡s ❡ ❛❧❣♦r✐t♠♦s✳

  ❖s ♠ét♦❞♦s ✐t❡r❛t✐✈♦s t❡♠ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❛✉t♦✲r❡❣✉❧❛r✐③❛çã♦✱ ♥♦ s❡♥t✐❞♦ q✉❡ ♦ tér♠✐♥♦ ❛♥t❡❝✐♣❛❞♦ ❞♦ ♣r♦❝❡ss♦ ✐t❡r❛t✐✈♦ t❡♠ ✉♠ ❡❢❡✐t♦ r❡❣✉❧❛r✐③❛❞♦r✱ ♦✉ s❡❥❛✱ ♦ í♥❞✐❝❡ ❞❛ ✐t❡r❛çã♦ ❢❛③ ❛ ❢✉♥çã♦ ❞♦ ♣❛râ♠❡tr♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ α ❡✱ ♦ ❝r✐tér✐♦ ❞❡ ♣❛r❛❞❛ ❢❛③ ❛ ❢✉♥çã♦ ❞♦ ♠ét♦❞♦ ❞❡ s❡❧❡çã♦ ❞♦ ♣❛râ♠❡tr♦✳ P❛r❛ ✈❡r ✉♠ ❡st✉❞♦ ❞❡t❛❧❤❛❞♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ r❡❣✉❧❛r✐③❛çã♦ q✉❡ ♣♦ss✉❡♠ ♦s ♠ét♦❞♦s ✐t❡r❛t✐✈♦s ✈❡❥❛ ❬✽✱✶✻❪✳

  ✶✳✷✳✶ ▼ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r

  ❆ ❝♦♥❞✐çã♦ ❞❡ ♦t✐♠❛❧✐❞❛❞❡ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ♠í♥✐♠♦s q✉❛❞r❛❞♦s ♥ã♦ ❧✐♥❡❛r

  1 2 min , kF (x) − yk

  x∈X

  2 é ❛ ❡q✉❛çã♦ ♥♦r♠❛❧

  

′ ∗ ′ ∗

  F (x) F (x) = F (x) y, ✭✶✳✸✮

  ′ ′ ∗

  (x) (x)

  ♦♥❞❡ F é ❛ ❞❡r✐✈❛❞❛ ❞❡ ❋ré❝❤❡t ❞❡ F ❡♠ x ∈ X ❡ F é ♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ♦♣❡r❛❞♦r ❛❞❥✉♥t♦✳ ❚r❛♥s❢♦r♠❛♥❞♦ ❛ ❡①♣r❡ssã♦ ✭✶✳✸✮ ❡♠ ✉♠❛ ✐t❡r❛çã♦ ❞❡ ♣♦♥t♦ ✜①♦ ♦❜té♠✲s❡

  

′ ∗

  x = x (x) (F (x) δ δ − F − y), ✭✶✳✹✮ ❧♦❣♦ ❞❡♥♦t❛♥❞♦ ♣♦r x ❛s ✐t❡r❛çõ❡s ♥♦ ❝❛s♦ ❞❡ ❞❛❞♦s ❝♦♠ r✉í❞♦ y ✭❡ n ♣♦r x n ♥♦ ❝❛s♦ ❞❡ ❞❛❞♦s ❡①❛t♦s y✮ ♦❜té♠✲s❡ ❛ ✐t❡r❛çã♦ ❞❡ ▲❛♥❞✇❡❜❡r δ δ δ δ δ

  

′ ∗

x = x (x ) (F (x ) ). n+1 n − F n n − y ✭✶✳✺✮

  ❯♠ ❡st✉❞♦ ❞❡t❛❧❤❛❞♦ ❞❡st❡ ♠ét♦❞♦ ♣❛r❛ ♦s ❝❛s♦s ❡♠ q✉❡ ♦ ♦♣❡r❛❞♦r F é ❧✐♥❡❛r ♦✉ ♥ã♦ ❧✐♥❡❛r ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✽✱✶✻❪✳

  ✼

  

✶✳✷✳✷ ▼ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❈❧áss✐❝♦ ❡

  ♦ ▼ét♦❞♦ ▲♦♣✐♥❣✲▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③✭ ▲❑✮ ❡♠ ❊s♣❛ç♦s ❞❡ ❍✐❧❜❡rt

  ❖ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ❛t✉❛ s♦❜r❡ ♣r♦❜❧❡♠❛s ❞❛ ❢♦r♠❛ ❞❛ ❡①♣r❡s✲ sã♦ ✭✶✳✶✮✱ ♦✉ s❡❥❛✱ ♣❛r❛ ✉♠❛ ❡q✉❛çã♦✳ ◆♦ ❝❛s♦ ❞❡ t❡r ✉♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s✱ ✉♠❛ ❛❧t❡r♥❛t✐✈❛ s❡r✐❛ ❛❣r✉♣❛r ❡❧❛s ♥✉♠ ♦♣❡r❛❞♦r ♣♦r ❜❧♦❝♦s ♦✉ ❝♦♠ ❡str✉t✉r❛ ✈❡t♦r✐❛❧ ✭❝♦♠ ✈❛❧♦r❡s ♥✉♠ ❡s♣❛ç♦ ♣r♦❞✉t♦✮✳ ❊ss❛ ❛❧✲ t❡r♥❛t✐✈❛ ✐♠♣❧✐❝❛ q✉❡ ❡♠ ❝❛❞❛ ✐t❡r❛çã♦ t❡♠ q✉❡ ❛✈❛❧✐❛r t♦❞♦s ♦s ♦♣❡r❛✲ ❞♦r❡s ❡♥✈♦❧✈✐❞♦s s✐♠✉❧t❛♥❡❛♠❡♥t❡ ♥❛ ❛♣r♦①✐♠❛çã♦ ❛t✉❛❧✱ ❛t✐✈✐❞❛❞❡ q✉❡ ♣❛r❛ ✉♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ❡q✉❛çõ❡s t♦r♥❛ ♦ ♣r♦❝❡ss♦ ♣♦✉❝♦ ❡❝♦♥ô♠✐❝♦ ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡✳ ❆ s❡❣✉✐r s❡rá ❞❡✜♥✐❞♦ ♦ ♣r♦❜❧❡♠❛ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s✳

  ❈♦♥s✐❞❡r❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❡t❡r♠✐♥❛r ❛❧❣✉♠❛ q✉❛♥t✐❞❛❞❡ ❢ís✐❝❛ x ∈

i p

p−1

  X ❛ ♣❛rt✐r ❞❡ p ∈ N ❞❛❞♦s y ⊂ Y ♦s q✉❛✐s ❡stã♦ ❢✉♥❝✐♦♥❛❧♠❡♥t❡ i=0 r❡❧❛❝✐♦♥❛❞♦s ♣♦r i

  F (x) = y , i = 0, . . . , p i ✭✶✳✻✮ − 1

  : D ♦♥❞❡ F i i

  ⊆ X → Y sã♦ ♦♣❡r❛❞♦r❡s ❡♥tr❡ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt X ❡ Y ❝♦♠ r❡s♣❡❝t✐✈♦s ♣r♦❞✉t♦s ✐♥t❡r♥♦s h·, ·i X ✱ k · k Y ✳ ◗✉❛♥❞♦ ♦ ❝♦♥t❡①t♦ é ❝❧❛r♦ s❡rã♦ ✉s❛❞♦s h·, ·i ❡ k · k X ✱ h·, ·i Y ✱ ❡ ❛s ♥♦r♠❛s ❛ss♦❝✐❛❞❛s k · k ♣♦r s✐♠♣❧✐❝✐❞❛❞❡✳ ❈♦♠♦ ❥á ♦❜s❡r✈❛❞♦✱ ♥❛ ♣rát✐❝❛✱ ♦s ❞❛❞♦s ♥ã♦ sã♦ ❝♦♥❤❡❝✐❞♦s ❞❡ ♠❛♥❡✐r❛ ❡①❛t❛✱ ♦✉ s❡❥❛✱ só sã♦ ❞✐s♣♦♥í✈❡✐s ❛♣r♦①✐♠❛çõ❡s δ i y

  ❞♦s ❞❛❞♦s ❡①❛t♦s s❛t✐s❢❛③❡♥❞♦✱ δ i i i , i = 0, . . . , p

  ✭✶✳✼✮ ky − y k < δ − 1. ❆♦ ❧♦♥❣♦ ❞❡st❡ tr❛❜❛❧❤♦ s❡ ❛ss✉♠❡ q✉❡ ♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ✭✶✳✻✮

  T

  p−1 ∗

  D t❡♠ ♣❡❧♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦ x i ✳ ❖ ♣r♦♣ós✐t♦ ❞♦ ♣r❡s❡♥t❡ ∈ i=0 tr❛❜❛❧❤♦ é ❡st✉❞❛r s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❞♦ t✐♣♦ ✭✶✳✻✮ ♣❛r❛ ♦s q✉❛✐s ❛ i

  ∗

  s♦❧✉çã♦ x ♥ã♦ ❞❡♣❡♥❞❡ ❝♦♥t✐♥✉❛♠❡♥t❡ ❞♦s ❞❛❞♦s y ✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦ ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ❞❡✜♥✐❞♦ ♣❡❧♦ s✐st❡♠❛ ✭✶✳✻✮ é ♠❛❧ ♣♦st♦ ❡ ♣♦rt❛♥t♦ ♣r❡❝✐s❛ s❡r r❡❣✉❧❛r✐③❛❞♦✳

  ❊①✐st❡♠ ♣❡❧♦ ♠❡♥♦s ❞♦✐s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ♠❛❧ ♣♦st♦s ❞❛ ❢♦r♠❛ ❞♦ s✐st❡♠❛ ✭✶✳✻✮✿ ▼ét♦❞♦s ❞❡ ❘❡❣✉❧❛r✐③❛çã♦ ■t❡✲ r❛t✐✈♦s ❡ ▼ét♦❞♦s ❞❡ ❘❡❣✉❧❛r✐③❛çã♦ ❞❡ ❚✐❦❤♦♥♦✈✱ ♠❛s ❛♠❜♦s ♠ét♦❞♦s

  ′ ∗

  (x) (x) sã♦ ✐♥❡✜❝✐❡♥t❡s ♣❛r❛ p ❣r❛♥❞❡ ♦✉ s❡ ❛s ❛✈❛❧✐❛çõ❡s ❞❡ F i ❡ F sã♦ i ❝✉st♦s❛s ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡✳ ❯♠❛ ❛❧t❡r♥❛t✐✈❛ sã♦ ♦s ♠ét♦❞♦s ❞❡ t✐♣♦ ❑❛❝③♠❛r③ ✭✈❡❥❛ ❬✶✺❪✮✱ q✉❡ ❝♦♥s✐❞❡r❛♠ ❞❡ ❢♦r♠❛ ❝í❝❧✐❝❛ ❝❛❞❛ ❡q✉❛çã♦ ♥♦ s✐st❡♠❛ ✭✶✳✻✮ ❞❡ ❢♦r♠❛ s❡♣❛r❛❞❛ ❡✱ ♣♦rt❛♥t♦✱ sã♦ ♠✉✐t♦ ♠❛✐s rá♣✐❞♦s ❡ sã♦ ❢r❡q✉❡♥t❡♠❡♥t❡ ♦s ♠ét♦❞♦s ❡s❝♦❧❤✐❞♦s ♥❛ ♣rát✐❝❛✳

  ✽ ❖ ♠ét♦❞♦ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❝❧áss✐❝♦ é ❞❡✜♥✐❞♦ ♣♦r δ δ δ δ δ [n]

  ′ ∗

  x = x (x ) (F (x ) ), n+1 n − F [n] n n − y [n] ✭✶✳✽✮ ♦♥❞❡ [n] := n mod p ∈ {0, 1, . . . , p − 1}✳ ◆♦t❡ q✉❡ ♣❛r❛ p = 1 t❡♠✲s❡ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✳ ❖❜s❡r✈❛çã♦ ✶✳✶✳ ➱ ✐♠❡❞✐❛t♦ ♦❜s❡r✈❛r q✉❡ ❛ ❡str❛té❣✐❛ t✐♣♦ ❑❛❝③♠❛r③ ♣♦❞❡ s❡r ✉s❛❞❛ ❝♦♥❥✉♥t❛♠❡♥t❡ ❝♦♠ q✉❛❧q✉❡r ♦✉tr♦ ♠ét♦❞♦ ✐t❡r❛t✐✈♦ ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ♠❛❧ ♣♦st♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ♠ét♦❞♦s t✐♣♦ ◆❡✇t♦♥✱ ❡t❝✳ ❊ss❡♥❝✐❛❧♠❡♥t❡✱ ❛♣❧✐❝❛✲s❡ ✉♠❛ ✐t❡r❛çã♦ ❞♦ ♠ét♦❞♦ ❡s❝♦❧❤✐❞♦ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s ❡q✉❛çõ❡s ❞♦ s✐st❡♠❛ ❞❡ ❢♦r♠❛ ❝í❝❧✐❝❛✳

  P❛r❛ r❡s♦❧✈❡r ♦ s✐st❡♠❛ ✭✶✳✻✮ ❝♦♠ ❛ r❡❧❛çã♦ ✭✶✳✼✮ s❡rá ✉s❛❞♦ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ✐♥❝♦r♣♦r❛♥❞♦ ✉♠ ♣❛râ♠❡tr♦ ❞❡ r❡❧❛①❛♠❡♥t♦ ❜❛♥❣✲❜❛♥❣ ❛♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❝❧áss✐❝♦ ✭✈❡❥❛ ❬✷✵❪✮✱ ❝♦♠✲ ❜✐♥❛❞♦ ❝♦♠ ✉♠ ♥♦✈♦ ♣r✐♥❝í♣✐♦ ❞❛ ❞✐s❝r❡♣â♥❝✐❛✱ ✐st♦ é✱ δ δ δ δ δ [n]

  

′ ∗

  x = x n F (x ) (F (x ) ), n+1 n − ω [n] n [n] n − y ✭✶✳✾✮ ❝♦♠ δ 1, (x ) [n] δ δ [n] [n] kF n − y k > τδ

  ω := ω (δ, y ) = n n ✭✶✳✶✵✮

  0, ❝❛s♦ ❝♦♥trár✐♦✳

  ♦♥❞❡ τ > 2 é ✉♠❛ ❝♦♥st❛♥t❡ ❡s❝♦❧❤✐❞❛ ❛♣r♦♣r✐❛❞❛♠❡♥t❡✳ ❆ ✐t❡r❛çã♦ δ δ i i (x ) ,

  ❛❝❛❜❛ q✉❛♥❞♦ kF i n − y k ≤ τδ ∀ i ∈ {0, 1, . . . , p − 1}✳ ❊ss❛ ✈❛✲ r✐❛♥t❡ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❝❧áss✐❝♦ s❡rá ❝❤❛♠❛❞♦ ❞❡ i

  ▲ = 0,

  ❧♦♣✐♥❣ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ✭ ▲❑✮✳ P❛r❛ ❞❛❞♦s ❡①❛t♦s ✭δ ∀i ∈ n = 1,

  {0, 1, . . . , p − 1}✮ t❡♠✲s❡ ω ∀n ∈ N✱ ❡ ❛ss✐♠ ♥❡st❡ ❝❛s♦ ♣❛rt✐✲

  ▲

  ❝✉❧❛r ♦ ♠ét♦❞♦ ▲❑ é ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❝❧áss✐❝♦✳ ❆ s❡❣✉✐r✱ ❡ ♣❛r❛ ❝♦♥t❡①t✉❛❧✐③❛r ♦ tr❛❜❛❧❤♦ ❢❡✐t♦ ♥♦s ♣ró①✐♠♦s ❝❛♣ít✉❧♦s✱ ❛♣r❡s❡♥t❛✲s❡ ✉♠ ❜r❡✈❡ r❡s✉♠♦ s♦❜r❡ ♦s r❡s✉❧t❛❞♦s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡ t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♦❜t✐❞♦s ♥♦s ú❧t✐♠♦s ❛♥♦s✳

  

✶✳✷✳✸ ❘❡❧❛çã♦ ❍✐stór✐❝❛ ❞❡ ❘❡s✉❧t❛❞♦s ❖❜t✐❞♦s ♣❛r❛

▼ét♦❞♦s ❚✐♣♦ ❑❛❝③♠❛r③

  ❆ ♣❡sq✉✐s❛ ❞♦s ♠ét♦❞♦s ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ♣❛r❛ ♣r♦❜❧❡♠❛s ♠❛❧ ♣♦st♦s ❝♦♠❡ç♦✉ ❝❡r❝❛ ❞❡ ❞♦③❡ ❛♥♦s ❛trás ✭✈❡❥❛ ❬✷✵❪✮✱ ♦♥❞❡ ❛ ❝♦♥✲ ✈❡r❣ê♥❝✐❛ ✭s❡♠ t❛①❛s✮ ❢♦✐ ♣r♦✈❛❞❛ ♣❛r❛ ❞❛❞♦s ❡①❛t♦s✱ ❡ ❛ ♣r♦✈❛ ❞❡ ❝♦♥✲ ✈❡r❣ê♥❝✐❛ ♣❛r❛ ❞❛❞♦s ✐♥❡①❛t♦s ❢♦✐ ✐♥❝♦♠♣❧❡t❛✳ ❆ ♣r♦✈❛ ❝♦♠♣❧❡t❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ❞❛❞♦s ✐♥❡①❛t♦s ✭s❡♠ t❛①❛s✮ ❢♦✐ ❢❡✐t❛ ❡♠ ❬✶✸❪✳

  ✾ ❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛✲s❡ ✉♠ ❜r❡✈❡ r❡s✉♠♦ ❞♦s r❡s✉❧t❛❞♦s ❞❡ ❛♥á❧✐s❡

  ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♠ét♦❞♦s t✐♣♦ ❑❛❝③♠❛r③✱ ♣❛r❛ ♦s ❝❛s♦s ❧✐♥❡❛r ❡ ♥ã♦ ❧✐♥❡❛r✿

  ❬✷✵✵✻❪ ■t❡r❛t✐✈❡❧②✲❘❡❣✉❧❛r✐③❡❞✲●❛✉ss✲◆❡✇t♦♥✲❑❛❝③♠❛r③ ❬✸❪❀ ❝♦♥✈❡r❣ê♥❝✐❛ ❝♦♠ t❛①❛s❀ ❬✷✵✵✼❪ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❬✶✶✱ ✶✸❪❀ ❝♦♥✈❡r❣ê♥❝✐❛ s❡♠ t❛①❛s❀ ❬✷✵✵✽❪ ❙t❡❡♣❡st✲❉❡s❝❡♥t✲❑❛❝③♠❛r③ ❬✻❪❀ ❝♦♥✈❡r❣ê♥❝✐❛ s❡♠ t❛①❛s❀ ❬✷✵✵✾❪ ❊①♣❡❝t❛t✐♦♥✲▼❛①✐♠✐③❛t✐♦♥✲❑❛❝③♠❛r③ ❬✶✷❪❀ ❝♦♥✈❡r❣ê♥❝✐❛ s❡♠ t❛①❛s❀ ❬✷✵✵✾❪ ❇❧♦❝❦✲▼❛①✐♠✐③❛t✐♦♥✲❑❛❝③♠❛r③ ❬✶✵❪❀ ❝♦♥✈❡r❣ê♥❝✐❛ s❡♠ t❛①❛s ♣❛r❛ s✐st❡♠❛s ❧✐♥❡❛r❡s❀ ❬✷✵✶✵❪ ▲❡✈❡♥❜❡r❣✲▼❛rq✉❛r❞t✲❑❛❝③♠❛r③ ❬✷❪❀ ❝♦♥✈❡r❣ê♥❝✐❛ s❡♠ t❛①❛s❀ ❬✷✵✶✶❪ ■t❡r❛t❡❞ ❚✐❦❤♦♥♦✈✲❑❛❝③♠❛r③ ❬✺❪❀ ❝♦♥✈❡r❣ê♥❝✐❛ s❡♠ t❛①❛s❀ ❬✷✵✶✶❪ P❛r❛❧❡❧❧✲❘❡❣✉❧❛r✐③❡❞✲◆❡✇t♦♥✲❑❛❝③♠❛r③ ❬✶❪❀ ❝♦♥✈❡r❣ê♥❝✐❛ s❡♠ t❛①❛s✳ ◆♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛✲s❡ ❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦

  ▲

  ♠ét♦❞♦ ▲❑✳

  ✶✵

  ❈❛♣ít✉❧♦ ✷ ❆♥á❧✐s❡ ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ❞♦ ▼ét♦❞♦ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③

  ◆❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ♦s r❡s✉❧t❛❞♦s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦

  ▲

  ♠ét♦❞♦ ▲❑✱ ✐st♦ é✱ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ❞❛❞♦s ❡①❛t♦s ✭❚❡♦r❡♠❛ ✷✳✶✮ ❡ ♥♦

  ▲

  ❝❛s♦ ❞❡ ❞❛❞♦s ❝♦♠ r✉í❞♦ ✭❚❡♦r❡♠❛ ✷✳✷✮ ♦ ♠ét♦❞♦ ▲❑ ♠❛✐s ♦ ❝r✐tér✐♦ ❞❡ ♣❛r❛❞❛ ✭❛ s❡r ❞❡✜♥✐❞♦✮✱ é ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦✳

  ❆♦ ❧♦♥❣♦ ❞♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ s❡rã♦ ❛ss✉♠✐❞❛s ❤✐♣ót❡s❡s q✉❡ sã♦ ♣❛✲ ❞rã♦ ♥❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ ♠ét♦❞♦s ❞❡ r❡❣✉❧❛r✐③❛çã♦ ✐t❡r❛t✐✈♦s✳ ◆♦ ❝♦♥t❡①t♦ ❞♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ✭✶✳✻✮✱ s✉♣õ❡✲s❡ q✉❡ ♦s ♦♣❡r❛❞♦r❡s F , i = 0, 1, . . . , p i

  − 1 sã♦ ❋ré❝❤❡t ❞✐❢❡r❡♥❝✐á✈❡✐s ❡ q✉❡ ∃ρ > 0 t❛❧ q✉❡ ∀i = 0, 1, . . . , p − 1✱ t❡♠✲s❡

  p−1

  \

  ′

  (x) ρ (x ) D i kF i k ≤ 1, x ∈ B ⊂ D := ✭✷✳✶✮ i=0 ❡✱

  ′ i i (e Y i i (e Y (x) x) (x)(x x) η (x) x) kF − F − F i − e k ≤ kF − F k

  ✭✷✳✷✮ x (x ) ρ ∀x, e ∈ B ⊂ D,

  (x ) = ❝♦♠ 0 < η < 1/2 ❡ B ρ

  {x ∈ X : kx−x k < ρ}✳ ❆❧é♠ ❞♦ ❛♥t❡r✐♦r✱ ♦ ♣❛râ♠❡tr♦ τ ❞❡ ✭✶✳✶✵✮ s❛t✐s❢❛③✱ 1 + η

  τ > 2 > 2.

  ✭✷✳✸✮

  ✶✷ ❈♦♠♦ ❢♦✐ ❛♣♦♥t❛❞♦ ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✱ ♣❛r❛ ❞❛❞♦s ❧✐✈r❡s ❞❡ r✉í❞♦ ♦

  ▲

  ♠ét♦❞♦ ▲❑ é ❡q✉✐✈❛❧❡♥t❡ ❛♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❝❧áss✐❝♦✱ ♠❛s ♥♦ ❝❛s♦ ❞❡ ❞❛❞♦s ❝♦♠ r✉í❞♦✱ ♦s ♠ét♦❞♦s ❞❡ r❡❣✉❧❛r✐③❛çã♦ ✐t❡r❛t✐✈♦s ♣r❡❝✐s❛♠ t❡r♠✐♥❛r ❛s ✐t❡r❛çõ❡s ❛♥t❡❝✐♣❛❞❛♠❡♥t❡✱ ❢❛t♦ q✉❡ s❡ ❝♦♥s❡❣✉❡ ❝♦♠ ✉♠ ❝r✐tér✐♦ ❞❡ ♣❛r❛❞❛ ❛❞❡q✉❛❞♦✳ ◆♦ s❡❣✉✐♥t❡ ❧❡♠❛✱ ♦❜té♠✲s❡ ✉♠❛ δ ❡st✐♠❛t✐✈❛ r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ❛ ♠♦♥♦t♦♥✐❝✐❞❛❞❡ ❞♦ ❡rr♦✱ ke n k := kx n −

  ∗

  x k✱ r❡s✉❧t❛❞♦ q✉❡ ♠♦t✐✈❛ ❛ ❞❡✜♥✐çã♦ ❞♦ ❝r✐tér✐♦ ❞❡ ♣❛r❛❞❛ ♣❛r❛ ♦

  ▲

  ♠ét♦❞♦ ▲❑✳

  ∗

  (x ) ▲❡♠❛ ✷✳✶✳ ❙❡❥❛ x ✉♠❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ✭✶✳✻✮ ♥❛ ❜♦❧❛ B ρ/2 ✱

  (x ) ❝♦♠ ♦s ♦♣❡r❛❞♦r❡s F i ❋ré❝❤❡t ❞✐❢❡r❡♥❝✐á✈❡✐s ❡♠ B ρ s❛t✐s❢❛③❡♥❞♦ δ ✭✷✳✶✮ ❡ ✭✷✳✷✮✳ ❙❡❥❛ {x ❛ s❡q✉ê♥❝✐❛ ❞❡✜♥✐❞❛ ♣♦r ✭✶✳✾✮✱ ✭✶✳✶✵✮ ❡ n } n∈N δ

  ∗

  (x ) (x ) ❛✐♥❞❛ s✉♣♦♥❤❛ q✉❡ ♣❛r❛ ❝❡rt♦ n ∈ N✱ x ρ/2 ρ ✳ ❊♥tã♦ δ 2 δ n ∈ B ⊂ B 2

  ∗ ∗ n (x ) 2(1 + η)δ (x ) , δ δ [n] δ δ [n] [n] kx n+1 − x k − kx n − x k

  ≤ ω kF [n] n − y k − (1 − 2η)kF [n] n − y k ✭✷✳✹✮

  ♦♥❞❡ [n] := n mod p ∈ {0, 1, . . . , p − 1}✳ δ δ = 0 = x

  Pr♦✈❛✿ P❛r❛ ω n ❛ ❞❡s✐❣✉❛❧❞❛❞❡ s❡❣✉❡ ❞❡ x ✱ ❧♦❣♦ ♣❛r❛ n+1 n ω n

  6= 0 t❡♠✲s❡ δ 2 δ 2

  ∗ ∗

  = kx n+1 − x k − kx n − x k δ δ δ δ

  ∗ ∗ ∗ ∗

  = , x , x hx − x n+1 − x n+1 i − hx − x n − x n i δ δ δ 2 δ δ δ δ

  ∗ ∗

  = 2 , x , x , x + hx n i − hx n+1 i − kx n k hx n+1 ± x n n+1 ± x n i δ δ δ δ 2 δ δ δ

  ∗

  = 2 , x + 2 , x hx n − x n+1 i + kx n+1 − x n k hx n+1 − x n n i δ δ δ δ δ 2

  ∗

  = 2 , x (1.9) hx − x n n − x n+1 i + kx n+1 − x n k δ δ δ δ δ δ [n] 2

  ∗ ′ ∗

  = 2ω , F (x ) (F (x ) ) n [n] hx − x n [n] n n − y i + kx n+1 − x n k δ δ δ δ δ δ [n] 2

  ′ ∗

  = 2ω (x )(x ), F (x ) n [n] hF [n] n − x n n − y i + kx n+1 − x n k δ δ δ δ δ δ [n] [n]

  ′ ∗

  = 2ω (x )(x ) (x ) ), F (x ) n [n] [n] δ δ hF [n] n − x n ± (F n − y n − y i 2 kx n+1 − x n k δ δ δ δ [n]

  • ∗ ′ ∗

  = 2ω (x ) (x ) (x )(x ) + y , n [n] [n] δ δ δ δ hF n − F − F [n] n n − x − y [n] [n] 2 δ δ 2 ❝✳s✱ (∗) (2.2) [n] n − y i − 2ω kF [n] n − y k kx n+1 − x n k δ δ δ δ [n] [n] + F (x ) n (x )

  ∗ n [n] [n] [n] η (x ) (x ) (x )

  ≤ 2ω kF n − F k + ky − y k kF n − y k

  δ δ [n] 2 ✶✸ n [n] (x )

  − ω kF n − y k δ δ [n] [n] δ δ [n] [n] n [n] [n] η (x ) + δ (x ) ≤ 2ω kF n − y k + δ kF n − y k δ δ [n] 2 n [n] (x )

  − ω kF n − y k δ δ [n] δ δ [n] [n] = ω (x ) 2(η + 1)δ (x ) , n [n] [n] kF n − y k − (1 − 2η)kF n − y k

  ♦❜t❡♥❞♦ ❛ss✐♠ ❛ ❡st✐♠❛t✐✈❛ ❞❡s❡❥❛❞❛✳ δ δ ❊♠ (∗) ❢♦✐ ✉s❛❞♦ ✭✶✳✾✮ ❡ ✭✷✳✶✮ ♣❛r❛ ♦❜t❡r✱ 2 2 δ δ δ [n] [n] 2 δ δ 2

  ′ ∗ = ω (x ) (F (x ) ) (x ) . [n] n [n] kx n+1 −x n k n kF [n] n n −y k ≤ ω kF n −y k

  ❆ s❡❣✉✐r✱ ❡ ♠♦t✐✈❛❞♦ ♣❡❧♦ ▲❡♠❛ ✷✳✶✱ ❞❡✜♥❡✲s❡ ♦ í♥❞✐❝❡ ❞❡ ♣❛r❛❞❛

  ▲

  ♣❛r❛ ♦ ♠ét♦❞♦ ▲❑✳ δ δ δ := n (y )

  ❉❡✜♥✐çã♦ ✷✳✶✳ ❉❡✜♥❡✲s❡ n ✱ ♦ í♥❞✐❝❡ ❞❡ ♣❛r❛❞❛ ♣❛r❛ ♦

  ∗ ∗ ▲

  ♠ét♦❞♦ ▲❑✱ ❝♦♠♦ ♦ ♠❡♥♦r ✐♥t❡✐r♦ ♠ú❧t✐♣❧♦ ❞❡ p ∈ N t❛❧ q✉❡ δ δ δ x δ = x δ = δ . n n +1 · · · = x n +p ∗ ∗ ∗ ✭✷✳✺✮ ❆❣♦r❛✱ ✉s❛♥❞♦ ♦ ▲❡♠❛ ✷✳✶ ❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ í♥❞✐❝❡ ❞❡ ♣❛r❛❞❛ ♣❛r❛ ♦

  ▲

  ♠ét♦❞♦ ▲❑✱ t❡♠✲s❡ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ❞❡ ♠♦♥♦t♦♥✐❝✐❞❛❞❡ ❞♦ ❡rr♦ δ n ∗ ke k := kx n − x k✿ δ δ

  ∗

  , i = 0, . . . , p ▲❡♠❛ ✷✳✷✳ ❙❡❥❛♠ x ✱ F i ❝♦♠♦ ♥♦ ▲❡♠❛ ✷✳✶ ❡ n

  − 1 ❡ x n

  ∗

  ❝♦♠♦ ❞❡✜♥✐❞♦ ❡♠ ✭✷✳✺✮✳ ❊♥tã♦✱ δ δ δ

  ∗ ∗ n+1 n ✭✷✳✻✮ , kx − x k ≤ kx − x k, n = 0, . . . , n ∗ δ

  = 0 ❡ ✭✷✳✺✮ ✐♠♣❧✐❝❛ ω ✱ ♣❛r❛ t♦❞♦ i ∈ {0, . . . , p − 1}✱ ✐st♦ é n +i i (x ) , i = 0, . . . , p δ δ i δ i kF − y k ≤ τδ − 1. ✭✷✳✼✮ n

  = 0 Pr♦✈❛✿ ◆❛s ❤✐♣ót❡s❡s ❞♦ ▲❡♠❛ ✷✳✶✱ s❡ ω n ✱ ❡♥tã♦ ✭✷✳✻✮ é s❛t✐s❢❡✐t♦ δ δ

  = x ❥á q✉❡ ♥❡ss❡ ❝❛s♦ x ✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ω n ❡♠ n+1 n ✭✶✳✶✵✮ ❡ ❞❡ τ ❡♠ ✭✷✳✸✮ t❡♠✲s❡ [n] δ δ [n] [n] 2(1 + η)

  (x ) > δ , kF [n] n − y k > τδ

  ✶✹ ♦ q✉❡ ✐♠♣❧✐❝❛ [n] δ δ [n]

  2(1 + η)δ (x ) [n] − (1 − 2η)kF n − y k < 0,

  ❡ ❞❡ss❛ ❢♦r♠❛ ♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✹✮ é ♥ã♦ ♣♦s✐t✐✈♦ ♦❜t❡♥❞♦ ❛ss✐♠ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✻✮ ♣❛r❛ ❡ss❡ n✳ ❆ ♣r♦✈❛ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✻✮ δ

  ∗

  (x ) é ❝♦♠♣❧❡t❛❞❛ ♣♦r ✐♥❞✉çã♦ s♦❜r❡ n✳ P❛r❛ n = 0✱ ❝♦♠♦ x ∈ B ρ/2 ✱ δ

  ∗

  (x ) ρ (x ) ❡♥tã♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✻✮ ✐♠♣❧✐❝❛ x δ δ 1 ∈ B ρ/2 ⊂ B ✳ ❆❣♦r❛✱

  ∗

  (x ) ♣❛r❛ 0 < n < n t❛❧ q✉❡ x n ∈ B ρ/2 ✱ t❡♠✲s❡ ♥♦✈❛♠❡♥t❡ ♣♦r ✭✷✳✻✮ δδ

  ∗

  (x ) ρ (x ) q✉❡ x n+1 ∈ B ρ/2 ⊂ B ✱ ❧♦❣♦ é ✈á❧✐❞❛ ♣❛r❛ t♦❞♦ n = 0, . . . , n ✱

  ♦ q✉❡ ❝♦♠♣❧❡t❛ ❛ ♣r♦✈❛✳ P❛r❛ ♣r♦✈❛r ✭✷✳✼✮✱ ✉s❛✲s❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✹✮ δ δ

  • i ❝♦♠ n = n ✱ i ∈ {0, . . . , p − 1}✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ n ✱ t❡♠✲s❡ q✉❡
  • δ δ δ

    ∗ ∗

    x = x + i] = i δ δ n +i n ∗ ∗ ∗ δ ❡ [n ✱ ❛ss✐♠ δ δ i δ δ i i n +i i i (x δ ) 2(1 + η)δ (x δ ) ,

      ≤ ω kF n − y k − (1 − 2η)kF n − y k ∗ ∗ δ ♣❛r❛ i ∈ {0, . . . , p − 1}✳ ❙❡ ω n +i 6= 0 ♣❛r❛ ❛❧❣✉♠ i ∈ {0, . . . , p − 1}✱ ❡♥tã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ i δ δ i

      2(1 + η)δ (x δ ) i − (1 − 2η)kF n − y k ≥ 0, δ δ i i

      (x ) ♦ q✉❡ ✐♠♣❧✐❝❛ ♣♦r ✭✷✳✸✮ q✉❡ kF i δ ♦ q✉❡ ❝♦♥tr❛❞✐③ ❛ δ δ n − y k < τδ

      = 0 ❞❡✜♥✐çã♦ ❞❡ ω ✱ ❧♦❣♦ ω ✱ ♣❛r❛ t♦❞♦ i ∈ {0, . . . , p − 1} ❡ ❛ss✐♠ n +i n +i ∗ ∗ ♦❜té♠✲s❡ ✭✷✳✼✮✳ δ δ δ

      = x ◆♦t❡ q✉❡ ♣❛r❛ n > n ✱ t❡♠✲s❡ ω n δ ✱ ✐st♦ é✱ ♦

      ≡ 0 ❡ ❛ss✐♠ x n

      ∗ n δ

      ♠ét♦❞♦ ▲❑ t♦r♥❛✲s❡ ❡st❛❝✐♦♥ár✐♦ ❛♣ós ♦ í♥❞✐❝❡ n ✳

      ∗

      ❆❣♦r❛✱ ✉s❛♥❞♦ ♦ r❡s✉❧t❛❞♦ ❛♥t❡r✐♦r t❡♠✲s❡ ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✳ Pr♦♣♦s✐çã♦ ✷✳✶✳ ◆❛s ❤✐♣ót❡s❡s ❞♦ ▲❡♠❛ ✷✳✶✱ ♦❜té♠✲s❡ ❛s s❡❣✉✐♥t❡s ❡st✐♠❛t✐✈❛s n δ −1 2

      X [n] δ i 2 δ δ 2 kx − x k τ n (τ min (δ )) ω (x ) , n [n] ≤ ky − F n k ≤

      ∗ i

      (1 n=0 − 2η)τ − 2(1 + η) δ i i ✭✷✳✽✮ = y ,

      ❡ ♣❛r❛ ❞❛❞♦s ❡①❛t♦s✱ ✐st♦ é✱ y ∀i = 0, . . . , p − 1✱ t❡♠✲s❡

      ∞

      X [n] 2 (x n ) < n=0 ky − F [n] k ∞. ✭✷✳✾✮

      δ ✶✺

      = 1 Pr♦✈❛✿ P❛r❛ n = 0, . . . , n n ✱ ♣♦r ❞❡✜♥✐çã♦ ✐♥❢❡r❡✲s❡

      − 1✱ ❝♦♠♦ ω [n] δ δ [n]

    −1 ∗

    < τ (x ) = x q✉❡ δ [n] n n ❡ ✉s❛♥❞♦ ky − F n k✳ ❈❤❛♠❛♥❞♦ ❞❡ e − x

      ✭✷✳✹✮ t❡♠✲s❡✱ 2 2 δ δ [n] 2

    n+1 n n (x ) 2τ (1 + η) + 2η ,

    −1 ke k − ke k ≤ ω ky − F [n] n k − 1

      ♦ q✉❡ ✐♠♣❧✐❝❛ ♣♦r ✭✷✳✸✮✱ [n] n n+1 2 2 [n] 2 δ δ 2 ke k − ke k (τ δ ) (x ) , n [n]

      ≤ ω ky − F n k ≤

      −1

      1 (1 + η) − 2η − 2τ

      ❡ ❛ss✐♠✱ [n] 2 2 i 2 δ δ 2 ke k − ke k τ ( ) n n+1 (τ min (δ )) n (x ) . i ≤ ω ky − F [n] n k ≤ (1

      − 2η)τ − 2(1 + η) δ ❙♦♠❛♥❞♦ ♥❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡♠ n = 0, . . . , n

      − 1✱ t❡♠✲s❡✱ n δ −1 2 2 X [n] δ τ ( ) δ i 2 δ δ 2 ∗ ke k − ke n k n (τ min (δ )) ω n (x ) ,

      ≤ ky − F [n] n k ≤

      ∗ i

      (1 n=0 − 2η)τ − 2(1 + η) ♦❜t❡♥❞♦ ✜♥❛❧♠❡♥t❡✱ n δ −1 2

      X [n] δ i 2 δ δ 2 kx − x k τ n (τ min (δ )) ω n (x ) . ≤ ky − F [n] n k ≤

      ∗ i

      (1 n=0 i − 2η)τ − 2(1 + η) = 0,

      ❆❣♦r❛✱ ♣❛r❛ ❞❛❞♦s ❡①❛t♦s δ ∀i = 0, . . . , p − 1✱ ❞❡ ✭✷✳✹✮ s❡ ❞❡❞✉③✱ 2 2 [n] 2

      ∗ ∗ n+1 n [n] n (x ) , kx − x k − kx − x k ≤ (2η − 1)kF − y k

      ❡ s♦♠❛♥❞♦ ♦❜té♠✲s❡ ❛ ❡st✐♠❛t✐✈❛✱ N 2 2

      ∗ ∗

      X [n] 2 kx − x k − kx − x k N +1 (x n ) kF [n] − y k ≤

      1 n=0 2 − 2η

      ∗

      kx − x k ,

      ≤ ∀N ∈ N,

      1 − 2η

      ❞❡ ♦♥❞❡ ✐♥❢❡r❡✲s❡ q✉❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ s♦♠❛s ♣❛r❝✐❛✐s é ❧✐♠✐t❛❞❛✱ ❡ ❛ss✐♠ ✜♥❛❧♠❡♥t❡ ♣♦❞❡✲s❡ t♦♠❛r ❧✐♠✐t❡ q✉❛♥❞♦ N → ∞ ♣❛r❛ ♦❜t❡r

      ∞ 2

      X [n] 2 kx − x k (x n ) < kF [n] − y k ≤ ∞,

      1 − 2η

      ✶✻ ❞❡ ♦♥❞❡ ♦❜té♠✲s❡ ♦ r❡s✉❧t❛❞♦✳

      ❉❛ Pr♦♣♦s✐çã♦ ✷✳✶✱ ❡s♣❡❝✐✜❝❛♠❡♥t❡ ♣♦r ✭✷✳✽✮✱ ✐♥❢❡r❡✲s❡ q✉❡ ♥❡❝❡s✲ δ < s❛r✐❛♠❡♥t❡ n

      ∞ ❡ ❛ss✐♠ ❛ ✐t❡r❛çã♦ t❡r♠✐♥❛ ♥✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡

      ∗

      ♣❛ss♦s✳ ◆❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✱ ❛♣r❡s❡♥t❛♠✲s❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s

      (x ) s♦❧✉çõ❡s ❞❡ ✭✶✳✻✮ ♥✉♠❛ ❜♦❧❛ B ρ ✳ ❆ ♣r♦✈❛ ❢❡✐t❛ ♥♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ♠❡❧❤♦r❛ ❛ ♣r♦✈❛ ❢❡✐t❛ ❡♠ ❬✷✵❪ ✭Pr♦♣♦s✐çã♦ ✹✳✶✮✱ q✉❡ é ✐♥❝♦♠♣❧❡t❛✳

      , i = 0, . . . , p Pr♦♣♦s✐çã♦ ✷✳✷✳ ❙❡❥❛♠ ♦s ♦♣❡r❛❞♦r❡s F i

      − 1 s❛t✐s❢❛③❡♥❞♦

      ∗

      (x ) ✭✷✳✶✮ ❡ ✭✷✳✷✮✳ ❙❡ x ρ é ✉♠❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ✭✶✳✻✮✱ ❡♥tã♦

      ∈ B

      

    ρ (x )

      ✶✳ t♦❞❛ ♦✉tr❛ s♦❧✉çã♦ ❞❡ ✭✶✳✻✮✱ ex ∈ B s❛t✐s❢❛③✱

      ∗ ∗ ′ ∗

      x x (x )), − e ∈ N (F i ∀i = 0, . . . , p − 1,

      ♦♥❞❡ N (A) é ♦ ♥ú❝❧❡♦ ❞♦ ♦♣❡r❛❞♦r ❧✐♥❡❛r A✱

      † †

      (x ) ✷✳ ❡①✐st❡ ✉♠❛ x ✲s♦❧✉çã♦ ❞❡ ♥♦r♠❛ ♠í♥✐♠❛✱ x ρ ✱ ✐st♦ é✱ x

      ∈ B é ✉♠❛ s♦❧✉çã♦ ❞❡ ✭✶✳✻✮ ❡ s❛t✐s❢❛③

      †

      kx − x k = inf kx − x k

      x∈S

      (x ) : x ♦♥❞❡ S := {x ∈ B ρ s❛t✐s❢❛③ ✭✶✳✻✮}

      †

      ✸✳ ❛ x ✲s♦❧✉çã♦ ❞❡ ♥♦r♠❛ ♠í♥✐♠❛ x é ú♥✐❝❛ ❡ s❛t✐s❢❛③ ! ⊥

      

    p−1

      \

      

    † ′ †

    x (x )) .

      − x ∈ N (F i i=0 Pr♦✈❛✿

      (x ) ✶✳ ❉❡ ✭✷✳✷✮ t❡♠✲s❡ q✉❡ ♣❛r❛ t♦❞♦ i ∈ {0, . . . , p − 1} ❡ ∀x, ex ∈ B ρ i (x) i x) (x)(x x) i (x) i x)

      |kF − F (e k − kF i − e k| ≤ ηkF − F (e k, ❡ ❧❡♠❜r❛♥❞♦ q✉❡ η < 1/2✱ ✐♥❢❡r❡✲s❡✱

      1

      ′ i i (e (x) x) (x)(x x) kF − F k ≤ kF i − e k.

      1 − η

      ❙✐♠✐❧❛r♠❡♥t❡ t❡♠✲s❡✱

      1

      ′

      (x)(x x) (x) x) i i (e kF i − e k ≤ kF − F k,

      ✶✼ ❡ ❛ss✐♠✱

      1

      1

      

    ′ ′

      (x)(x x) (x) x) (x)(x x) i i (e kF i − e k ≤ kF − F k ≤ kF i − e k, 1 + η

      1 − η

      ✭✷✳✶✵✮

      ∗ ∗ ′ ∗ ∗ ∗

      x ρ (x ) (x )(x x ) ❧♦❣♦ ♣❛r❛ x , e ∈ B s♦❧✉çõ❡s✱ t❡♠✲s❡ kF − e k = 0✱ i ♣❛r❛ t♦❞♦ i ∈ {0, . . . , p − 1}✱ ♣r♦✈❛♥❞♦ ❛ss✐♠ ♦ ✐t❡♠ ✶✳

      ✷✳ ❉❡✜♥❛ m := inf kx − x k,

      

    x∈S

      ❡ ❡s❝♦❧❤❛ σ > 0 t❛❧ q✉❡ m ≤ kx − x k < σ < ρ✳ ❉❡✜♥❛ ❛❣♦r❛ σ σ s❛t✐s❢❛③ ✭✶✳✻✮} ⊂ S, := (x ) : x

      S {x ∈ B ❡♥tã♦ ✉s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞♦ í♥✜♠♦ t❡♠✲s❡✱ inf kx − x k = m. k σ x∈S σ

      ❙❡❥❛ {x } k∈N ⊂ S t❛❧ q✉❡ k k→∞ kx − x k −→ m.

      (x ) ❈♦♠♦ {x k ρ ✱ ❡♥tã♦ é ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❡ ♣♦r✲

      } k∈N ⊂ B t❛♥t♦ ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❢r❛❝❛♠❡♥t❡ ❝♦♥✈❡r❣❡♥t❡✱ q✉❡ s❡ ❞❡♥♦t❛rá ❞❛ ♠❡s♠❛ ♠❛♥❡✐r❛✱ ❡ ❝✉❥♦ ❧✐♠✐t❡ ❢r❛❝♦ s❡rá ❞❡♥♦t❛❞♦ i ♣♦r ex✳ ❈♦♠♦ ♦s ♦♣❡r❛❞♦r❡s F sã♦ ❋ré❝❤❡t ❞✐❢❡r❡♥❝✐á✈❡✐s✱ ❡♥tã♦

      ′ ∗

      (x ) ♦s ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s F sã♦ ❧✐♠✐t❛❞♦s ❡ ♣♦rt❛♥t♦ i

      ′ ∗ ∗ ′ ∗ ∗

      F (x )(x ) ⇀ F (x x ), i − x i − x ∀i = 0, . . . , p − 1. k )(e ❉❛ s❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ ✐♥❢❡r✐♦r ❞❛ ♥♦r♠❛ ♥❛ t♦♣♦❧♦❣✐❛ ❢r❛❝❛ ♥✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt s❡❣✉❡ q✉❡✱ x k ✭✷✳✶✶✮ ke − x k ≤ lim inf kx − x k,

      k→∞ ′ ∗ ∗ ′ ∗ ∗

      (x x ) (x )(x k ) kF i )(e − x k ≤ lim inf kF i − x k. ✭✷✳✶✷✮

      k→∞ ∗

      , x ❈♦♠♦ x k

      ∈ S, ∀k ∈ N✱ ♣❡❧♦ ✐t❡♠ ✶ ❞❛ ♣r❡s❡♥t❡ ♣r♦♣♦s✐çã♦✱ t❡♠✲s❡

      ′ ∗ ∗

      (x )(x ) k kF i − x k = 0, ∀i = 0, . . . , p − 1, ∀k ∈ N,

      ✶✽

      ′ ∗ ∗

      (x x ) ❡ ❛ss✐♠ ♣♦r ✭✷✳✶✷✮✱ kF )(e i − x k = 0, ∀i = 0, . . . , p − 1✳ P♦r

      (x ) (x ) ✭✷✳✶✶✮✱ s❡❣✉❡ q✉❡ ex ∈ B σ ρ ✱ ❧♦❣♦ ♣♦r ✭✷✳✶✵✮ ex é

      ⊂ B s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ✭✶✳✻✮✱ ✐st♦ é✱ ex ∈ S σ ❡ ❛ss✐♠✱ x ke − x k ≥ inf kx − x k.

      x∈S σ

      ❋✐♥❛❧♠❡♥t❡✱ ❝♦♠♦ (2.11) inf k k x kx − x k = lim kx − x k = lim inf kx − x k ≥ ke − x k,

      x∈S σ k→∞ k→∞

      ✐♥❢❡r❡✲s❡ q✉❡ ex é ✉♠❛ x ✲s♦❧✉çã♦ ❞❡ ♥♦r♠❛ ♠í♥✐♠❛✳ ✸✳ ❉❡✜♥❛ ♦ ♦♣❡r❛❞♦r✱

      p−1

      \ p F := (F , . . . , F ) : D = D , i

      p−1 ⊂ X → Y i=0

      ❡♥tã♦✱ ♦ ♣r♦❜❧❡♠❛ ✭✶✳✻✮ ♣♦❞❡ s❡ ❢♦r♠✉❧❛r ❝♦♠♦ F (x) = y✱ ♦♥❞❡

      p−1

      y = (y , . . . , y ) ✳ ◆❡st❡ ❝❛s♦✱

      p−1

      \

      ′ ′

      (x)) = (x)), N (F N (F i i , i = 0, . . . , p i=0

      ❡ ❝♦♠♦ ♦s ♦♣❡r❛❞♦r❡s F − 1 s❛t✐s❢❛③❡♠ ✭✷✳✷✮✱ t❡♠✲s❡✱ 2

      ′ p

      x) (x)(x x) kF (x) − F (e − F − e k Y

      p−1

      X 2

      ′

      = (x) x) (x)(x x) i i (e i=0 kF − F − F i − e k Y

      p−1 2 X 2 2 2 p i i (e (x) x) = η x) ,

      ≤ η kF − F k kF (x) − F (e k Y i=0 ρ (x ) ♣❛r❛ t♦❞♦ x, ex ∈ B ✱ ♦✉ s❡❥❛✱ ♦ ♦♣❡r❛❞♦r F s❛t✐s❢❛③ ❛s ❤✐♣ó✲

      †

      t❡s❡s ❞❛ Pr♦♣♦s✐çã♦ ✷✳✶ ❞❡ ❬✶✻❪✱ ❡ ❛ss✐♠ x é ❛ ú♥✐❝❛ x ✲s♦❧✉çã♦ ρ (x ) ❞❡ ♥♦r♠❛ ♠í♥✐♠❛ ♥❛ ❜♦❧❛ B ❡ s❛t✐s❢❛③✱

      ! ⊥

      p−1

      \

      

    † ′ ′

      x (x))) = (x)) , − x ∈ (N (F N (F i i=0

      ❝♦♠♣❧❡t❛♥❞♦ ❛ ♣r♦✈❛✳

      ✶✾ ❆❣♦r❛ ❛♣r❡s❡♥t❛✲s❡ ❛ ♣r♦✈❛ ❞♦ ♣r✐♠❡✐r♦ ❞♦s r❡s✉❧t❛❞♦s ♣r✐♥❝✐♣❛✐s ❞♦

      ♣r❡s❡♥t❡ ❝❛♣ít✉❧♦✱ ✐st♦ é✱ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❧♦❝❛❧ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞❡✇❡❜❡r✲ ❑❛❝③♠❛r③ ♣❛r❛ ❞❛❞♦s ❡①❛t♦s✳

      , i = 0, . . . , p ❚❡♦r❡♠❛ ✷✳✶✳ ❆ss✉♠❛ q✉❡ ♦s ♦♣❡r❛❞♦r❡s F i

      − 1✱ sã♦ (x )

      ❋ré❝❤❡t ❞✐❢❡r❡♥❝✐á✈❡✐s ♥❛ ❜♦❧❛ B ρ ✱ s❛t✐s❢❛③❡♥❞♦ ✭✷✳✶✮✱ ✭✷✳✷✮ ❡ q✉❡ δ i (x ) =

      ✭✶✳✻✮ t❡♠ ✉♠❛ s♦❧✉çã♦ ❡♠ B ρ/2 ✳ ❊♥tã♦✱ ♣❛r❛ ❞❛❞♦s ❡①❛t♦s y i y , i = 0, . . . , p n ❣❡r❛❞❛ ♣♦r ✭✶✳✾✮ ❝♦♥✈❡r❣❡ ♣❛r❛ −1✱ ❛ s❡q✉ê♥❝✐❛ {x } n∈N

      †

      ✉♠❛ s♦❧✉çã♦ ❞❡ ✭✶✳✻✮✳ ❆✐♥❞❛ ♠❛✐s✱ s❡ x ❞❡♥♦t❛ ❛ ú♥✐❝❛ x ✲s♦❧✉çã♦ ❞❡ (x )

      ♥♦r♠❛ ♠í♥✐♠❛ ❞❡ ✭✶✳✻✮ ❡♠ B ρ ❡✱

      ′ † ′

      (x )) (x)), (x ), ρ ✭✷✳✶✸✮ N (F i ⊂ N (F i ∀i = 0, . . . , p − 1, ∀x ∈ B

      ❡♥tã♦✱

      

    n→∞

      x . n −→ x

      ∗

      (x ) Pr♦✈❛✿ ❙❡❥❛ ex ✉♠❛ s♦❧✉çã♦ ❞❡ ✭✶✳✻✮ ❡♠ B ρ/2 ✳ ❉❡✜♥❛

      

    e x . n := e n

      − x P❡❧♦ ▲❡♠❛ ✷✳✷✱ ❛ s❡q✉ê♥❝✐❛ {ke n é ♠♦♥ót♦♥❛ ❞❡❝r❡s❝❡♥t❡ ❧✐♠✐t❛❞❛ k} n∈N ✐♥❢❡r✐♦r♠❡♥t❡ ♣♦r ✵✱ ❧♦❣♦ é ❝♦♥✈❡r❣❡♥t❡ ♣❛r❛ ❛❧❣✉♠ ε ≥ 0✳ ❆ ♣r♦✈❛ ❝♦♥s✐st❡ ❡♠ ♠♦str❛r q✉❡ {e n é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤②✱ ♦ q✉❡

      } n∈N ❛ s✉❛ ✈❡③ ✐♠♣❧✐❝❛ q✉❡ {x n é ✉♠❛ s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡✱ ❛ss✐♠

      } n∈N ❢❡✐t♦ ✐ss♦✱ só r❡st❛ ♣r♦✈❛r q✉❡ ♦ ❧✐♠✐t❡ ❞❡ {x n é ✉♠❛ s♦❧✉çã♦ ❞♦

      } n∈N ♣r♦❜❧❡♠❛ ✭✶✳✻✮✳ ❙❡❥❛♠ j, k ∈ N t❛✐s q✉❡ k ≤ j✳ P❛r❛ k = k 1 ❡ j = j 1

      · p + k · p + j , j

      ❝♦♠ k 1 1 s❛t✐s❢❛③❡♥❞♦✱ ∈ {0, . . . , p − 1}✱ s❡❥❛ k ≤ l ≤ j

      p−1 p−1

      X s s

      X s l s m (x ) (x ) . s=0 s=0 ky − F ·p+s k ≤ ky − F ·p+s k, ♣❛r❛ k ≤ m ≤ j ✭✷✳✶✹✮

      := p ❉❡✜♥❛ l 1 1 ✳ ❊♥tã♦✱

      − 1 ❡ l := l · p + l j k j l l k ke − e k ≤ ke − e k + ke − e k, ❡ 2 j l j l j l = , e ke − e k he − e − e i 2 2

      = , e j j l l ke k − 2he i + ke k 2 2 = , e , e j l j l l l ke k − ke k − 2he i + 2he i 2 2

      = + 2 , e j l l j l ke k − ke k he − e i.

      ✷✵ ❆♥❛❧♦❣❛♠❡♥t❡ ♦❜té♠✲s❡✱ l k = k l + 2 l k , e l 2 2 2 ke − e k ke k − ke k he − e i. 2 2 2 2

      ) ) P❛r❛ k → ∞✱ (ke j l ❡ (ke k l t❡♥❞❡♠ ❛ ③❡r♦✳ ❆ss✐♠✱ k − ke k k − ke k

      k→∞

      , e ♣❛r❛ ♠♦str❛r q✉❡ ke j k l j l

      − e k −→ 0✱ é s✉✜❝✐❡♥t❡ ♣r♦✈❛r q✉❡ he − e i ❡ l k l , e he −e i t❡♥❞❡♠ ❛ ③❡r♦ q✉❛♥❞♦ k → ∞✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ✭✶✳✽✮✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r ❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ t❡♠✲s❡✱

      l−1

      X

      ∗ l k l i i+1 ), e l , e (x x

      |he − e i| = − x − x i=k

      l−1

      D E

      X [i]

      ′ ∗ ∗

      = F (x ) (F (x ) x i=k [i] − y − x i [i] i ), e l

      l−1

      D E

      X [i]

      ′ ∗

      = F (x ) , F (x x ) [i] i i )(e l i=k − y [i] − x

      l−1

      X [i]

      ′ ∗

      (x i ) (x i x l ) ≤ kF [i] − y kkF [i] )(e − x k. i=k

      ′ ∗

      (x x ) ❆❣♦r❛ é ♣r❡❝✐s♦ ❡st✐♠❛r kF i )(e l [i] − x k✳ ❯s❛♥❞♦ ✭✷✳✷✮ ❡ ❛ ❞❡s✐❣✉❛❧✲ ❞❛❞❡ tr✐❛♥❣✉❧❛r t❡♠✲s❡✱

      ′ ∗ ′ ∗

      (x i x l ) (x i x i + x i l ) kF )(e − x k = kF )(e − x − x k [i] [i]

      ∗ ∗ ′ ∗

      = x ) x ) (x i ) (x i x i ) kF [i] (e − (F [i] (e − F [i] − F )(e − x [i]

      ′ [i] l [i] i [i] l i i l (x ) (x ) (x ) (x )(x )

      − F − (F − F − F [i] − x k

      ∗ ∗ [i] (e [i] l [i] (e [i] i x ) (x ) x ) (x )

      ≤ kF − F k + ηkF − F k

      ∗ ∗

    • η (x ) x ) + F x ) (x ) [i] i [i] (e [i] (e [i] l kF − F − F k [i] [i] [i] l [i] i

      (x ) (x ) ≤ (1 + η)ky − F k + 2ηky − F k. ❈♦♠❜✐♥❛♥❞♦ ❛s ❞✉❛s ú❧t✐♠❛s ❡st✐♠❛t✐✈❛s ♦❜té♠✲s❡✱

      l−1

      X [i] [i] l k l [i] i [i] l , e (x ) (x ) |he − e i| ≤ (1 + η) ky − F kky − F k i=k

      ✭✷✳✶✺✮

      l−1

      X [i] 2 + 2η (x ) . [i] i ky − F k

      ✷✶ ■st♦ ♠♦str❛ q✉❡ he l − e k , e l i

      ·p+s − x l ·p+s+1 k

      )

      ·p+s

      (x l

      ′ s

      X s=i 1 kF

      1 − η l 1 −1

      1

      =

      X s=i 1 kx l

      (F s (x l

      1 − η l 1 −1

      1

      ) k ≤

      ·p+s − x l ·p+s+1

      )(x l

      ·p+s

      (x l

      ′ i 1

      ∗

      ·p+s

      −1

      X s=i 1 kF s (x l

      ·p+s

      X kF s (x l

      p−2

      ) k ≤

      ·p+i 1

      ❛ss✐♠✱ ❝♦♠♦ ky i 1 − F i 1 (x l

      ) − y s k,

      ·p+s

      p−2

      ) − y s

      1 − η

      1

      =

      ) − y s k

      ·p+s

      X s=i 1 kF s (x l

      1 − η l 1 −1

      1

      ) k ≤

      X s=i 1 kF

      1 − η l 1

      k→∞

      P❛r❛ ♣r♦✈❛r ✭✷✳✶✻✮✱ é ♥❡❝❡ssár✐♦ ♣r✐♠❡✐r♦ ❛❝❤❛r ✉♠❛ ❡st✐♠❛t✐✈❛ ♣❛r❛ ky [i] − F [i] (x l ) k✳ ❋❛③❡♥❞♦ i = i · p + i 1 ❝♦♠ i 1 ∈ {0, . . . , p − 1}✱ t❡♠✲s❡✱ ky [i] − F [i]

      ) − F i 1

      ·p+i 1

      ) k + kF i 1 (x l

      ·p+i

    1

      (x l

      ) k ≤ ky i 1 − F i 1

      ·p+l

    1

      (x l ) k = ky i 1 − F i 1 (x l

      X i=k ky [i] − F [i] (x i ) k 2 k→∞ −→ 0.

      ·p+l 1

      l−1

      ❥á q✉❡ ♣♦r ✭✷✳✾✮ t❡♠✲s❡✱

      −→ 0, ✭✷✳✶✻✮

      k→∞

      (x l ) k

      X i=k ky [i] − F [i] (x i ) kky [i] − F [i]

      l−1

      −→ 0✱ s❡

      (x l

      ) k, ❡ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ✉s❛♥❞♦ ✭✷✳✶✵✮✱ ✭✷✳✶✮ ❡ ✭✶✳✽✮ ♦❜té♠✲s❡✱ kF i 1

      1

      ·p+s+1

      ) k ≤

      ·p+s+1

      (x l

      ) − F i 1

      ·p+s

      X s=i 1 kF i 1 (x l

      −1

      )) ≤ l 1

      (x l

      (x l

      ) − F i 1

      ·p+s

      X s=i 1 (F i 1 (x l

      −1

      ) k = l 1

      ·p+l 1

      (x l

      ) − F i 1

      ·p+i 1

      ) − y s k,

      ✷✷ t❡♠✲s❡✱

      p−2

      X [i] − η s

      2 (x l ) s (x l ) ky − F [i] k ≤ kF ·p+s − y k.

      1 − η s=i 1

      ❯s❛♥❞♦ ✭✷✳✶✹✮ ❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✱ t❡♠✲s❡✱

      l−1

      X [i] [i] [i] i [i] l (x ) (x ) i=k ky − F kky − F k l

      p−1

      X X i i 1 1 i m i l (x ) (x ) ≤ ky − F m =k i =0 1 1 ·p+i 1 kky − F 1 ·p+l 1 k l

      !

      p−1 p−2

      X X

      X

      2 − η i 1 s i (x m ) s (x l )

      ≤ ky − F 1 ·p+i 1 k ky − F ·p+s k

      1 − η m =k i 1 =0 s=i 2 1 l p−1 !

      X X

      2 − η i 1 i 1 m (x ) 1

      ≤ ky − F ·p+i k

      1 − η m =k i =0 l 1

      p−1

      X X

      2 − η i 1 2 p i (x m )

      ≤ ky − F 1 ·p+i 1 k

      1 − η m =k i l 1 =0

      X

      2 − η [i] 2 p (x ) . [i[ i

      ≤ ky − F k

      1 − η 1

      i=k−k

      ✭✷✳✶✼✮

      k→∞

      ) ❈♦♠♦ (k − k 1

      −→ ∞✱ ✉s❛♥❞♦ ✭✷✳✾✮ ♣r♦✈❛✲s❡ ✭✷✳✶✻✮ ❡ ❛ss✐♠ ♣♦r ✭✷✳✶✺✮ , e t❡♠✲s❡ q✉❡ he l k l

      − e i → 0 q✉❛♥❞♦ k → ∞✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ❡ t❡♥❞♦ ❝✉✐❞❛❞♦ ❡♠ tr♦❝❛r ♦s í♥❞✐❝❡s ❞♦s s♦♠❛tór✐♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ♥♦ ❝❛s♦ k = j , e

      ✮✱ ♣r♦✈❛✲s❡ he l j l ≤ j ≤ l✭q✉❡ ❛❝♦♥t❡❝❡ q✉❛♥❞♦ l − e i → 0

      k→∞ k→∞

      q✉❛♥❞♦ k → ∞ ❡ ❛ss✐♠ ke j l l k − e k −→ 0 ❡ ke − e k −→ 0✱ ✐st♦ é✱ j k n k→∞ ke − e k −→ 0✳ ❈♦♠ ✐st♦ ✐♥❢❡r❡✲s❡ q✉❡ {e } n∈N é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ n

      ❈❛✉❝❤②✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ {x } n∈N é ❝♦♥✈❡r❣❡♥t❡✳ ❙❡❥❛ x ♦ ❧✐♠✐t❡✳ [n] (x n )

      ◆♦✈❛♠❡♥t❡ ♣♦r ✭✷✳✾✮✱ ♦❜té♠✲s❡ q✉❡ ky − F [n] k → 0 ♣❛r❛ n → ∞✱ ♦ q✉❡ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✐♠♣❧✐❝❛✱ i i i (x ) i (x ) ∗ ky − F k = lim ky − F n·p+i k = 0, ∀i = 0, . . . , p − 1,

      n→∞ ∗

      ♦✉ s❡❥❛✱ x é ✉♠❛ s♦❧✉çã♦ ❞❡ ✭✶✳✻✮✳ ❆❣♦r❛✱ ♣❡❧♦ ✐t❡♠ ✸ ❞❛ Pr♦♣♦s✐çã♦

      ✷✸

      †

      ✷✳✷✱ ✭✶✳✻✮ t❡♠ ✉♠❛ ú♥✐❝❛ x ✲s♦❧✉çã♦ ❞❡ ♥♦r♠❛ ♠í♥✐♠❛ x q✉❡ s❛t✐s❢❛③✱ ! ⊥

      p−1

      \

      † ′ † x (x )) .

      − x ∈ N (F i ✭✷✳✶✽✮ i=0 P♦r ✭✶✳✾✮ ❡ ✭✷✳✶✸✮ t❡♠✲s❡✱ [n]

      ′ ∗ ′ ∗

      x n+1 n = F (x n ) (y (x n )) (x n ) ) − x − F [n] ∈ R(F [n] [n]

      ♦♥❞❡ R(A) é ❛ ✐♠❛❣❡♠ ❞♦ ♦♣❡r❛❞♦r A✱ ♦ q✉❡ ✐♠♣❧✐❝❛✱ ! ⊥

      p−1

      \

      ′ ⊥ ′ † ⊥ ′ †

      x (x )) (x )) (x )) , n+1 n n − x ∈ N (F [n] ⊂ N (F [n] ⊂ N (F i i=0

      ♣❛r❛ t♦❞♦ n ∈ N✱ ❧♦❣♦ ♣♦r ✐♥❞✉çã♦ ♦❜té♠✲s❡✱ !

      ⊥ p−1

      \

      ′ †

      x (x )) , n − x ∈ N (F i ∀n ∈ N. i=0

      ⊥

      T

      p−1 ′ †

      (x )) ❆ss✐♠✱ ❝♦♠♦ é ❢❡❝❤❛❞♦✱ t♦♠❛♥❞♦ ❧✐♠✐t❡ ❝♦♥❝❧✉í✲s❡✱ i=0 N (F i

      !

      ⊥ p−1

      \

      ∗ ′ † x (x )) .

      − x ∈ N (F i i=0 ❋✐♥❛❧♠❡♥t❡✱

      ! ⊥

      p−1

      \

      † ∗ † ∗ ′ †

      x = x + x (x )) , i − x − x − x ∈ N (F i=0

      T

      p−1 † ∗ ′ † ∗ †

      (x )) = x ❡ ❝♦♠♦ x ✱ ✐♥❢❡r❡✲s❡ q✉❡ x ✱ ❝♦♠♣❧❡t❛♥❞♦

      − x ∈ i=0 N (F i ❛ss✐♠ ❛ ♣r♦✈❛✳

      ❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛✲s❡ ♦ ú❧t✐♠♦ r❡s✉❧t❛❞♦ ❞♦ ♣r❡s❡♥t❡ ❝❛♣ít✉❧♦✱ ✐st♦

      ▲

      é✱ ♦ ♠ét♦❞♦ ▲❑ ✭✶✳✾✮✱ ✭✶✳✶✵✮ ❥✉♥t♦ ❛♦ ❝r✐tér✐♦ ❞❡ ♣❛r❛❞❛ ✭✷✳✺✮ é ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦✳ ❆ ♣r♦✈❛ ❢❡✐t❛ ♥♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ♠❡❧❤♦r❛ ❛ ♣r♦✈❛ ❢❡✐t❛ ❡♠ ❬✷✵❪ ✭❚❡♦r❡♠❛ ✹✳✹✮✱ q✉❡ é ✐♥❝♦♠♣❧❡t❛✳ ❚❡♦r❡♠❛ ✷✳✷✳ ❙♦❜ ❛s ❤✐♣ót❡s❡s ❞♦ ❚❡♦r❡♠❛ ✷✳✶✱ ♣❛r❛ ❞❛❞♦s s❛t✐s❢❛✲ δ ③❡♥❞♦ ✭✶✳✼✮ ❡ ❝♦♠ n ❞❡✜♥✐❞♦ ❝♦♠♦ ❡♠ ✭✷✳✺✮✱ ❛ ✐t❡r❛çã♦ ❞♦ ♠ét♦❞♦ δ ∗ ▲

      (x ) ▲❑✱ x δ ✱ ❝♦♥✈❡r❣❡ ♣❛r❛ ✉♠❛ s♦❧✉çã♦ ❞❡ ✭✶✳✻✮ ❡♠ B q✉❛♥❞♦ n ρ/2

      δ → 0✳ ❙❡ ❛❞✐❝✐♦♥❛❧♠❡♥t❡ ✈❛❧❡ ✭✷✳✶✸✮✱ ❡♥tã♦ δ δ→0

      † x δ . n −→ x

      ✷✹

      ∗

      (x ) Pr♦✈❛✿ ❙❡❥❛ x ρ/2 ♦ ❧✐♠✐t❡ ❞❛ s❡q✉ê♥❝✐❛ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲

      ∈ B p ❑❛❝③♠❛r③ ♣❛r❛ ❞❛❞♦s ❡①❛t♦s ✭❚❡♦r❡♠❛ ✷✳✶✮ ❡ s❡❥❛ {δ k ✉♠❛

      } k∈N ⊂ R + i

      k→∞ p−1 p δ k δ k

      := (y ) , k s❡q✉ê♥❝✐❛ t❛❧ q✉❡ δ k ✳ ❉❡✜♥❛ y δ k −→ 0 ∈ R i=0 ∈ N✱ ❛ s❡q✉ê♥❝✐❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❞❡ ✈❡t♦r❡s ❞❡ ❞❛❞♦s ❝♦♠ r✉í❞♦✱ ❡

      {y } k∈N δ δ k k := n (y ) s❡❥❛ n k ♦ ❝♦rr❡s♣♦♥❞❡♥t❡ í♥❞✐❝❡ ❞❡ ♣❛r❛❞❛ ❞❛❞♦ ♣♦r ✭✷✳✺✮

      ∗ δ k

      , y ) ❛♣❧✐❝❛❞♦ ❛♦ ♣❛r (δ k ✳ k

      Pr✐♠❡✐r♦ s✉♣♦♥❤❛ q✉❡ {n k∈N t❡♠ ✉♠ ♣♦♥t♦ ❞❡ ❛❝✉♠✉❧❛çã♦ ✭❝♦♠ } r❡❧❛çã♦ à t♦♣♦❧♦❣✐❛ ❞❛ ♦r❞❡♠ ❡♠ N✮ ✜♥✐t♦ n ∈ N✱ ❡♥tã♦ s❡♠ ♣❡r❞❛ ❞❡ k = n

      ❣❡♥❡r❛❧✐❞❛❞❡ ♣♦❞❡✲s❡ ❛ss✉♠✐r q✉❡ ♣❛r❛ k ❣r❛♥❞❡ n ✳ ❆ss✐♠✱ ♣❛r❛ k ❣r❛♥❞❡✱ ♣♦r ✭✷✳✼✮ t❡♠✲s❡✱ δ δ i k i k i ✭✷✳✶✾✮ (x ) , ky − F n k ≤ τδ k ∀i = 0, . . . , p − 1. i , F (x), i = 0, . . . , p ρ (x )

      P❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ F i − 1✱ ♥❛ ❜♦❧❛ B ✱ ♣❛r❛ n δ δ k k ✜①♦✱ x n ❞❡♣❡♥❞❡ ❝♦♥t✐♥✉❛♠❡♥t❡ ❞❡ y ✱ ❡ ❛ss✐♠ ♣♦r ✭✷✳✶✾✮✱ δ k→∞ δ δ k→∞ k k k i x (x ) n −→ x − F n k −→ 0, ∀i = 0, . . . , p − 1, n ❡ ky i

      ♦✉ s❡❥❛✱ δ k→∞ i k F (x ) (x ) = y , i i n n −→ F ∀i = 0, . . . , p − 1, n

      ✐st♦ é✱ ❛ n✲és✐♠❛ ✐t❡r❛çã♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③✱ x ✱ ♣❛r❛ ❞❛❞♦s

      ∗

      = x n ❡①❛t♦s é ✉♠❛ s♦❧✉çã♦ ❞❡ ✭✶✳✻✮ ❡ ❛ss✐♠ ❛ ✐t❡r❛çã♦ ❛❝❛❜❛ ❝♦♠ x δ k ∗ k→∞

    k

    p

      ❡ x n −→ x ✱ ♣❛r❛ ❛ s❡q✉ê♥❝✐❛ {δ } k∈N ❝♦♥✈❡r❣✐♥❞♦ ♣❛r❛ 0 ∈ R k ❛r❜✐trár✐❛✳

      k→∞

      ❙ó ❢❛❧t❛ ❝♦♥s✐❞❡r❛r ❡♥tã♦ ♦ ❝❛s♦ n k −→ ∞✱ q✉❡ é ♦ ❝❛s♦ ♦❜s❡r✈❛❞♦

      ❣❡r❛❧♠❡♥t❡ ♥❛ ♣rát✐❝❛✳ ❙❡❥❛ n ∈ N ❛r❜✐trár✐♦✱ ❡♥tã♦ ♣❡❧♦ ▲❡♠❛ ✷✳✷ > n t❡♠✲s❡ q✉❡ ♣❛r❛ n k ✱ δ δ δ k ∗ k ∗ k ∗ n n kx n k − x k ≤ kx n − x k ≤ kx n − x k + kx − x k.

      := n (ǫ) ❉❛❞♦ ǫ > 0✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✷✳✶✱ ❡①✐st❡ n 1 1

      ∈ N t❛❧ q✉❡✱

      ∗ n 1 .

      kx − x k < ǫ/2, ∀n ≥ n δ k P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ ❥á ❢♦✐ ❛♣♦♥t❛❞♦✱ x ❞❡♣❡♥❞❡ ❝♦♥t✐♥✉❛♠❡♥t❡ ❞❡ n 1 δ δ k k k→∞ y

      := k (n ) ✱ ❧♦❣♦ x n n 1 −→ x ∈ N t❛❧ 1 ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ ❡①✐st❡ k 1 1 1 q✉❡✱ δ k n 1 1 , kx n 1 − x k < ǫ/2, ∀k > k

      ♣♦rt❛♥t♦✱ δ k ∗ 1 (n (ǫ)) : n > n (ǫ), 1 k 1 kx n − x k < ǫ, ∀k > k

      ✷✺ ✐st♦ é✱ δ k

    k→∞

    ∗ x . n −→ x k

      ❋✐♥❛❧♠❡♥t❡✱ s❡ ✭✷✳✶✸✮ ✈❛❧❡✱ ❡♥tã♦ ❝♦♠♦ ♥❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✷✳✶ t❡♠✲s❡✱ ! ⊥

      p−1 δ k ′ † \ x (x )) , n i k − x ∈ N (F ∀k ∈ N, i=0

      ♣♦rt❛♥t♦✱ ❢❛③❡♥❞♦ k → ∞ ❡ ✉s❛♥❞♦ ♦ ♠❡s♠♦ ❛r❣✉♠❡♥t♦ q✉❡ ♥♦ ❚❡♦r❡♠❛

      † ∗

      = x ✷✳✶ ♦❜té♠✲s❡ x ✱ ❝♦♥❝❧✉✐♥❞♦ ❛ss✐♠ ❛ ♣r♦✈❛✳

      ✷✻

      ❈❛♣ít✉❧♦ ✸ ❚❛①❛s ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛

      ◆❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ❞✉❛s ✈❡rsõ❡s ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲ ❑❛❝③♠❛r③ ♣❛r❛ ♦♣❡r❛❞♦r❡s ❡♠ ❜❧♦❝♦s✱ ❛ s❛❜❡r✱ ❛ ✈❡rsã♦ ♥ã♦ s✐♠étr✐❝❛ ✭LK✮ ❡ s✐♠étr✐❝❛ ✭sLK✮✳ P❛r❛ ❡ss❛s ✈❡rsõ❡s✱ ♦❜té♠✲s❡ t❛①❛s ❞❡ ❝♦♥✲ ✈❡r❣ê♥❝✐❛ ❡ t❛♠❜é♠ ❝♦♥❞✐çõ❡s s✉✜❝✐❡♥t❡s ♣❛r❛ ❣❛r❛♥t✐r ❡ss❛s t❛①❛s✳

      ✸✳✶ Pr♦❜❧❡♠❛s ■♥✈❡rs♦s ❡♠ ❆♥á❧✐s❡

      ❖s ♣r♦❜❧❡♠❛s ✐♥✈❡rs♦s ❛ s❡r❡♠ ❝♦♥s✐❞❡r❛❞♦s sã♦ ❛q✉❡❧❡s ♦♥❞❡ ♦ ♣r♦✲ ❜❧❡♠❛ ❞✐r❡t♦ ❡stá ❞❛❞♦ ♣♦r ♦♣❡r❛❞♦r❡s ❝♦♠ ❡str✉t✉r❛ ❞❡ ❜❧♦❝♦✳

      ❙❡❥❛♠ A : X , i = 0, . . . , p i i

      → Y − 1, p , i = 0, . . . , p ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ♦♥❞❡ X, Y i

      −1✱ sã♦ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt C r❡❛✐s✳ ◗✉❛♥❞♦ ❢♦r ♥❡❝❡ssár✐♦ s❡ ❞❡♥♦t❛rá ♣♦r X ❛ ✈❡rsã♦ ❝♦♠♣❧❡①✐✜❝❛❞❛ C = + ix : x , x

      ❞♦ ❡s♣❛ç♦ X✱ ✐st♦ é✱ X 1 2 1 2 {x ∈ X}✳ ❖ ♦❜❥❡t✐✈♦ ❛q✉✐ é r❡s♦❧✈❡r ♦ s✐st❡♠❛ ❞❡ p ❡q✉❛çõ❡s✱

      A x = y , i = 0, . . . , p i i ✭✸✳✶✮ − 1,

      ♦♥❞❡ ♦s ❞❛❞♦s y i ✱ ♥♦ ❝❛s♦ ❞❡ s❡r ❞❛❞♦s ❝♦♠ r✉í❞♦ s❡rã♦ ❞❡♥♦t❛❞♦s ♣♦r δ y i ✱ ❡ ❛ss✉♠❡✲s❡ q✉❡ ♦ s✐st❡♠❛ é ♠❛❧ ♣♦st♦ ♦✉ ♠❛❧ ❝♦♥❞✐❝✐♦♥❛❞♦✳ ◆❛ s❡q✉ê♥❝✐❛✱ ❛s ✈❛r✐á✈❡✐s ❡♠ ♥❡❣r✐t♦ ❞❡♥♦t❛rã♦ ❛q✉❡❧❛s ❝♦♠ ❡str✉t✉r❛ ❞❡ ❜❧♦❝♦✱ ✐st♦ é✱ ❡❧❡♠❡♥t♦s ❞❡ ❡s♣❛ç♦s ♣r♦❞✉t♦✳ P❛r❛ r❡s✉♠✐r ♦ s✐st❡♠❛ ❝♦♠♦ ✉♠❛ ❡q✉❛çã♦✱ ❞❡✜♥❡♠✲s❡ ♦ ♦♣❡r❛❞♦r b ❆ ❡ ♦ ✈❡t♦r ❞❡ ❞❛❞♦s b② ♣♦r

          A y

          b ❆ =   , b ② =   .

      ✳✳✳ ✳✳✳ A y

      ✷✽ , . . . , Y )

      ❆ss✐♠✱ b ❆ : X → ❨✱ ♦♥❞❡ ❨ é ♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❨ = (Y

      p−1

      ❡ ♣♦rt❛♥t♦ ❛❣♦r❛ ❛ ❡q✉❛çã♦ ❛ r❡s♦❧✈❡r é b ❆x = b②. ✭✸✳✷✮

      P❛r❛ ❝♦♥tr♦❧❛r ✉♠ ♣♦ssí✈❡❧ ♠❛❧ ❝♦♥❞✐❝✐♦♥❛♠❡♥t♦✱ ✐♥❝♦r♣♦r❛♠✲s❡ p ♦♣❡✲ : Y , i = 0, . . . , p r❛❞♦r❡s ❞❡ ♣ré✲❝♦♥❞✐❝✐♦♥❛♠❡♥t♦ M i i i

      → Y − 1✱ ❛✉t♦✲ ❛❞❥✉♥t♦s ❞❡✜♥✐❞♦s ♣♦s✐t✐✈♦s✱ ♣❛r❛ ❛ss✐♠ ♦❜t❡r ❛ ✈❡rsã♦ ♣ré✲❝♦♥❞✐❝✐♦♥❛❞❛ ❞♦ s✐st❡♠❛ ✭✸✳✶✮✱ 2 1 2 1 M A x = M y , i = 0, . . . , p i i − 1, i i ✭✸✳✸✮ s✐st❡♠❛ q✉❡ é ♦❜✈✐❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛♦ s✐st❡♠❛ ✭✸✳✶✮ ❥á q✉❡ M i é

      ❞❡✜♥✐❞♦ ♣♦s✐t✐✈♦ ♣❛r❛ i = 0, . . . , p − 1✳ ❆❧é♠ ❞♦ ❛♥t❡r✐♦r✱ r❡✉♥❡♠✲s❡ ♦s ♦♣❡r❛❞♦r❡s ❞❡ ♣ré✲❝♦♥❞✐❝✐♦♥❛♠❡♥t♦ ♥✉♠❛ ♠❛tr✐③ ❞✐❛❣♦♥❛❧ ♣♦r ❜❧♦❝♦s ▼✱

      ), i = 0, . . . , p ▼ = diag(M i − 1.

      ❈♦♠♦ s❡rá ❝♦♥✈❡♥✐❡♥t❡ tr❛❜❛❧❤❛r ❝♦♠ ♦♣❡r❛❞♦r❡s ❡ ❞❛❞♦s ♣ré✲❝♦♥❞✐❝✐♦♥❛❞♦s✱ ❞❡✜♥❡♠✲s❡

       2 1   2 1  1 M A M y 1 2     2 b    

      ❆ := ▼ ❆ = ② = b ✳✳✳ ✳✳✳

        , ② := ▼   , 1 2 2 1 M A M y

      p−1 p−1 p−1 p−1 δ δ δ 2 1 ✭✸✳✹✮

      = b ❡ ♦s ❞❛❞♦s ❝♦♠ r✉í❞♦ s❡rã♦ r❡♣r❡s❡♥t❛❞♦s ♣♦r b② ❡ ② ▼ ② ✳ ❆❧é♠ ❞♦ ❛♥t❡r✐♦r✱ ❝♦♠♦ é ✉s✉❛❧✱ ❛ss✉♠❡✲s❡ ✉♠❛ ❝♦t❛ s♦❜r❡ ♦ ♥í✈❡❧ ❞❡ r✉í❞♦✱ δ i i , i = 0, . . . , p ky i − y k ≤ δ − 1 ♦♥❞❡ δ i é ❛ q✉❛♥t✐❞❛❞❡ ❞❡ r✉í❞♦ ♥❛ i✲és✐♠❛ ❡q✉❛çã♦✳ ❈♦♠ ✐st♦✱ ❞❡✜♥❡✲s❡ ♦ ♥í✈❡❧ ❞❡ r✉í❞♦ ❣❧♦❜❛❧ ♣♦r δ✱

      

    p−1 p−1

    δ 2 δ

      X 2 X 2 2 = i δ =: δ , kb② − b②k ky i − y k ≤ i i=0 i=0

      ❡ ♦ ♥í✈❡❧ ❞❡ r✉í❞♦ ❣❧♦❜❛❧ ♥❛ ✈❡rsã♦ ♣ré✲❝♦♥❞✐❝✐♦♥❛❞❛ ♣♦r

      p−1 1 δ 2 X 2 δ 2 1 2 2 2 2 = (y i ) δ =: δ .

      k② − ②k kM i − y k ≤ k▼ k M ✭✸✳✺✮ i=0 i

      ✷✾

      

    ✸✳✷ ▼ét♦❞♦s ❚✐♣♦ ❑❛❝③♠❛r③ ❡ ❖♣❡r❛❞♦r❡s

    ❡♠ ❇❧♦❝♦

      ◆❡st❛ s❡çã♦ s❡rã♦ ❞❡✜♥✐❞❛s ❛s ✈❡rsõ❡s s✐♠étr✐❝❛ ❡ ♥ã♦ s✐♠étr✐❝❛ ❞❛ ✐t❡r❛çã♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❛♣❧✐❝❛❞❛ ❛♦ s✐st❡♠❛ ♣ré✲❝♦♥❞✐❝✐♦♥❛❞♦ ✭✸✳✸✮ ❡ s❡ ❡①♣r❡ss❛rã♦ ❝♦♠♦ ✐t❡r❛çõ❡s ❞❡ ▲❛♥❞✇❡❜❡r ♣ré✲❝♦♥❞✐❝✐♦♥❛❞❛s ♣❛r❛ ♦♣❡r❛❞♦r❡s ❡♠ ❜❧♦❝♦s✳ ❉❛q✉✐ ❡♠ ❞✐❛♥t❡ s❡rá r❡q✉❡r✐❞❛ ❛ s❡❣✉✐♥t❡ ❝♦t❛ ♣❛r❛ ❛ ♥♦r♠❛ ❞♦s ♦♣❡r❛❞♦r❡s ♣ré✲❝♦♥❞✐❝✐♦♥❛❞♦s

      ∗

      M A i i ✭✸✳✻✮ kA i k < 2, ∀i = 0, . . . , p − 1. P❛r❛ ♦♣❡r❛❞♦r❡s ❞❡ ♣ré✲❝♦♥❞✐❝✐♦♥❛♠❡♥t♦ ❧✐♠✐t❛❞♦s ❝♦♠♦ ♦s ✉s❛❞♦s ❛q✉✐✱ ❡st❛ ❝♦♥❞✐çã♦ é s❡♠♣r❡ s❛t✐s❢❡✐t❛ ✐♥tr♦❞✉③✐♥❞♦ ✉♠ t❛♠❛♥❤♦ ❞❡ ♣❛ss♦ τ > 0

      ✱ ✐st♦ é✱ ✉s❛♥❞♦ ♣ré✲❝♦♥❞✐❝✐♦♥❛❞♦r❡s τM i ❡♠ ✈❡③ ❞❡ M i ✳

      

    ✸✳✷✳✶ ▼ét♦❞♦ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ♥ã♦ ❙✐♠étr✐❝♦

    ❝♦♠ Pré✲❈♦♥❞✐❝✐♦♥❛♠❡♥t♦

      ❉❡✜♥❡✲s❡ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ✭LK✮ ❝❧áss✐❝♦ ♥ã♦ s✐✲ ♠étr✐❝♦ ❝♦♠ ♣ré✲❝♦♥❞✐❝✐♦♥❛♠❡♥t♦✱ ❝♦♠♦ ❛ s❡q✉ê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ❛♣r♦✲ ①✐♠❛❞❛s {x n ❣❡r❛❞❛ ♣♦r

      } n∈N δ

      ∗

      x = x M (A x ), [k] = k mod p k+1 k [k] [k] k − A [k] − y [k]

      ✭✸✳✼✮ x := x , n = 0, 1, . . . n np = x

      ♦♥❞❡ x é ♦ ❡❧❡♠❡♥t♦ ✐♥✐❝✐❛❧ ❡ ❛ ❛♣r♦①✐♠❛çã♦ ❞❡✈❡ s❡r ✐♥t❡r♣r❡t❛❞❛ δ = x

      ❝♦♠♦ x n ♥♦ ❝❛s♦ ❞❡ ✉s❛r ❞❛❞♦s ❝♦♠ r✉í❞♦✳ n ❖❜s❡r✈❛çã♦ ✸✳✶✳ P❛r❛ ♦s ♠ét♦❞♦s t✐♣♦ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③✱ ❝♦♠♦ ✭✸✳✼✮✱ ❛ ❝♦♥❞✐çã♦ ✭✸✳✻✮ ❛♣❛r❡❝❡ ❝♦♠♦ ✉♠❛ ❝♦♥❞✐çã♦ ♥❛t✉r❛❧ ✈✐s❛♥❞♦

      

      M A ❣❛r❛♥t✐r q✉❡ ♦s ♦♣❡r❛❞♦r❡s I − A i i s❡❥❛♠ ♥ã♦ ❡①♣❛♥s✐✈♦s✳ ◆♦t❡ i i q✉❡ ♥ã♦ s❡ ✐♠♣õ❡ ✉♠❛ ❝♦t❛ ❡s♣❡❝✐✜❝❛ ♣❛r❛ ❛ ♥♦r♠❛ ❞♦s ♦♣❡r❛❞♦r❡s M i

      ❡♠ s✐✱ s❡♥ã♦ só ❝♦t❛s ♣❛r❛ ❛ ♥♦r♠❛ ❞♦s ♦♣❡r❛❞♦r❡s M ❡♠ ❝♦♠❜✐♥❛✲ i çã♦ ❝♦♠ ♦s ♦♣❡r❛❞♦r❡s A ✳ ❉❡ ❢❛t♦✱ é ❞❡ ❡s♣❡r❛r q✉❡ ❣r❛♥❞❡ ♣❛rt❡ ❞❛ ♣r❡s❡♥t❡ ❛♥á❧✐s❡✱ ✭♣♦r ❡①❡♠♣❧♦✱ ♥♦ ❝❛s♦ ❞❡ ❞❛❞♦s ❡①❛t♦s✮ ❛✐♥❞❛ s❡❥❛ ✈á✲ i ❧✐❞❛ ♣❛r❛ ♦♣❡r❛❞♦r❡s ❛✉t♦✲❛❞❥✉♥t♦s ✐❧✐♠✐t❛❞♦s M q✉❡ s❛t✐s❢❛ç❛♠ ✭✸✳✻✮✳ ◆♦ ❡♥t❛♥t♦✱ ♥♦ ❝❛s♦ ❞❡ ❞❛❞♦s ❝♦♠ r✉í❞♦✱ ❡♠ ❣❡r❛❧✱ ♦ ✉s♦ ❞❡ ♦♣❡r❛❞♦✲ δ r❡s M i ✐❧✐♠✐t❛❞♦s ❣❡r❛ sér✐♦s ♣r♦❜❧❡♠❛s ✭♥♦t❡ q✉❡ M i ❛t✉❛ s♦❜r❡ y ✱ i q✉❡ ♣♦❞❡ ♥ã♦ ❡st❛r ❜❡♠ ❞❡✜♥✐❞♦✮✳ ◆❡ss❛ s✐t✉❛çã♦ ♣♦❞❡✲s❡ ❝♦♥s✐❞❡r❛r δ

      M (y ) i i ✭s❡ ❡stá ❞❡✜♥✐❞♦✮ ❝♦♠♦ ♥í✈❡❧ ❞❡ r✉í❞♦ ✐♥❞✐✈✐❞✉❛❧ ❡ ❡①✐❣✐r q✉❡ i −y ❡ss❛ q✉❛♥t✐❞❛❞❡ s❡❥❛ ❧✐♠✐t❛❞❛✳ P❛r❛ ❡✈✐t❛r ❡ss❛s ❞✐✜❝✉❧❞❛❞❡s✱ ❛♦ ❧♦♥❣♦ ❞♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❛ss✉♠❡✲s❡ q✉❡ ♦s ♦♣❡r❛❞♦r❡s M i sã♦ ❧✐♠✐t❛❞♦s✳

      ✸✵ ❙❡❥❛ ♦ ♦♣❡r❛❞♦r tr✐❛♥❣✉❧❛r ✐♥❢❡r✐♦r L : Y → Y ❞❡✜♥✐❞♦ ❝♦♠♦ ❛

      ∗

      ♣❛rt❡ ❞♦ ♦♣❡r❛❞♦r AA ❜❛✐①♦ ❛ ❞✐❛❣♦♥❛❧✱ ✐st♦ é✱  1 1

       2 2 

      ∗

      M A A M 11 · · · 

       L

      := ,

      ✭✸✳✽✮ 

       ✳✳✳

      ✳✳✳  2 1 2 1 2 1 1 2

      

    ∗ ∗

      M A A M A A M

      p−1 · · · M p−1

    p−1 p−1 p−2 p−2

      ❡♥tã♦✱ t❡♠✲s❡ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳ ❚❡♦r❡♠❛ ✸✳✶✳ ❙❡❥❛ x n ❛ ✐t❡r❛çã♦ ❣❡r❛❞❛ ♣❡❧♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲ ❑❛❝③♠❛r③ ✭✸✳✼✮✳ ❊♥tã♦✱ ❛ ✐t❡r❛çã♦ ✭✸✳✼✮ ♣♦❞❡ ❡①♣r❡ss❛r✲s❡ ❝♦♠♦ ✉♠ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ♣♦r ❜❧♦❝♦s ♣ré✲❝♦♥❞✐❝✐♦♥❛❞♦s ✭♥ã♦ s✐♠étr✐❝♦✮ ❞❛ ❢♦r♠❛ δ

      

      M x = x (Ax ) n+1 n B n ✭✸✳✾✮ − A − y

      ♦♥❞❡

      −1

      M B = (I + L) , ✭✸✳✶✵✮

      L ❝♦♠♦ ❡♠ ✭✸✳✽✮ ❡ I é ♦ ♦♣❡r❛❞♦r ✐❞❡♥t✐❞❛❞❡ ❡♠ Y✳

      Pr♦✈❛✿ ❆ ♣r♦✈❛ s❡rá r❡❛❧✐③❛❞❛ ❢❛③❡♥❞♦ ✐♥❞✉çã♦ s♦❜r❡ ♦ ♥ú♠❡r♦ ❞❡ ❜❧♦❝♦s p✳ P❛r❛ p = 1 ♦ r❡s✉❧t❛❞♦ é tr✐✈✐❛❧♠❡♥t❡ ❝❡rt♦✳ ❙❡❥❛ ❡♥tã♦ p > 1 (l) ❡ 0 < l < p✱ ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ❡ s❡❥❛ x ♦ ✈❡t♦r ❣❡r❛❞♦ ♣❡❧❛ n+1 ✐t❡r❛çã♦ ✭✸✳✼✮ ❛ ♣❛rt✐r ❞❡ x n ✉s❛♥❞♦ ❛♣❡♥❛s ♦s ♣r✐♠❡✐r♦s l ❜❧♦❝♦s✱ ❡ s✉♣♦♥❤❛ q✉❡ ❛ ❢ór♠✉❧❛ ✭✸✳✾✮ é ✈á❧✐❞❛ ♣❛r❛ ♦s ♣r✐♠❡✐r♦s l ❜❧♦❝♦s✱ ♦✉ s❡❥❛✱ (l) δ

      ∗

      x = x ] [M ] ([A] x ] ), n+1 − [A − [y n l B l l n l ♦♥❞❡ [T] l r❡♣r❡s❡♥t❛ ♦s l ♣r✐♠❡✐r♦s ❜❧♦❝♦s ❞♦ ♦♣❡r❛❞♦r T ♦✉ é ❛ ♣❛rt❡ (l) ♣r✐♥❝✐♣❛❧ ❞❛ ♦r❞❡♠ l × l✱ ❞❡♣❡♥❞❡♥❞♦ ❞❛ ❡str✉t✉r❛ ❞❡ T✳ ▲♦❣♦✱ x n+1 ♣♦❞❡ ❡s❝r❡✈❡r✲s❡ ♣♦r ✭✸✳✼✮ ❝♦♠♦ (l+1) (l) (l) δ

      

      x = x M (A x ), n+1 n+1 − A l n+1 − y l l l ❡ ❛ss✐♠ ❝♦♠❜✐♥❛♥❞♦ ❡st❛s ❞✉❛s ú❧t✐♠❛s ❡①♣r❡ssõ❡s ♦❜té♠✲s❡ (l+1) δ δ (l)

      

    ∗ ∗

      x = x n ] l [M B ] l ([A] l x n ] l ) M l (A l x ) n+1 − [A − [y − A l − y l δ n+1

      ∗

      = x ] [M ] ([A] x ] ) + n l B l l n l − [A − [y δ δ

      ∗ ∗

      M (A x [A ] [M ] ([A] x ] ) ) l l n l l B l l n l −A l − A − [y − y l 1

      ∗ ∗ 2

      = x ] A M ) n l − ([A l

      ✸✶ ·

      ] l+1 [M B ] l+1 A x n − y δ l+1

      M 1 2 l A l [A

      ∗

      ] l

      I !

      −1

      · A x n

      − y δ l+1 = x n

      − [A

      ∗

      ] l+1 [I + L]

      −1 l+1

      A x n − y δ l+1

      = x n − [A

      ∗

      = x n − [A

      ] l+1 [I + L]

      ∗

      ] l+1 [M B ] l+1 ([A] l+1 x n − y δ l+1

      ), q✉❡ ♣♦r ✐♥❞✉çã♦ ❞á ♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦✳ ❊♠ (∗) s❡ ✉s♦✉ ❛ s❡❣✉✐♥t❡ ✐❞❡♥t✐❞❛❞❡✿ ❙❡ C ❡ F sã♦ ♦♣❡r❛❞♦r❡s ✐♥✈❡rsí✈❡✐s✱ ❡♥tã♦

      C E F

      −1

      = C

      −1

      −F

      −1

      EC

      −1

      F

      −1 .

      ✸✳✷✳✷ ▼ét♦❞♦ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❙✐♠étr✐❝♦ ❝♦♠ Pré✲❈♦♥❞✐❝✐♦♥❛♠❡♥t♦

      

    −1

    l

      ∗

      [M B ] l [A] l x n − [y δ

      I !

      ] l M 1 2 l A l x n

      − M 1 2 l y δ l − M 1 2 l A l [A

      ∗

      ] l [M B ] l ([A] l x n − [y δ

      ] l ) !

      = x n − ([A

      ∗

      ] l A

      ∗ l

      M 1 2 l ) ·

      [M B ] l −M 1 2 l

      A l [A

      ∗

      ] l [M B ] l

      [A] l x n − [y δ

      − [A

      ] l M 1 2 l A l x n

      − M 1 2 l y δ l !

      (∗)

      = x n − [A

      ∗

      ] l+1 [M B ]

      

    −1

    l

      M 1 2 l A l [A

      ∗

      ] l

      I !

      −1

      · A x n

      − y δ l+1 = x n

      P❡❧♦ ✈✐st♦ ♥❛ s✉❜s❡❝çã♦ ❛♥t❡r✐♦r✱ ❛ ✐t❡r❛çã♦ ✭✸✳✼✮ é ❡q✉✐✈❛❧❡♥t❡ ❛ ✭✸✳✾✮✱ ♦♥❞❡ ♦ ♦♣❡r❛❞♦r ❞❡ ♣ré✲❝♦♥❞✐❝✐♦♥❛♠❡♥t♦ M B é ♥ã♦ s✐♠étr✐❝♦✱ ♣♦r ❡ss❛ r❛③ã♦ é q✉❡ ♦ ♠ét♦❞♦ ✭✸✳✼✮ é ❝❤❛♠❛❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲ ❑❛❝③♠❛r③ ♥ã♦ s✐♠étr✐❝♦✳ ❆❣♦r❛✱ s❡ ❡♠ ✭✸✳✼✮ s❡ ❢❛③ ✉♠ p✲❝✐❝❧♦ ✉s✉❛❧ s❡❣✉✐❞♦ ❞❡ ♦✉tr♦ p✲❝✐❝❧♦ ♦♥❞❡ ❛ ♦r❞❡♠ ❞❛s ❡q✉❛çõ❡s é ✐♥✈❡rt✐❞❛ ❝♦♠✲ ♣❧❡t❛♥❞♦ ❛ss✐♠ ✉♠ 2p✲❝✐❝❧♦✱ ❡♥tã♦ t❡♠✲s❡ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲ ❑❛❝③♠❛r③ s✐♠étr✐❝♦ ❝♦♠ ♣ré✲❝♦♥❞✐❝✐♦♥❛♠❡♥t♦ ✭sLK✮✱ ❝♦♠♦ ❛ s❡q✉ê♥✲

      ✸✷ ❝✐❛ ❞❡ s♦❧✉çõ❡s ❛♣r♦①✐♠❛❞❛s {x n ❣❡r❛❞❛ ♣♦r✱

      } n∈N  δ

      ∗

      x M (A x ), k [k] [k] k − A − y

       [k] [k]    s❡ 0 ≤ n ♠♦❞ 2p ≤ p − 1  x = k+1

      ✭✸✳✶✶✮  δ

      ∗

       x M (A x ), k k  − A p−1−[k] p−1−[k] − y

      p−1−[k] p−1−[k]

        s❡ p ≤ n ♠♦❞ 2p ≤ 2p − 1 x n := x n2p , n = 0, 1, . . . [n] = n

      ♠♦❞ p, = x

      ♦♥❞❡ x é ♦ ❡❧❡♠❡♥t♦ ✐♥✐❝✐❛❧ ❡ ❛ ❛♣r♦①✐♠❛çã♦ ❞❡✈❡ s❡r ✐♥t❡r♣r❡t❛❞❛ n = x δ ❝♦♠♦ x n ♥♦ ❝❛s♦ ❞❡ ✉s❛r ❞❛❞♦s ❝♦♠ r✉í❞♦✳ n+1 =

      ❆❣♦r❛✱ ♣❛r❛ s✐♠♣❧✐✜❝❛r ❛ ♥♦t❛çã♦✱ ✉s❛♥❞♦ ✭✸✳✾✮ ♦❜té♠✲s❡ x δ

      ∗ ∗

      M A M y (I n + C B B B = A B

      − G)x ✱ ♦♥❞❡ G = A ❡ C ✳ ❆♣❧✐❝❛♥❞♦ ❛ ✐t❡r❛çã♦ ✭✸✳✼✮ ❞✉r❛♥t❡ ✉♠ p✲❝✐❝❧♦✱ t❡♠✲s❡ q✉❡ ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❛s ✐t❡r❛çõ❡s ✭✸✳✼✮ ❡ ✭✸✳✾✮ ♣♦❞❡ ❡s❝r❡✈❡r✲s❡ ❝♦♠♦

      I )(I ) ), ✭✸✳✶✷✮

      − G = (I − G p−1 − G p−2 · · · (I − G

      ∗

      = A M A , i = 0, . . . , p ♦♥❞❡ G i i i i − 1. ❆❣♦r❛✱ ❝♦♠♦ ♥♦ ♠ét♦❞♦ sLK s❡ ❢❛③ ✉♠ p✲❝✐❝❧♦ s❡❣✉✐❞♦ ❞❡ ♦✉tr♦ p✲❝✐❝❧♦ ❡♠ ♦r❞❡♠ ✐♥✈❡rs❛✱ ❡ ❝♦♠♦ ♦s ♦♣❡r❛❞♦r❡s I −G i sã♦ ❛✉t♦✲❛❞❥✉♥t♦s✱ ❡♥tã♦✱ ✉s❛♥❞♦ ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦ q✉❡ ❡♠ ✭✸✳✶✷✮ ♦❜té♠✲s❡ q✉❡ ✉♠ 2p✲❝✐❝❧♦ ❞♦ ♠ét♦❞♦ sLK ♣♦❞❡ ❡s❝r❡✈❡r✲ s❡ ❝♦♠♦

      ∗

      x = (I ((I + C ) + e C n+1 n B B − G) − G)x

      ✭✸✳✶✸✮

      

    ∗ ∗

      = (I (I + (I C + e C , n B B − G) − G)x − G) δ

      ∗ ∗

      C = A M y ♦♥❞❡ e B ✳ ❆ss✐♠✱ ♣❛r❛ ♦ ♠ét♦❞♦ sLK t❡♠✲s❡ ♦ s❡❣✉✐♥t❡ B r❡s✉❧t❛❞♦✱ q✉❡ é s✐♠✐❧❛r ❛♦ ❚❡♦r❡♠❛ ✸✳✶✳ ❚❡♦r❡♠❛ ✸✳✷✳ ❙❡❥❛ x n ❛ ✐t❡r❛çã♦ ❣❡r❛❞❛ ♣❡❧♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲ ❑❛❝③♠❛r③ s✐♠étr✐❝♦ ✭✸✳✶✶✮✳ ❊♥tã♦✱ ❛ ✐t❡r❛çã♦ ✭✸✳✶✶✮ ♣♦❞❡ ❡①♣r❡ss❛r✲s❡ ❝♦♠♦ ✉♠ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ♣♦r ❜❧♦❝♦s ♣ré✲❝♦♥❞✐❝✐♦♥❛❞♦s ✭s✐♠é✲ tr✐❝♦✮ ❞❛ ❢♦r♠❛ δ

      ∗

      x = x M (Ax ), n+1 n SB n ✭✸✳✶✹✮ − A − y

      ♦♥❞❡ 2 1 2 1

      

    ∗ ∗

      M SB i B ✭✸✳✶✺✮ = M (2I A A M ))M , B − diag(M i i i ❡ L, I, M B sã♦ ❝♦♠♦ ♥♦ ❚❡♦r❡♠❛ ✸✳✶✳

    • (M

      )C B + e C B ✱ t❡♠✲s❡✱

      A

      )y δ =

      ∗ B

      M

      ∗

      M B + A

      ∗

      A )A

      ∗ B

      M

      ∗

      )C B + e C B = ((I − A

      ∗

      (I − G

      ∗

      (M B − M

      ✳ ▲♦❣♦✱ ❝❤❛♠❛♥❞♦ M SB := M

      A , ♦♥❞❡ D = diag(M 1 2 i

      A i A

      ∗ i

      M 1 2 i ) é ♦ ♦♣❡r❛❞♦r ❡♠ ❜❧♦❝♦s q✉❡ ❝♦♥t❡♠ só ❛

      ❞✐❛❣♦♥❛❧ ❞♦ ♦♣❡r❛❞♦r AA

      ∗

      ∗ B

      )C B + e C B . ❆❣♦r❛✱ ♣❛r❛ ♦ t❡r♠♦ (I − G

      [2I − D] M B

      ❡♠ ✭✸✳✶✸✮✱ t❡♠✲s❡ x n+1 = (I − A

      ∗

      M SB A )x n + (I

      − G

      ∗

      ∗

      ∗ B

      

    B

      ∗

      M SB y δ ,

      ∗

      = A

      (2I − D)M B y δ

      

    B

      M

      ∗

      y δ = A

      ∗

      − AA

      ∗

      (I + L

      

    B

      M

      )M B y δ (3.10) = A

      AA

      M

      ∗

      M B + M

      ∗ B

      )y δ =

      A

      ∗

      

    B

      −1 B

      ((M

      ∗ B

      )

      −1

      − AA

      ∗

      [2I − D] M B

      M

      ❡ ❛ss✐♠ ♦❜té♠✲s❡ ❛ ❡①♣r❡ssã♦ ✭✸✳✶✹✮ ❝♦♠♣❧❡t❛♥❞♦ ❛ ♣r♦✈❛✳

      M

      ∗

      AA

      ∗ B − MB

      (M B + M

      ∗

      I − A

      M B A =

      ∗

      AA

      ∗ B

      M

      ∗

      A + A

      ∗ B

      ∗

      =

      M

      ✸✸ Pr♦✈❛✿ P♦r ✭✸✳✶✷✮ ❡ ✭✸✳✶✸✮ t❡♠✲s❡✱ (I

      − G)

      ∗

      (I − G)

      = (I − A

      ∗

      

    B

      M B A − A

      A )(I − A

      ∗

      M B A )

      =

      I − A

      ∗

      M B )A

      I − A

      ∗

      M

      −1

      − 1 2 AA

      ∗

      ]M B A (3.10) =

      I − A

      ∗

      

    B

      ∗ B

      [2I + L + L

      ∗

      − AA

      ∗

      ] M B A =

      I − A

      )

      −1 B − 1 2 AA ∗

      ∗

      ∗

    B

      [M

      

    B

      (I − 1 2 AA

      ∗

      M B ) +

      (I − 1 2 M

      AA

      [M

      ∗

      )M B ]A =

      I − A

      ∗

      M

      

    B

    • M
    • I + L)M B

      ✸✹

      

    ✸✳✸ ❚❛①❛s ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ▼ét♦❞♦

    ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ♥ã♦ ❙✐♠étr✐❝♦

      ◆❛ ♣r❡s❡♥t❡ s❡❝çã♦ s❡ ❛♣r❡s❡♥t❛ ❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡ s❡ ❡st❛✲ ❜❡❧❡❝❡♠ ❛s t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ♠ét♦❞♦ LK ✭✸✳✼✮✳ ❆ ♣r✐♥❝✐♣❛❧

      ∗

      M A ❞✐✜❝✉❧❞❛❞❡ ❡♠ ❝♦♠♣❛r❛çã♦ ❛♦ ♠ét♦❞♦ sLK✱ é q✉❡ ♦ ♦♣❡r❛❞♦r A B ♥ã♦ é ❛✉t♦✲❛❞❥✉♥t♦✱ s❛❧✈♦ ❡♠ ❝❛s♦s tr✐✈✐❛✐s✳ ❆ss✐♠✱ ❛ ❛♥á❧✐s❡ ❝❧áss✐❝❛ ❜❛s❡❛❞❛ ❡♠ ♦♣❡r❛❞♦r❡s ❛✉t♦✲❛❞❥✉♥t♦s ♥ã♦ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞❛✳

      ❈♦♠♦ ❥á ❢♦✐ ❛♣♦♥t❛❞♦✱ ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❛s ✐t❡r❛çõ❡s ✭✸✳✼✮ ❡ ✭✸✳✾✮ ❡stá ❞❛❞❛ ♣♦r ✭✸✳✶✷✮ ❡✱ ❛❧é♠ ❞♦ ❛♥t❡r✐♦r✱ ❞❡✜♥❡✲s❡ ❛ ❡①t❡♥sã♦ ♥❛t✉r❛❧ C ❞♦ ♦♣❡r❛❞♦r G ♣❛r❛ X ❝♦♠♦ G(u) = G(x + iy) := G(x) + iG(y), ∀u = C x + iy

      ∈ X ✳ ❆ss✐♠✱ t❡♠✲s❡ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳ ▲❡♠❛ ✸✳✶✳ ❙❡ ✭✸✳✻✮ é ✈á❧✐❞❛✱ ❡♥tã♦ G é ✉♠ ♦♣❡r❛❞♦r ❛❝r❡t✐✈♦✱ ✐st♦ é✱ s❛t✐s❢❛③ C Re C . X hGx, xi ≥ 0, ∀x ∈ X i = A M i A i

      Pr♦✈❛✿ P♦r ❞❡✜♥✐çã♦✱ ∀i = 0, . . . , p − 1✱ ♦ ♦♣❡r❛❞♦r G i é s✐♠étr✐❝♦✱ s❡♠✐❞❡✜♥✐❞♦ ♣♦s✐t✐✈♦ ❝♦♠ ♥♦r♠❛ ❧✐♠✐t❛❞❛ ♣♦r ✷✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ kI − G i k ≤ 1✱ ❧♦❣♦ ♣♦r ✭✸✳✶✷✮ t❡♠✲s❡ kI − Gk ≤ 1✳ ❆ss✐♠

      Re X C = X C X C hGx, xi hx, xi − Reh(I − G)x, xi c.s 2 2 ≥ kxk X C − kI − Gkkxk X C ≥ 0,

      ♦♥❞❡ c.s s✐❣♥✐✜❝❛ q✉❡ ❢♦✐ ✉s❛❞❛ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✳ ❖❜s❡r✈❛çã♦ ✸✳✷✳ ❙❡❣✉❡ ❞♦ ▲❡♠❛ ✸✳✶ q✉❡ ♦ ❡s♣❡❝tr♦ ❞❡ G ❡stá ❝♦♥t✐❞♦ ♥♦ s❡♠✐ ❡s♣❛ç♦ ♣♦s✐t✐✈♦✱ σ(G) ⊂ {λ ∈ C : Re(λ) ≥ 0} ❡ q✉❡ ❛ ❝♦♥❤❡❝✐❞❛ ❡st✐♠❛t✐✈❛

      1

      −1

      , k(G + tI) k ≤ C ∀t > 0, t

      é ✈á❧✐❞❛ ✭✈❡❥❛ ♦ ❚❡♦r❡♠❛ ✸✳✷ ❞♦ ❝❛♣ít✉❧♦ ✸ ❞❡ ❬✶✽❪✮✱ ✐st♦ é✱ G é ✉♠ ♦♣❡r❛❞♦r ❢r❛❝❛♠❡♥t❡ s❡t♦r✐❛❧ ✭✈❡❥❛ ❬✷✷✱ ✷✸❪✮✳ P❛r❛ ❡ss❡s ♦♣❡r❛❞♦r❡s ❛s α ♣♦t❡♥❝✐❛s ❢r❛❝✐♦♥❛✐s G ♣❛r❛ α > 0 ❡stã♦ ❜❡♠ ❞❡✜♥✐❞❛s ♠❡❞✐❛♥t❡ ✉♠❛ ✐♥t❡❣r❛❧ ❞❡ t✐♣♦ ❉✉♥❢♦r❞✲❙❝❤✇❛rt③✳

      ◆♦ s❡❣✉✐♥t❡ ❧❡♠❛ ♦❜té♠✲s❡ ✉♠❛ ❡st✐♠❛t✐✈❛ ♣❛r❛ ♦ ❡rr♦ ♣r♦♣❛❣❛❞♦✳ δ ▲❡♠❛ ✸✳✷✳ ❙❡ ✭✸✳✻✮ é ✈á❧✐❞❛✱ ❡ x é ♦ ✈❡t♦r ❣❡r❛❞♦ ♣❡❧♦ ♠ét♦❞♦ ✭✸✳✾✮ n ❝♦♠ ❞❛❞♦s ♣❡rt✉r❜❛❞♦s ❡ x n é ❣❡r❛❞♦ ♣♦r ✭✸✳✾✮ ❝♦♠ ❞❛❞♦s ❡①❛t♦s✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C = C(kAk, kM B δ k) t❛❧ q✉❡ n M ✭✸✳✶✻✮ . kx n − x k ≤ Cnδ

      ✸✺ ❙❡ ❛❞✐❝✐♦♥❛❧♠❡♥t❡ t❡♠✲s❡ n

      ∗

      AA sup B ) 1 , k(I − M k ≤ C ✭✸✳✶✼✮

      n∈N

      ❝♦♠ C 1 ❝♦♥st❛♥t❡✱ ❡♥tã♦ δ n nδ M , √ kx n − x k ≤ C ✭✸✳✶✽✮

      −1

      , ♦♥❞❡ C = C(C 1 B kM k, kM B k) ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ n✳

      Pr♦✈❛✿ ❆♣❧✐❝❛♥❞♦ ✐t❡r❛❞❛♠❡♥t❡ ✭✸✳✾✮ ♦❜té♠✲s❡

      

    n−1

    δ n j δ

      X

      

    ∗ ∗ ∗

      M A M A A M y x = (I ) x (I ) n − A − A B B B j=0

    • ♦ q✉❡ ✐♠♣❧✐❝❛

      n−1 δ j δ

      X

      ∗ ∗

      M A A M x = (I ) (y n − x − A − y), n B B j=0 ❧♦❣♦ ❝♦♠♦ I − G é ♥ã♦ ❡①♣❛♥s✐✈♦✱

      n−1 δ j δ

      X

      

      M n B M , kx n − x k ≤ kI − Gk kA kky − yk ≤ Cnδ j=0

      ∗

      M ♦♥❞❡ C = kA B k. ❆❣♦r❛✱ ♣❛r❛ ♣r♦✈❛r ✭✸✳✶✽✮✱ ❞❡♥♦t❛♥❞♦

      n−1

      X j g (x) = (1 , L j=0 − x) t❡♠✲s❡✱ δ n 2 kx n − x k δ δ

      ∗ ∗ ∗ ∗

      = (A M A )A M (y (A M A )A M (y L B B L B B hg − y), g − y)i δ δ

      ∗ ∗ ∗

      AA AA = g L (M B )M B (y L (M B )M B (y hAA − y), g − y)i δ δ

      −1 ∗ ∗ ∗

      = (M AA g (M AA )) M (y (M AA )M (y B L B B L B B hM B − y), g − y)i

      (∗)

    −1 n δ δ

    ∗ ∗

      = (I AA ) ) M (y (M AA )M (y B B L B B c.s hM − (I − M − y), g − y)i B n δ 2

      −1 ∗ ∗ B L B B AA ) (M AA ) (y

      ≤ kM B kkI − (I − M kkg kkM − y)k

      ✸✻   (3.17) n−1

      X j 2 δ 2

      −1 ∗ 1 ) B ) B  AA

      ≤ kM k(1 + C k(I − M k  kM k ky − yk B 2 j=0 2 2 2

      −1 B (1 + C )C nδ = C nδ , 1 1

      ≤ kM B kkM k M M q

      −1

      )C ❝♦♠ C := kM B 1 1 ✱ ❞❡ ♦♥❞❡ ♦❜té♠✲s❡ ✭✸✳✶✽✮ ❝♦♠✲ k kM B k(1 + C

      ♣❧❡t❛♥❞♦ ❛ss✐♠ ❛ ♣r♦✈❛✳ ❊♠ (∗) ❢♦✐ ✉s❛❞❛ ❛ ✐❞❡♥t✐❞❛❞❡

      n−1

      X j n x (1 = 1 . j=0 − x) − (1 − x)

      †

      ❆❣♦r❛✱ ♣❛r❛ ❡st✐♠❛r ♦ ❡rr♦ ❞❡ ❛♣r♦①✐♠❛çã♦ kx n − x k✱ é ♥❡❝❡ssár✐♦

      ♦ s❡❣✉✐♥t❡ ❧❡♠❛ q✉❡ ❡st❛❜❡❧❡❝❡ q✉❡ I − G é ✉♠❛ ❝♦♥tr❛çã♦ ♣❛r❛ ♦s ❡❧❡♠❡♥t♦s q✉❡ ♥ã♦ ♣❡rt❡♥❝❡♠ ❛♦ ♥ú❝❧❡♦ ❞❡ G✳ C

      = 1 ▲❡♠❛ ✸✳✸✳ ❙❡❥❛♠ η > 0 ❡ x ∈ X ❝♦♠ kxk X C ✳ ❙✉♣♦♥❤❛ q✉❡ ✭✸✳✻✮

      é ✈á❧✐❞❛✳ ❙❡ Re X C hGx, xi ≥ η, 1 2 p−1 ∗

      M ❡♥tã♦ ❡①✐st❡ 0 < γ < 1 ❞❡♣❡♥❞❡♥❞♦ ❞❡ η, p ❡ t❛❧ q✉❡ 2 kA i i k i=0 k(I − G)xk X C ≤ 1 − γ. ✭✸✳✶✾✮

      Pr♦✈❛✿ ❙❡❥❛ {E λ,i λ ❛ ❢❛♠í❧✐❛ ❡s♣❡❝tr❛❧ ❛ss♦❝✐❛❞❛ ❝♦♠ ♦ ♦♣❡r❛❞♦r }

      ∗ C

      A M A , i = 0, . . . , p i − 1✱ ❞❡✜♥✐❞♦s ❡♠ X i i ✳ ◆❡st❛ ♣r♦✈❛✱ ❛ ♥♦r♠❛ C ❡♠ X ❡ ❛ ♥♦r♠❛ ❞❡ ♦♣❡r❛❞♦r❡s s❡rã♦ ❞❡♥♦t❛❞❛s ♣♦r k · k✳ ❆❧é♠ ❞♦ ❛♥t❡r✐♦r✱ ♣❛r❛ ❝❛❞❛ ξ > 0✱ ❝♦♥s✐❞❡r❡ ♦s ♣r♦❥❡t♦r❡s ♦rt♦❣♦♥❛✐s

      Z Z P = dE Q = I = dE . ξ,i λ,i ξ,i ξ,i λ,i − P λ>ξ

    ξ,i ξ,i ξ,i x ξ,i x = ,

    λ≤ξ 2 2 2 C 2 2

      ◆♦t❡ q✉❡ kP k = kQ k = 1 ❡ kP k kQ k kxk ∀x ∈ X ✳

      M i A i , ❙❡❥❛ 0 < ξ < 1 t❛❧ q✉❡ (1−ξ) ≥ (1−kA i k) ∀i = 0, . . . , p−1✳ P♦r 1 2

      ∗

      M

      2 ✭✸✳✻✮✱ t❡♠✲s❡ q✉❡ max i ❡ ♣♦rt❛♥t♦ ξ ♣♦❞❡ s❡r ❡s❝♦❧❤✐❞♦ kA i i k <

      ). ❞❡ ✉♠ ✐♥t❡r✈❛❧♦ (0, ξ 2 2

      = 2ξ > ξ > 0 ❆❣♦r❛✱ ❞❡✜♥❛ θ := 1 − (1 − ξ) ✳ ❯s❛♥❞♦ ❝á❧❝✉❧♦

      − ξ

      ✸✼ ❡s♣❡❝tr❛❧✱ ♣❛r❛ ♦ ξ ❡s❝♦❧❤✐❞♦ t❡♠✲s❡ 2 Z Z 2 2 2 2

      ∗

    • λ≤ξ
    • 2 2 2 2 ξ,i x + max , (1 M i A i ξ,i x

        M i A i )x = (1 d λ,i x (1 d λ,i x λ>ξ k(I − A i k − λ) kE k − λ) kE k

        ≤ kP k {(1 − ξ) − kA i k) }kQ k 2 2 2 2 = x + (1 ( x ) ξ,i ξ,i kP k − ξ) kxk − kP k 2 2 2 2

        = (1 ) x + (1 ξ,i − (1 − ξ) kP k − ξ) kxk 2 2

        = θ x + (1 , ξ,i kP k − θ)kxk ✭✸✳✷✵✮

        ❡ 2 Z Z 2 2 2 2

        ∗

      • λ≤ξ
      • 1

        2 2 2 4 2 ✭✸✳✷✶✮

          M i A i = λ d λ,i x λ d λ,i x λ>ξ kA i k kE k kE k

        • ξ,i x A i ξ,i x ≤ ξ kP k kM k kQ k
        • 2 2 i 2 2 ξ,i ξ,i x + 4( x ).

            ≤ ξ kP k kxk − kP k ❆❣♦r❛✱ ❞❡✜♥❛ ♣❛r❛ ❝❛❞❛ k ≤ p − 1 ♦s ♦♣❡r❛❞♦r❡s k

            Y

            ∗ ∗ ∗ H = (I M A ) = H M A H , H = (I M A ). k i i k k k i=0 −A i ⇔ H k−1 −A k k−1 −A k

            ➱ ❝❧❛r♦✱ ♣♦r ✭✸✳✻✮✱ q✉❡ kH k ≤ 1✱ ❡ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ❞❛ ❡①♣r❡ssã♦ r❡❝✉r✲ s✐✈❛

            ∗ ∗

            H M A (H M A k k k k k − I = H k−1 − I − A k k−1 − I) − A k

            

          ∗ ∗

            = (I M A )(H M A k k k k − A k k−1 − I) − A k C s❡ ✐♥❢❡r❡✱ ✉s❛♥❞♦ ✐♥❞✉çã♦✱ q✉❡ ♣❛r❛ x ∈ X q✉❛❧q✉❡r✱ k

            X

            ∗ ∗ k k k i i M A x M A x k(H − I)xk ≤ k(H k−1 − I)xk + kA k k ≤ kA i k. i=0

            ❉❛❞♦ q✉❡ ✭♣♦r ✭✸✳✶✷✮✮ G = G − I + I = I − H ✱ t❡♠✲s❡

            p−1 p−1

            X

            

            M i A i x kGxk ≤ kA i k. ✭✸✳✷✷✮

            ✸✽ ❆ss✐♠✱ ❛♣❧✐❝❛♥❞♦ ✭✸✳✷✵✮✱ ♦❜té♠✲s❡ ❛ ❡st✐♠❛t✐✈❛ 2 2 2

            ∗ k k k ξ,k x = M A )H x H x + (1 x kH k k(I − A k k−1 k ≤ θkP k−1 k − θ)kH k−1 k 2 2 x + θ( ξ,k x ξ,k (I )x

            ≤ (1 − θ)kH k−1 k kP k + kP − H k−1 k) 2 !

            k−1 2 X ∗

            x + θ ξ,k x M i A i x ≤ (1 − θ)kH k−1 k kP k + kA i k i=0 2

            !

            k−1 2 X ∗ + θ ξ,k x M i A i x .

            ≤ (1 − θ)kxk kP k + kA i k i=0 ✭✸✳✷✸✮

            ❉❡✜♥❛ ❛❣♦r❛ ❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s

            k−1

            X p D = 5, D = (9 + 8 D ), k = 1, . . . , p k i i=0 − 1.

            ❆ ♣r♦✈❛ ❞♦ ▲❡♠❛ ✸✳✸ é✱ ❛ ✜♥❛❧✱ ♣♦r ❝♦♥tr❛❞✐çã♦✳ ❆ss✉♠❛ q✉❡ ♦ r❡s✉❧t❛❞♦ ♥ã♦ é ✈á❧✐❞♦✳ ❊♥tã♦ ❡①✐st❡ ❛❧❣✉♠ η > 0 t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ ǫ > 0 ❡①✐st❡ C ✉♠ x ∈ X t❛❧ q✉❡ X C 2 X C ✭✸✳✷✹✮ |h(I − G)x, xi | ≥ (1 − ǫ), RehGx, xi ≥ η, kxk = 1.

            ❙❡❥❛ 0 < ǫ < 1 s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ t❛❧ q✉❡ p ! 2

            1 ǫ

            ≤ P ✭✸✳✷✺✮ √

            

          p−1

            1 + D i=0 i2 2

            ∗

            (1 ǫ) max (1 M i A i i ✭✸✳✷✻✮ − ≥ − kA k)

            i=0,...,p−1 p +1

            ! 2 η ǫ < .

            ✭✸✳✷✼✮ P

            √

            p−1 C i=0 D i

            ❈♦♠♦ |h(I − G)x, xi X | ≤ k(I − G)xk ❡ x x x k(I − G)xk = kH p−1 k ≤ kH p−2 k ≤ · · · ≤ kH k,

            ❡♥tã♦✱ ♣❛r❛ ❡ss❡ ǫ ❡①✐st❡ ✉♠ x ❝♦♠♦ ❡♠ ✭✸✳✷✹✮ ❝♦♠ k x 2 kH k ≥ 1 − ǫ, ∀k = 0, . . . , p − 1.

            √ ǫ

            P♦r ✭✸✳✷✻✮ ❛ ❡s❝♦❧❤❛ ❞❡ ξ = ♣♦❞❡ s❡r ✉s❛❞❛ ❡♠ ✭✸✳✷✵✮ ❡ ✭✸✳✷✸✮✳ √ √

            ǫ ǫ P♦rt❛♥t♦✱ ❝♦♠ θ = 2 − ǫ > ✱ ♦❜té♠✲s❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ 2

            !

            k−1

            X

            ∗ √

            1 x M i A i , − ǫ ≤ (1 − θ) + θ kP ǫ,k k + kA i k ∀k = 0, . . . , p − 1,

            ✸✾ q✉❡ ♣♦r s✉❛ ✈❡③ ✐♠♣❧✐❝❛ ❛ ❡st✐♠❛t✐✈❛ r

            k−1

            X ǫ ǫ

            √

            ∗ √

            x M A i i 1 ǫ, kP ǫ,k k+ kA i k ≥ − ≥ 1− ≥ 1− ∀k = 0, . . . , p−1. i=0 θ θ ✭✸✳✷✽✮

            ❆❣♦r❛✱ ✉s❛♥❞♦ ✭✸✳✷✶✮✱ ♦❜té♠✲s❡ 2 2 2 2

            ∗ √ √ √

            M A x x + 4(1 x ) x ). k k ǫ,k ǫ,k ǫ,k kA k k ≤ ǫkP k − kP k ≤ ǫ + 4(1 − kP k ✭✸✳✷✾✮

            P❛r❛ k = 0✱ ♦❜té♠✲s❡ ❞❡ ✭✸✳✷✽✮ ❡ ✭✸✳✷✾✮ 2 √ √ √ 2

            ∗ √

            x ǫ, M A x ǫ ǫ. kP ǫ,0 k ≥ 1 − kA k ≤ ǫ + 4 ≤ D

            Pr♦❝❡❞❡♥❞♦ ♣♦r ✐♥❞✉çã♦ s❡ ♠♦str❛ q✉❡ 2 1

            ∗

          2k+1

            M A x ǫ , k k k ✭✸✳✸✵✮ kA k k ≤ D ∀k = 0, . . . , p − 1. ❯s❛♥❞♦ ✭✸✳✷✽✮ ♠❛✐s ❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦ ♣❛r❛ k − 1, k ≥ 1✱ ♦❜té♠✲s❡

            !

            k−1 k−1 1 X 1 p X p

            √

            √ 2i+2 2k+1 x ǫ ǫ D 1 + D . i i kP ǫ,k k ≥ 1 − − ≥ 1 − ǫ i=0 i=0

            ◆♦t❡ q✉❡✱ ♣♦r ✭✸✳✷✺✮✱ ♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❡st❛ ❞❡s✐❣✉❛❧❞❛❞❡ é ♣♦s✐t✐✈♦✳ P♦r✲ t❛♥t♦✱ ♣♦r ✭✸✳✷✾✮ t❡♠✲s❡ 2

            ∗

            M k A k x kA k k   2

            !! 1 k−1 2k+1 X p  1 1 + D i

            ≤ ǫ + 4 1 − − ǫ i=0

             ! !! 2 1 1 k−1 k−1 2k+1 2k+1 2k+1 X p 1 X p

            

          • 4 2ǫ 1 + D ǫ 1 + D i i

            ≤ ǫ − i=0 i=0 ! 1 k−1 1 2k+1 2k+1 X p 9 + 8 D = ǫ D , i k

            ≤ ǫ i=0 ♦ q✉❡ ♣r♦✈❛ ✭✸✳✸✵✮✱ ♣❛r❛ t♦❞♦ k = 0, . . . , p−1✳ ❋✐♥❛❧♠❡♥t❡✱ ❝♦♥s❡❣✉❡✲s❡ ✉♠❛ ❝♦♥tr❛❞✐çã♦ ♣❛r❛ ✭✸✳✷✹✮✳ ❯s❛♥❞♦ ✭✸✳✷✷✮ ❡ ✭✸✳✷✼✮ ♦❜té♠✲s❡

            η X C ≤ RehGx, xi

            

          p−1 p−1 p−1

            X X 1 p 1 X p

            ∗ 2i+2 2p+1

            M i A i x ǫ D i D i < η, ≤ kGxk ≤ kA i k ≤ ≤ ǫ

            ✹✵ ❛ss✐♠ ✭✸✳✷✹✮ ♥ã♦ ♣♦❞❡ s❡r ✈á❧✐❞❛ ♣❛r❛ ❡ss❛ ❡s❝♦❧❤❛ ❞❡ ǫ✱ ❝♦♠♣❧❡t❛♥❞♦ ❛ss✐♠ ❛ ♣r♦✈❛✳ ❖❜s❡r✈❛çã♦ ✸✳✸✳ ❯♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞✐r❡t❛ ❞❛ ♣r♦✈❛ ❞♦ ▲❡♠❛ ✸✳✸✱ é q✉❡ γ ❡♠ ✭✸✳✶✾✮ ♣♦❞❡ s❡r ❡s❝♦❧❤✐❞♦ ❝♦♠♦ ♦ ♠❛✐♦r ♥ú♠❡r♦ ǫ t❛❧ q✉❡ ✭✸✳✷✺✮✲✭✸✳✷✼✮ ♥ã♦ sã♦ s❛t✐s❢❡✐t❛s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ η s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ ✭t❛❧ q✉❡ ✭✸✳✷✼✮ ✐♠♣❧✐❝❛ ✭✸✳✷✺✮ ❡ ✭✸✳✷✻✮✮✱ ♦ ▲❡♠❛ ✸✳✸ é ✈á❧✐❞♦ ❝♦♠ ❛ s❡❣✉✐♥t❡ ❡s❝♦❧❤❛ ❞❡ γ p +1

            ! 2 η γ = .

            P √

            p−1 i=0 D i

            ❆❣♦r❛✱ é ♣♦ssí✈❡❧ ❡st✐♠❛r ♦ ❡rr♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ❛ss✉♠✐♥❞♦ q✉❡ ✉♠❛ ❝♦♥❞✐çã♦ ❞❡ ❢♦♥t❡ é s❛t✐s❢❡✐t❛✳ ❈♦♠♦ ✐♥❞✐❝♦✉✲s❡ ❛♥t❡r✐♦r♠❡♥t❡✱ ❛ ❛♥á❧✐s❡ ❝❧áss✐❝❛ ✈✐❛ t❡♦r✐❛ ❡s♣❡❝tr❛❧ ♥ã♦ é ♣♦ssí✈❡❧ ♣♦rq✉❡ ♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ♦♣❡r❛❞♦r ♥ã♦ é s✐♠étr✐❝♦✳ P❛r❛ s✉♣❡r❛r ❡ss❡ ♦❜stá❝✉❧♦✱ ✉s❛✲s❡ ♦ ❝á❧❝✉❧♦ ❢✉♥❝✐♦♥❛❧ ♥♦ r❛♥❣♦ ♥✉♠ér✐❝♦✳ C C ❉❡✜♥✐çã♦ ✸✳✶✳ ❉❡✜♥❡✲s❡ ♦ r❛♥❣♦ ♥✉♠ér✐❝♦ ❞♦ ♦♣❡r❛❞♦r G : X

            → X ❝♦♠♦ C

            W (G) := X C : x , X C = 1 {hGx, xi ∈ X kxk }.

            ❈♦♠ ❡st❛ ❞❡✜♥✐çã♦✱ ♣♦❞❡✲s❡ ❡♥✉♥❝✐❛r ♦ ❧❡♠❛ ♣r✐♥❝✐♣❛❧✱ ♥❡❝❡ssár✐♦ ♣❛r❛ ❛❝❤❛r ❛s ❡st✐♠❛t✐✈❛s ❞♦ ❡rr♦ ❞❡ ❛♣r♦①✐♠❛çã♦✳ ▲❡♠❛ ✸✳✹✳ ❙❡ ✭✸✳✻✮ é ✈á❧✐❞❛ ❡ ❡①✐st❡ h > 0 t❛❧ q✉❡

            π

            −1

            W (G) (h) < ✭✸✳✸✶✮

            ⊂ Σ := {λ ∈ C : | arg(λ)| ≤ tan },

            2 2 1 p−1

            ∗

            M ❡♥tã♦✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C ❞❡♣❡♥❞❡♥❞♦ ❞❡ α, h, p ❡ ✱ kA i i k i=0 t❛❧ q✉❡ n α C

            G , k(I − G) k ≤ ∀α > 0. α (n + 1) C

            Pr♦✈❛✿ ❙❡❥❛ η > 0✱ ❡♥tã♦✱ ♣❡❧♦ ▲❡♠❛ ✸✳✶✱ ♣❛r❛ x ∈ X ✱ t❡♠✲s❡ Re < η. X C X C X C hGx, xi ≥ η ♦✉ 0 ≤ RehGx, xi X C = 1 ❙❡ RehGx, xi ✱ ❡♥tã♦ ♣❡❧♦ ▲❡♠❛ ✸✳✸✱ ❡①✐st❡

            ≥ η ❝♦♠ kxk 0 < γ < 1 t❛❧ q✉❡ p X C X C X C

            1 |1 − hGx, xi | = |h(I − G)x, xi | ≤ k(I − G)xk ≤ − γ,

            ✹✶ ❡ ❛ss✐♠ t❡♠✲s❡

            W (G) ⊂ Σ ∩ ({λ ∈ C : 0 ≤ Re(λ) < η}∪{λ ∈ C : η ≤ Re(λ), p

            1 |1 − λ| < − γ})

            ❡ ♣♦rt❛♥t♦ W (G)

            ⊂ (Σ ∩ {λ ∈ C : 0 ≤ Re(λ) < η}) ∪{λ ∈ C : η ≤ Re(λ), p

            1 2 |1 − λ| < − γ}.

            −1

            (ψ) (h) ❆❣♦r❛✱ ✜①❡ η = cos ✱ ♦♥❞❡ ψ = tan ❡ t♦♠❡ γ ❝♦♠♦ ❡♠ ✭✸✳✶✾✮✳ ❯s❛♥❞♦ ♦ ❈á❧❝✉❧♦ ❋✉♥❝✐♦♥❛❧ ❞❡ ❈r♦✉③❡✐① ✭♣❛r❛ ❞❡t❛❧❤❡s ✈❡❥❛ ❬✹❪✮✱ ❝♦♥❝❧✉í✲s❡ q✉❡ ♣❛r❛ t♦❞♦ α > 0 n α n α

            G sup λ C k(I − G) k ≤ C |(1 − λ) |,

            λ∈Σ

            ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ C C ≤ 11.08✳ P❛r❛ ♦s ❡❧❡♠❡♥t♦s ❞❡ W (G) q✉❡

            ♣❡rt❡♥❝❡♠ ❛ Σ ∩ {λ ∈ C : 0 ≤ Re(λ) < η} t❛❧ q✉❡ 0 < Re(λ) < η✱ t❡♠✲s❡✱ ❞❡✈✐❞♦ ❛ q✉❡ ❛ ❢✉♥çã♦ t❛♥❣❡♥t❡ é ❝r❡s❝❡♥t❡ 2 2 Im(λ) Re(λ) + Im(λ) 2 tan(arg(λ)) = + 1, ≤ h ⇒ ≤ h 2 Re(λ) Re(λ)

            ♦✉ s❡❥❛ p 2 η 1 + h < = cos(ψ) ( |λ| ≤ Re(λ) †). 2 cos(ψ) 2

            = (1 P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❡ |1 − λ| − λ)(1 − λ) = 1 + |λ| − 2Re(λ) ❡ ❞❛ ❢♦r♠❛ ♣♦❧❛r ❞❡ λ✱ t❡♠✲s❡ 2 2

            = 1 + |1 − λ| |λ| − 2|λ| cos(arg(λ)). (††) π

            ❆❣♦r❛✱ ❝♦♠♦ λ ∈ Σ✱ ❡♥tã♦ | arg(λ)| ≤ ψ < ❡ ❝♦♠♦ ❛ ❢✉♥çã♦ ❝♦ss❡♥♦ π 2 ]

            é ❞❡❝r❡s❝❡♥t❡ ❡♠ [0, t❡♠✲s❡ cos(ψ) ≤ cos(| arg(λ)|) = cos(arg(λ)) ♦ 2 q✉❡ ✐♠♣❧✐❝❛ ♣♦r (††) 2 2 2 |1 − λ| ≤ 1 + |λ| − 2|λ| cos(ψ) ♠❛s ♣♦r (†) |λ|

            ≤ |λ| cos(ψ)✱ ❡ ❛ss✐♠ t❡♠✲s❡ 2 |1 − λ| ≤ 1 − |λ| cos(ψ).

            ❈♦♠ ❡ss❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣♦❞❡✲s❡ ❡st✐♠❛r n α 2 α α 2 n n 2 λ )

            |(1 − λ) | = (|1 − λ| |λ| ≤ |1 − |λ| cos(ψ)| |λ|

            ✹✷ n

            1 2 α .

            ≤ |1 − |λ| cos(ψ)| |λ cos(ψ)| α cos (ψ) ❆❣♦r❛✱ ❢❛③❡♥❞♦ z = |λ| cos(ψ) ♥❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡✱ t❡♠✲s❡ z ≥ 0✱ 2 n/2 α z (ψ) z

            ❛t✐♥❣❡ ♦ s❡✉ ≤ η = cos ≤ 1✱ ❡ ❛ ❢✉♥çã♦ r❡❛❧ f(z) = (1 − z) n

            ) ♠á①✐♠♦ ✭♣❛r❛ 0 ≤ z ≤ 1✮ ❡♠ z = α/(α + ✱ ❧♦❣♦ n α α 2 n 2 n/2 α 2 α α

            (1 z , − z) ≤ ≤ ∀ 0 ≤ z ≤ 1, n n n

            α + α + α + 2 2 2 ♦ q✉❡ ❝❧❛r❛♠❡♥t❡ ✐♠♣❧✐❝❛✱ α

            −α

            n n/2 α α α (1 z ,

            − z) ≤ ≤ α ∀ α > 0, n α + 2

            2 ❡ ♣♦rt❛♥t♦✱ α

            −α n α α n

            λ , |(1 − λ) | ≤ ∀ α > 0. α cos (ψ) 2 α α α n −α 2

            , = α P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ (n + 1) ≤ (2n) ∀n ∈ N✱ ❡♥tã♦ ≤ (n+1) 4 α α ✱ ❡ ❛ss✐♠ t❡♠✲s❡ α 2 n n α

            1 λ , |(1 − λ) | ≤ ∀ α > 0. α cos(ψ) (n + 1)

            ❆❣♦r❛✱ ♣❛r❛ ♦s ❡❧❡♠❡♥t♦s ❞❛ ♦✉tr❛ ♣❛rt❡ ❞♦ r❛♥❣♦ ♥✉♠ér✐❝♦ t❡♠✲s❡ 1/2 |1 − λ| ≤ (1 − γ) ≤ 1 ♦ q✉❡ ✐♠♣❧✐❝❛ |λ| ≤ 2✱ ❛ss✐♠ n α n α α 2 n

            λ (1 , |(1 − λ) | = |1 − λ| |λ| ≤ 2 − γ) ∀λ : Re(λ) ≥ η. n 2 α n→∞ (n + 1)

            ❈♦♠♦ (1 − γ) −→ 0✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C✱ q✉❡ n

          2

          −α ,

            ❞❡♣❡♥❞❡ ❞❡ γ ❡ α✱ t❛❧ q✉❡ (1 − γ) ≤ C(n + 1) ∀n ∈ N✱ ❡ ❛ss✐♠ 4α α α , 2 C

            ♦❜té♠✲s❡ ♦ r❡s✉❧t❛❞♦ ❡s❝♦❧❤❡♥❞♦ C = max{ cos(ψ) }✳ ❖❜s❡r✈❛çã♦ ✸✳✹✳ P❛r❛ ♣♦❞❡r ❛♣❧✐❝❛r ♦ ▲❡♠❛ ✸✳✹ é ♣r❡❝✐s♦ ❢♦❝❛r ❛ ❛t❡♥çã♦ ♥❛ ❝♦♥❞✐çã♦ ✭✸✳✸✶✮✳ ❯s❛♥❞♦ ❛ ♣❛rt❡ r❡❛❧ ❡ ✐♠❛❣✐♥ár✐❛ ❞❡ G(u) ♣❛r❛ u = x + iy ∈ W (G)✱ ❝♦♠ G ❝♦♠♦ ❛♥t❡s✱ t❡♠✲s❡ X C = hG(x + iy), x + iyi hGx, xi + hGy, yi + i(hGy, xi − hGx, yi)

            ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ ✉♠❛ ❝♦♥❞✐çã♦ ❡q✉✐✈❛❧❡♥t❡ ❛ ✭✸✳✸✶✮ é ❛ ❡①✐stê♥❝✐❛ ❞❡ h > 0 t❛❧ q✉❡

            |hGy, xi − hGx, yi| ≤ h(hGx, xi + hGy, yi),

            ✹✸ ♦✉ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡

            ∗ ∗ ∗ ∗ ∗

            M A M A )x, y M A x, x M A y, y B B B |h(A B − A i| ≤ h(hA i + hA i) 2 2 2

            = + = 1 ♣❛r❛ t♦❞♦ x, y ∈ X t❛❧ q✉❡ kx + iyk X kxk kyk C ✳ ❋❛③❡♥❞♦

            A A z = M B x B y ❡ v = M ✱ ✐♥❢❡r❡✲s❡ q✉❡ ❛ ❝♦♥❞✐çã♦ ❛♥t❡r✐♦r é s❛t✐s❢❡✐t❛ s❡✱ ♣❛r❛ t♦❞♦ z, v ∈ Y✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡

            

          −1 −1 ∗ −1 −1

            ) )z, v z, z v, v |h(M B − (M B i| ≤ h hM B i + hM B i

            é ✈á❧✐❞❛✳ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ M B ✭✸✳✶✵✮✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ é s❛t✐s❢❡✐t❛ s❡ ♣❛r❛ t♦❞♦ z, v ∈ Y✱ 2 2

            ∗

          • )z, v

            ✭✸✳✸✷✮ |h(L − L i| ≤ h(kzk kvk hLz, zi + hLv, vi). ❊st❛ ♦❜s❡r✈❛çã♦ ❧❡✈❛ ❛♦ s❡❣✉✐♥t❡ ❧❡♠❛✳ ▲❡♠❛ ✸✳✺✳ ❙❡ q é t❛❧ q✉❡ kLk ≤ q < 1, ❡♥tã♦✱ ✭✸✳✸✶✮ é ✈á❧✐❞❛ ❝♦♠ h = q/(1 − q). Pr♦✈❛✿ P❡❧❛ ❖❜s❡r✈❛çã♦ ✸✳✹ é s✉✜❝✐❡♥t❡ ♣r♦✈❛r ✭✸✳✸✷✮✳ ◆♦t❡ q✉❡ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ t❡♠✲s❡

            ∗

            )z, v |h(L − L i| ≤ |hLz, vi| + |hz, Lvi| ≤ kLzkkvk + kzkkLvk hLz, zi + hLv, vi ≥ −kLzkkzk − kLvkkvk

            ♣♦rt❛♥t♦ é s✉✜❝✐❡♥t❡ ♣r♦✈❛r 2 2

          • kLzkkvk + kzkkLvk ≤ h(kzk kvk − kLzkkzk − kLvkkvk),

            ♦✉ s❡❥❛ 2 2 ). kLzkkvk + kzkkLvk + h(kLzkkzk + kLvkkvk) ≤ h(kzk kvk

          • ▼❛s✱ ❝♦♠♦ kLk ≤ q✱ t❡♠✲s❡ kLzkkvk + kzkkLvk + h(kLzkkzk + kLvkkvk)
          • 2 2<
          • h ) ≤ q(2kzkkvk + hkzk kvk
          • 2 2 ),

              ≤ q(1 + h)(kzk kvk

              ✹✹ 2 2 ❥á q✉❡ kzk kvk ≥ 2kzkkvk✳ ▲♦❣♦✱ ❝♦♠♦ h = q/(1 − q)✱ ❡♥tã♦

            • q(1 + h) = h

              ✱ ♦ q✉❡ ❝♦♠♣❧❡t❛ ❛ ♣r♦✈❛✳ P❛r❛ ✜♥❛❧✐③❛r✱ ❜❛st❛ ❛❝❤❛r ❝♦♥❞✐çõ❡s s✉✜❝✐❡♥t❡s ♣❛r❛ ❣❛r❛♥t✐r ✭✸✳✶✼✮

              ❡ ❛ss✐♠ ♦❜t❡r ✭✸✳✶✽✮✳ P❛r❛ ❝♦♥s❡❣✉✐r ✐st♦✱ é ♥❡❝❡ssár✐♦ ♦ s❡❣✉✐♥t❡ t❡✲ ♦r❡♠❛ q✉❡ s❡ ❛♣r❡s❡♥t❛ s❡♠ ❞❡♠♦str❛çã♦✳ P❛r❛ ♦s ❞❡t❛❧❤❡s ❞❛ ♣r♦✈❛ ✈❡❥❛ ❬✷✶❪✳ ❚❡♦r❡♠❛ ✸✳✸✳ ❙❡❥❛ T ✉♠ ♦♣❡r❛❞♦r ❧✐♠✐t❛❞♦ s♦❜r❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛✲ ♥❛❝❤ ❝♦♠♣❧❡①♦✳ ❙❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C t❛❧ q✉❡

              C

              −1

              , k(T − λI) k ≤ ∀|λ| &gt; 1, λ ∈ C, |λ − 1|

              ❡♥tã♦✱ n sup kT k &lt; ∞.

              n∈N ∗

              AA ❯s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✸✳✸ ❝♦♠ T = I − M B ✱ ♣♦❞❡✲s❡ ♦❜t❡r ✭✸✳✶✼✮ s❡ ❝♦♥s❡❣✉❡✲s❡ ♣r♦✈❛r

              C

              ∗ −1 B , AA k(M − λI) k ≤ ∀|λ − 1| &gt; 1, λ ∈ C, ✭✸✳✸✸✮

              |λ| t❡♥❞♦ ❛ss✐♠ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳ ▲❡♠❛ ✸✳✻✳ ❙❡

              1

              ∗

              ✭✸✳✸✹✮ kLk + kAA k &lt; 1,

              2 ❡♥tã♦ ✭✸✳✸✸✮ é ✈á❧✐❞❛✳

              ∗

              Pr♦✈❛✿ ❉❡✜♥❛ ♦ ♦♣❡r❛❞♦r S(λ) := (AA − λI)✱ ♣❛r❛ λ ∈ C✳ P❡❧❛

              ❞❡✜♥✐çã♦ ❞❡ M B ✭✸✳✶✵✮✱ ♣❛r❛ ❝❛❞❛ λ ∈ C t❛❧ q✉❡ |λ − 1| &gt; 1✱ t❡♠✲s❡

              

            ∗ −1 ∗ −1 −1

            B AA = M k(M − λI) k k(AA − λ(I + L)) B k −1 ∗ −1

              ≤ kM B kk(AA − λ(I + L)) k

              −1 −1

              = kM B kk(S(λ) − λL) k

              −1 −1 −1 −1

              = λL) S (λ) kM B kk(I − S(λ) k

              −1 −1 −1 −1

              λL) ≤ kM B kkS(λ) kk(I − S(λ) k (L.B) −1 −1 kM B k |λ|kS(λ) k

              ≤

              −1

              1 |λ| − |λ|kS(λ) kkLk

              −1

              λL s❡♠♣r❡ q✉❡ kS(λ) k &lt; 1✱ ❥á q✉❡ ❛ss✐♠ é ♣♦ssí✈❡❧ ❛♣❧✐❝❛r ♦ ▲❡♠❛ ❞❡

              ∗ −1

              AA ❇❛♥❛❝❤✭L.B✮ ❡ ❣❛r❛♥t✐r ❛ ❡①✐stê♥❝✐❛ ❞❡ (M B ✳ ❆❣♦r❛✱ ♣❛r❛

              −λI)

              ✹✺ ♣r♦✈❛r ❡ss❛ ❝♦♥❞✐çã♦✱ ✉s❛✲s❡ ❛ ❡st✐♠❛t✐✈❛ ♣❛r❛ ♦♣❡r❛❞♦r❡s ❛✉t♦✲❛❞❥✉♥t♦s

              1

              −1 sup .

              kS(λ) k ≤ |λ − t|

              

            t∈[0,kAA k]

              ❆ss✐♠✱

              2 |λ|

              −1

              sup sup sup = , |λ|kS(λ) k ≤

              2

              λ,|λ−1|&gt;1 λ,|λ−1|&gt;1 t∈[0,kAA k] |λ − t| − kAA k 2

            −1

              λL ∗ ❡ ❞❛ ❤✐♣ót❡s❡ ❝♦♥❝❧✉í✲s❡ q✉❡ kS(λ) k ≤ kLk &lt; 1✳ ❋✐♥❛❧✲

              2−kAA k 1 2kM k B

              ♠❡♥t❡✱ r❡✉♥✐♥❞♦ ❛s ❝♦t❛s ❛♥t❡r✐♦r❡s ♥❛ ❝♦♥st❛♥t❡ C =

              2−kAA k−2kLk

              ♦❜té♠✲s❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✸✳✸✸✮✱ ❝♦♠♣❧❡t❛♥❞♦ ❛ss✐♠ ❛ ♣r♦✈❛✳ ❆ s❡❣✉✐r✱ ♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ❞❛ s❡❝çã♦ ✸✳✸✱ ♦♥❞❡ ❛s t❛①❛s ❞❡ ❝♦♥✲

              ✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ sã♦ ❞❡r✐✈❛❞❛s✳ ❚❡♦r❡♠❛ ✸✳✹✳ ❙❡❥❛♠ L, A s❛t✐s❢❛③❡♥❞♦ ✭✸✳✸✹✮ ❡ s✉♣♦♥❤❛ q✉❡ ✭✸✳✻✮ é ✈á❧✐❞❛✳ ❆❧é♠ ❞✐ss♦✱ ❛ss✉♠❛ q✉❡ ❛ ❝♦♥❞✐çã♦ ❞❡ ❢♦♥t❡ ν

              †

              x = G w, ♣❛r❛ ❛❧❣✉♠ 0 &lt; ν &lt; ∞,

              − x

              ∗

              M A é s❛t✐s❢❡✐t❛ ❝♦♠ G = A B ✳ ❊♥tã♦✱ ❛ ✐t❡r❛çã♦ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲ ❑❛❝③♠❛r③ ✭✸✳✼✮ s❛t✐s❢❛③ ❛ ❡st✐♠❛t✐✈❛ ❞♦ ❡rr♦

              1 δ

              † 1 M nδ + C , 2 kx n − x k ≤ C ν n

              , C ♣❛r❛ ❛❧❣✉♠❛s ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s C 2ν+1 1 2 2 ✳ ❊♠ ♣❛rt✐❝✉❧❛r ❛ r❡❣r❛ ❞❡

              ❡s❝♦❧❤❛ ❞❡ ♣❛râ♠❡tr♦ ❛✲♣r✐♦r✐ n ∼ δ ❢♦r♥❡❝❡ ❛ t❛①❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ M ❞❡ ♦r❞❡♠ ót✐♠❛ δ 2ν+1

              †

              kx n − x k ∼ δ M ♦✉ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡ δ 1 2ν

              † 2 2ν+1 .

              kx n − x k ≤ CkM kδ Pr♦✈❛✿ ❆ ♣r♦✈❛ é ❛ ❝♦❧❡çã♦ ❞♦s r❡s✉❧t❛❞♦s ♣ré✈✐♦s✳ ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧✲ ❞❛❞❡ tr✐❛♥❣✉❧❛r t❡♠✲s❡ δ δ

              † † n n kx n − x k ≤ kx n − x k + kx − x k. δ

              P❛r❛ ♦ ❡rr♦ ♣r♦♣❛❣❛❞♦ kx n n −x k✱ s❡ L ❡ A s❛t✐s❢❛③❡♠ ✭✸✳✸✹✮✱ ❡♥tã♦ ❝♦♠♦ ♦❜s❡r✈❛❞♦ ♥♦ ▲❡♠❛ ✸✳✻✱ ❡ss❛ é ✉♠❛ ❝♦♥❞✐çã♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ❣❛r❛♥t✐r ✭✸✳✶✼✮ ❡ ♣❡❧♦ ▲❡♠❛ ✸✳✷ t❡♠✲s❡ δn 1 M nδ . kx n − x k ≤ C

              ✹✻

              †

              ❆❣♦r❛✱ ♣❛r❛ ♦ ❡rr♦ ❞❡ ❛♣r♦①✐♠❛çã♦ kx n − x k✱ ✉s❛♥❞♦ ❛ ❝♦♥❞✐çã♦ ❞❡

              † †

              x ❢♦♥t❡ ❡ ❞❛❞♦s ❡①❛t♦s y = Ax ✱ ❝♦♠ x ✲s♦❧✉çã♦ ❞❡ ♥♦r♠❛ ♠í♥✐♠❛ ❞❡ ✭✸✳✶✮✱ t❡♠✲s❡

              ∗ †

              M x = x (Ax ) n+1 n B n − A − Ax

              †

              = x ), n n − G(x − x

              ❞❡ ♦♥❞❡ ♦❜té♠✲s❡ ν

              † x = (I w. 1

              − x − G)G ■♥❞✉t✐✈❛♠❡♥t❡ s❡ ❝❤❡❣❛ à ❡①♣r❡ssã♦ n ν

              †

              x = (I G w n − x − G)

              ♦ q✉❡ ✐♠♣❧✐❝❛ n ν

              † n G kx − x k ≤ k(I − G) kkwk.

              ❆❣♦r❛✱ ♣♦r ✭✸✳✸✹✮✱ L s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞♦ ▲❡♠❛ ✸✳✺ ❡ ❛ss✐♠ ♠❡❞✐✲ ❛♥t❡ ❛ ❖❜s❡r✈❛çã♦ ✸✳✹ ♣♦❞❡✲s❡ ✉s❛r ♦ ▲❡♠❛ ✸✳✹✱ ♦❜t❡♥❞♦ n ν C C 2 2 G &lt; k(I − G) k ≤ ν ν

              (n + 1) n ♦✉ s❡❥❛

              1 δ

              † 1 M nδ + C . 2 kx n − x k ≤ C ν n

              ❋✐♥❛❧♠❡♥t❡✱ é ❝❧❛r♦ q✉❡ ❝♦♠ ❛ r❡❣r❛ ❞❡ ❡s❝♦❧❤❛ ❞❡ ♣❛râ♠❡tr♦ ❛✲♣r✐♦r✐ 2ν+1 2 n t❡♠✲s❡

              ∼ δ M √ 2ν+1

              −ν

              nδ M , ∼ δ ∼ n M

              ❡ ❛ss✐♠ δ 2ν+1

              †

              , kx n − x k ∼ δ M ♦✉ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡ δ 1 2ν

              † 2 2ν+1

              , kx n − x k ≤ CkM kδ 2 1 − 2ν+1 1 ♦♥❞❡ C ❞❡♣❡♥❞❡ ❞❡ kM ✳ k

              ◆♦t❡ q✉❡✱ s❛❧✈♦ ❛ ❝♦♥❞✐çã♦ ❞❡ ❢♦♥t❡✱ s❡♠♣r❡ é ♣♦ssí✈❡❧ ❛t✐♥❣✐r ❛s ❤✐♣ót❡s❡s ❞❡st❡ t❡♦r❡♠❛ ♠❡❞✐❛♥t❡ ❛ ✐♥tr♦❞✉çã♦ ❞❡ ✉♠ ♣❛râ♠❡tr♦ τ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♦s ♦♣❡r❛❞♦r❡s M i ✳

              ✹✼ ❖❜s❡r✈❛çã♦ ✸✳✺✳ ◆♦ ❚❡♦r❡♠❛ ✸✳✹✱ ❛s t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ sã♦ ❡s✲ ν

              † ∗

              M A ) ) t❛❜❡❧❡❝✐❞❛s s♦❜ ❛s ❝♦♥❞✐çã♦ ❞❡ ❢♦♥t❡ x B ✳ ❙❡r✐❛ − x ∈ R((A

              ✐♥t❡r❡ss❛♥t❡ s✉❜st✐t✉✐r ❡ss❛ ❝♦♥❞✐çã♦ ♣❡❧❛ ❝♦♥❞✐çã♦ ❞❡ ❢♦♥t❡ ✉s✉❛❧ ❝♦♠ ν

              ∗

              A ) ) ✐♠❛❣❡♠ R((A ✳ ▼❛s✱ ♥ã♦ é ❝❧❛r♦ s❡ ✐st♦ ♣♦❞❡ s❡r ❢❡✐t♦ ❞❛❞❛s ❛s ♠❡s♠❛s s✉♣♦s✐çõ❡s ❞♦ ❚❡♦r❡♠❛ ✸✳✹✳ ❯♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❛s ❝♦♥❞✐✲ çõ❡s ❞❡ ❢♦♥t❡ ♣♦❞❡ ♣r♦✈❛✈❡❧♠❡♥t❡ s❡r ♠♦str❛❞❛ s❡

              

            ∗ ∗ ∗

              A M A A d 1 x B x 2 x kA k ≤ kA k ≤ d kA k, é ✈á❧✐❞❛ ✉♥✐❢♦r♠❡♠❡♥t❡ ♣❛r❛ ❛❧❣✉♠❛s ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s d 1 ❡ d 2 ✳ P❛r❛ ❡ss❡ ♣r♦♣ós✐t♦✱ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❑❛t♦✲❍❡✐♥③ ♣❛r❛ ♦♣❡r❛❞♦r❡s ❛❝r❡t✐✈♦s ✭✈❡❥❛ ❬✶✼❪✮ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞❛✳ P❛r❛ ♠❛✐♦r❡s ❞❡t❛✲ ❧❤❡s ✈❡❥❛ ❬✶✾❪✳ ❖❜s❡r✈❛çã♦ ✸✳✻✳ ❖ r❡s✉❧t❛❞♦ ❞♦ ❚❡♦r❡♠❛ ✸✳✹✱ ✉s❛ ❛ ❝❧áss✐❝❛ ❡s❝♦❧❤❛ 2ν+1 2

              ❞❡ ♣❛râ♠❡tr♦ ❛✲♣r✐♦r✐ α ∼ δ ✳ ◆♦ ❡♥t❛♥t♦✱ ✉s❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ❞❡ M P❧❛t♦ ❡ ❍ä♠❛r✐❦ ✭✈❡❥❛ ❬✷✷❪✮✱ é ♣♦ssí✈❡❧ ❡st❛❜❡❧❡❝❡r t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ r❡❣r❛s ❞❡ ❡s❝♦❧❤❛ ❞❡ ♣❛râ♠❡tr♦s ❛✲♣♦st❡r✐♦r✐✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ✉s❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ❞✐s❝r❡♣â♥❝✐❛✳

              ❆ ❝♦♥❞✐çã♦ ♣r✐♥❝✐♣❛❧ ♣❛r❛ ♦❜t❡r ❛s t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ é ✭✸✳✸✹✮✳ ❆ s❡❣✉✐r✱ ✉♠❛ ❝♦♥❞✐çã♦ s✉✜❝✐❡♥t❡ ♠❛✐s s✐♠♣❧❡s ♣❛r❛ ❡st✐♠❛r ❛ ♥♦r♠❛ ❞❡ L é ❡st❛❜❡❧❡❝✐❞❛✳

              p×p

              ❉❡✜♥❛ ❛ ♠❛tr✐③ |L| ∈ R ✱ ❝♦♠♦ ❛ ♠❛tr✐③ tr✐❛♥❣✉❧❛r ✐♥❢❡r✐♦r ❝♦♠ ❞✐❛❣♦♥❛❧ ♥✉❧❛ s❡ j ≥ i i,j = , i, j = 0, . . . , p

              ✭✸✳✸✺✮ |L| i,j ♦✉tr♦ ❝❛s♦ − 1. kL k

              ◆♦t❡ q✉❡ ♦s í♥❞✐❝❡s ❞❛s ❡♥tr❛❞❛s ♥❛ ♠❛tr✐③ |L| ❝♦♠❡ç❛♠ ❞❡ ✵✳ ◆❛ s❡q✉ê♥❝✐❛✱ |L| ♣♦❞❡ s❡r s✉❜st✐t✉í❞❛ ♣♦r q✉❛❧q✉❡r ♦✉tr❛ ♠❛tr✐③ tr✐❛♥❣✉❧❛r ✐♥❢❡r✐♦r ❝♦♠ ❞✐❛❣♦♥❛❧ ♥✉❧❛ s❛t✐s❢❛③❡♥❞♦ i,j i,j ✭✸✳✸✻✮ , i &lt; j, i, j = 0, . . . , p kL k ≤ |L| − 1.

              ❊♥tã♦✱ t❡♠✲s❡ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳ ▲❡♠❛ ✸✳✼✳ ❈♦♠ |L| ❝♦♠♦ ❡♠ ✭✸✳✸✺✮ ♦✉ ✭✸✳✸✻✮✱ t❡♠✲s❡ max ( kLk ≤ σ |L|),

              (N ) ♦♥❞❡ σ max ❞❡♥♦t❛ ♦ ♠❛✐♦r ✈❛❧♦r s✐♥❣✉❧❛r ❞❛ ♠❛tr✐③ N✳

              ✹✽ Pr♦✈❛✿ P❛r❛ z ∈ Y✱ ❛ i✲és✐♠❛ ❝♦♠♣♦♥❡♥t❡ ❞❡ Lz ♣♦❞❡ s❡r ❧✐♠✐t❛❞❛ ♣♦r

              

            p−1

            i Y i,j j

              X k(Lz) k i ≤ |L| kz k. j=0 ❆ss✐♠✱ t♦♠❛♥❞♦ ❛ s♦♠❛ ❞❡ q✉❛❞r❛❞♦s ❡ ♥♦t❛♥❞♦ q✉❡ σ max é ♦ ♦♣❡r❛❞♦r ♥♦r♠❛✱ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳

              ❆❣♦r❛✱ ❧❡♠❜r❛♥❞♦ q✉❡

              p−1 p−1 1 2 X 2 2 X ∗ ∗

              = A i = M i A i kAA k = kAk kM k kA i k, i=0 i=0 i t❡♠✲s❡ ♦ s❡❣✉✐♥t❡ ❝♦r♦❧ár✐♦ ❞♦ ❧❡♠❛ ❛♥t❡r✐♦r✳ ❈♦r♦❧ár✐♦ ✸✳✶✳ ❙❡❥❛ |L| ❝♦♠♦ ❡♠ ✭✸✳✸✺✮ ♦✉ ✭✸✳✸✻✮✳ ❆s t❛①❛s ❞❡ ❝♦♥✲ ✈❡r❣ê♥❝✐❛ ❞♦ ❚❡♦r❡♠❛ ✸✳✹ sã♦ ✈á❧✐❞❛s s❡

              

            p−1

              X

              1

              ∗

              σ ( M A max i i ✭✸✳✸✼✮ |L|) + kA i k &lt; 1.

              2 i=0 ❖❜s❡r✈❛çã♦ ✸✳✼✳ ❈♦♥s✐❞❡r❛♥❞♦ ♦ s✐♠♣❧❡s ♣r❡❝♦♥❞✐❝✐♦♥❛❞♦r ❞❛ ❢♦r♠❛ M = τ I i ✱ ❝♦♠ τ &gt; 0 ♦ ♣❛râ♠❡tr♦ ❞❡ t❛♠❛♥❤♦ ❞❡ ♣❛ss♦✱ é ❝❧❛r♦ q✉❡ ❛ ✐t❡r❛çã♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ t❡♠ t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ t♦❞♦ τ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✳ ❈♦♠ ❡❢❡✐t♦✱ ♣❛r❛ ❡ss❛ ❡s❝♦❧❤❛ ❞❡ M i t❡♠✲s❡

              1 τ &lt; ,

              ✭✸✳✸✽✮ 1 p−1 P

              ∗

              σ ( A max i |L|) + kA i k 2 i=0

              ♦♥❞❡ ❛q✉✐ |L| é ❛ ♠❛tr✐③ tr✐❛♥❣✉❧❛r ✐♥❢❡r✐♦r ❝♦♠ ❞✐❛❣♦♥❛❧ ♥✉❧❛ ❡ ❝♦♠

              ∗ i,j i = A

              |L| kA j k✱ ❛ss✐♠ ❝♦♠ τ s❛t✐s❢❛③❡♥❞♦ ✭✸✳✸✽✮ ♣♦❞❡✲s❡ ❛♣❧✐❝❛r ♦ ❝♦r♦❧ár✐♦ ❛♥t❡r✐♦r✳ ❖ ♠❡s♠♦ ❛❝♦♥t❡❝❡ ✉s❛♥❞♦ ♣r❡❝♦♥❞✐❝✐♦♥❛❞♦r❡s M i ✳ ❙✉❜st✐t✉✐♥❞♦ ❡❧❡s ♣♦r τM i ✱ s❡♠♣r❡ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ ♣♦ssí✈❡✐s ✈❛❧♦r❡s ❞❡ τ t❛❧ q✉❡ ❛s ❤✐♣ót❡s❡s ❞♦ ❈♦r♦❧ár✐♦ ✸✳✶ sã♦ s❛t✐s❢❡✐✲ t❛s✳

              ✹✾

              

            ✸✳✹ ❚❛①❛s ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ▼ét♦❞♦

            ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❙✐♠étr✐❝♦

              P❛r❛ ❝♦♠♣❧❡t❛r✱ ❡ ♣♦r r❛③õ❡s ❞❡ ❝♦♠♣❛r❛çã♦✱ ❛❣♦r❛ s❡ ❡st❛❜❡❧❡✲ ❝❡♠ ❛s t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ❛ ✈❡rsã♦ s✐♠étr✐❝❛ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ✈✐st❛ ♥❛ s❡❝çã♦ ✭✸✳✷✳✷✮✳ ❆ss✉♠❛ q✉❡ ✭✸✳✻✮ é ✈á✲ ❧✐❞❛✱ ❡♥tã♦ ❞❡✜♥✐♥❞♦✱ 1 2 1 2 1

              ∗ 2 B M

              := (2I A i A M )) B , − diag(M i i i t❡♠✲s❡

              ∗

              M B SB = B .

              ✭✸✳✸✾✮ ❈♦♠ ✐st♦✱ ❛ ✐t❡r❛çã♦ ❞♦ ♠ét♦❞♦ sLK ✭✸✳✶✹✮ é ❡q✉✐✈❛❧❡♥t❡ à ✐t❡r❛çã♦ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ❛♣❧✐❝❛❞❛ ❛♦ s✐st❡♠❛ δ

              BA x = By , δ ✭✸✳✹✵✮ =

              ♦♥❞❡ s❡ ✉s❛ ♦ ♥í✈❡❧ ❞❡ r✉í❞♦ δ S kB(y − y)k✳ ❈♦♠ ❡❢❡✐t♦✱ ❢❛③❡♥❞♦ ✐st♦✱ t❡♠✲s❡ δ

              ∗

              x = x (BAx ) n+1 n − (BA) − By δ

              

            ∗ ∗

              = x B B (Ax ) n − A − y δ

              ∗

              M = x n SB (Ax ). − A − y

              ❆ss✐♠✱ ❛ t❡♦r✐❛ ❝❧áss✐❝❛ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ✐♠❡❞✐❛t❛♠❡♥t❡ ❢♦r♥❡❝❡ ✭✈❡❥❛ ❬✽❪✮✿ ν

              † ∗

              M A = (A ) w

              ❚❡♦r❡♠❛ ✸✳✺✳ ❙✉♣♦♥❤❛ q✉❡ ❛s ❝♦♥❞✐çõ❡s ✭✸✳✻✮ ❡ x SB −x sã♦ ✈á❧✐❞❛s✳ ❊♥tã♦✱ ♣❛r❛ ♦ ♠ét♦❞♦ sLK ❡ ❛ ❡s❝♦❧❤❛ ❞❡ ♣❛râ♠❡tr♦ ❛✲ 2

              − 2ν+1

              ♣r✐♦r✐ n ∼ δ ✱ t❡♠✲s❡ ❛ t❛①❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ ♦r❞❡♠ ♦t✐♠❛❧ S 2ν+1

              † n .

              kx − x k ≤ Cδ S ❖❜s❡r✈❛çã♦ ✸✳✽✳ ❙❡❣✉♥❞♦ ❬✶✾❪✱ ✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍❡✐♥③ ✭✈❡❥❛ ❬✽❪✮✱ ♥♦ ❚❡♦r❡♠❛ ✸✳✺ ♣♦❞❡ s❡r ✉s❛❞❛ ✉♠❛ ❝♦♥❞✐çã♦ ❞❡ ❢♦♥t❡ ✉s✉❛❧✱ ❥á q✉❡ s❡ ❛ ❝♦♥❞✐çã♦ ✭✸✳✻✮ é ✈á❧✐❞❛✱ ❡♥tã♦ ❡①✐st❡♠ ❝♦♥st❛♥t❡s m 1 ❡ m 2 t❛✐s q✉❡✱ Y Y m , 1 2 kyk ≤ kByk ≤ m kyk ∀y ∈ Y. 1

              ❊♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ 0 ≤ ν ≤ ✱ ❡♠ ❝❛❞❛ ✉♠ ❞❡ss❡s ❝❛s♦s ❛s ❝♦♥❞✐✲ 2 ν ν

              † ∗ † ∗

              = (A M A ) w = (A A ) w çõ❡s ❞❡ ❢♦♥t❡ x SB ❡ x sã♦

              − x − x ❡q✉✐✈❛❧❡♥t❡s✳

              ✺✵ ❆ss✐♠✱ ♦ r❡s✉❧t❛❞♦ ❞♦ ❚❡♦r❡♠❛ ✸✳✺ é ✈á❧✐❞♦ ❝♦♠ ❛ ❝♦♥❞✐çã♦ ❞❡ ❢♦♥t❡ ν 1

              † ∗

              = (A A ) w ✉s✉❛❧ x ♣❛r❛ 0 ≤ ν ≤ ❡✱ ❝❧❛r❛♠❡♥t❡✱ ❝♦♠ δ S ❡♠

              − x 2 ✈❡③ ❞❡ δ✳ ❆ss✐♠✱ ❢♦r❛♠ ❡st❛❜❡❧❡❝✐❞❛s ❡①❛t❛♠❡♥t❡ ❛s ♠❡s♠❛s t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ s♦❜ ❛ ♠❡s♠❛ ❝♦♥❞✐çã♦ ❞❡ ❢♦♥t❡ q✉❡ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ♣♦r ❜❧♦❝♦s ♦r❞✐♥ár✐♦✳

              ❈♦♠♣❛r❛♥❞♦ ♦ ♠ét♦❞♦ sLK ❝♦♠ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ♣♦r ❜❧♦❝♦s =

              ✉s✉❛❧✱ ❞á ♣❛r❛ ♥♦t❛r ✉♠❛ ❞✐❢❡r❡♥ç❛✿ ❆♦ ✉s❛r ♣r❡❝♦♥❞✐❝✐♦♥❛❞♦r❡s M i sLK τ I

              ✱ ❡♥tã♦✱ ♣♦r ✭✸✳✻✮ τ = τ ♣♦❞❡ s❡r ❡s❝♦❧❤✐❞♦ ❝♦♠♦

              2 τ sLK &lt; .

              ∗

              max A i

              i=0,...,p−1 kA i k

              P♦r ♦✉t♦ ❧❛❞♦✱ ❝♦♠ ❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❡s❝♦❧❤❛ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡✲ 2 &lt; 2

              ❜❡r ♣♦r ❜❧♦❝♦s✱ τ ❞❡✈❡ s❡r ❡s❝♦❧❤✐❞♦ ❞❡ t❛❧ ♠❛♥❡✐r❛ q✉❡ τkAk ✱ ♦✉ s❡❥❛✱

              2

              2 τ &lt; &lt; . L P

              p−1 ∗ ∗

              max A A i i i=0,...,p−1 kA i k i=0 kA i k

              ❯♠❛ s✐t✉❛çã♦ s✐♠✐❧❛r ❛❝♦♥t❡❝❡ ❝♦♠ ♦ ♠ét♦❞♦ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ♥ã♦ s✐♠étr✐❝♦✳ ❊♠❜♦r❛ ♣❛r❛ ♦❜t❡r t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ s❡❥❛ ♥❡❝❡ssár✐♦ ✭✸✳✸✽✮✱ ♣❛r❛ ♦❜t❡r só ❝♦♥✈❡r❣ê♥❝✐❛✱ ❛ ❝♦♥❞✐çã♦

              2 τ LK &lt;

              ∗

              max A i

              i=0,...,p−1 kA k i

              é s✉✜❝✐❡♥t❡ ✭❞❡t❛❧❤❡s ❡♠ ❬✼❪✮✱ ♣❡❧♦ ♠❡♥♦s ❡♠ ❞✐♠❡♥sã♦ ✜♥✐t❛✳ ❆ss✐♠✱ ❛s ✐t❡r❛çõ❡s t✐♣♦ ❑❛❝③♠❛r③ ❛❧é♠ ❞❡ s❡r ♠❛✐s ❢á❝❡✐s ❞❡ ✐♠♣❧❡♠❡♥t❛r ❞♦ q✉❡ ❛s ✐t❡r❛çõ❡s t✐♣♦ ▲❛♥❞✇❡❜❡r ♣♦r ❜❧♦❝♦s✱ t❛♠❜é♠ ♣❡r♠✐t❡♠ ❡s❝♦❧❤❡r ✉♠ t❛♠❛♥❤♦ ❞❡ ♣❛ss♦ ♠❛✐♦r✳

              ❈❛♣ít✉❧♦ ✹ ❊①♣❡r✐ê♥❝✐❛s ◆✉♠ér✐❝❛s

              ◆❡st❡ ú❧t✐♠♦ ❝❛♣ít✉❧♦✱ s❡rá ❝♦♠♣❛r❛❞♦ ♦ ❞❡s❡♠♣❡♥❤♦ ❞❡ ❛❧❣✉♥s ♠é✲ t♦❞♦s ✐t❡r❛t✐✈♦s ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❞❛ ❢♦r♠❛ ❞♦ s✐st❡♠❛ ✭✸✳✶✮ ❞♦ ❈❛♣ít✉❧♦ ✸✳ ❖s ❝r✐tér✐♦s ❞❡ ❝♦♠♣❛r❛çã♦ s❡rã♦ ♦ ❡rr♦ ♥❛ ✐t❡r❛çã♦✱ ♦ ❡rr♦ ♥❛ s♦❧✉çã♦ ❡♥❝♦♥tr❛❞❛ ❡ ♦ t❡♠♣♦ ♠é❞✐♦ ❞❡ ❡①❡❝✉çã♦✳

              

            ✹✳✶ ▼ét♦❞♦s ■t❡r❛t✐✈♦s ❛ ❙❡r❡♠ ❈♦♠♣❛r❛✲

            ❞♦s

              ❖s ♠ét♦❞♦s ❡st✉❞❛❞♦s ♥♦s ❝❛♣ít✉❧♦s ✷ ❡ ✸ ❢♦r❛♠ ✈❛r✐❛♥t❡s ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③✱ ❛ s❛❜❡r✱ ♦s ♠ét♦❞♦s LK ❡ sLK ♣❛r❛ ♦♣❡r❛✲ n ❞♦r❡s ❝♦♠ ❡str✉t✉r❛ ❞❡ ❜❧♦❝♦ ✐♥❝♦r♣♦r❛♥❞♦ ♦ ♣❛râ♠❡tr♦ ω ✭✶✳✶✵✮ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ❞❛❞♦s ❝♦♠ r✉í❞♦✳ ❆ ✐❞❡✐❛ é ❝♦♠♣❛r❛r ❡ss❡s ♠ét♦❞♦s ♥♦s ❝❛s♦s ❞❡ ❞❛❞♦s ❡①❛t♦s ❡ ❞❛❞♦s ❝♦♠ r✉í❞♦✱ ❝♦♠ ♠ét♦❞♦s ❜❡♠ ❡st❛❜❡❧❡❝✐❞♦s✳ ❖s t❡st❡s ❢❡✐t♦s ♥❛ ♣ró①✐♠❛ s❡❝çã♦ s❡rã♦ ♣❛r❛ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❡♠ ❞✐♠❡♥sã♦ ✜♥✐t❛✳

              ❖ ♣r✐♠❡✐r♦ ❞♦s ♠ét♦❞♦s ✉t✐❧✐③❛❞♦ ♣❛r❛ ❛ ❝♦♠♣❛r❛çã♦ s❡rá ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ❞❡s❝r✐t♦ ♥♦ ❈❛♣ít✉❧♦ ✶ ♣❛r❛ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s✱

              ∗

              x = x (Bx n+1 n n − B − y). ❖✉tr♦ ❞♦s ♠ét♦❞♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛ ❝♦♠♣❛r❛çã♦ é ♦ ♠ét♦❞♦ ❞❡

              N ×N

              ▲❛♥❞✇❡❜❡r ❝♦♠ ❜✉s❝❛ ❧✐♥❡❛r✳ ❙❡❥❛ B ∈ R ❝♦♠ B &gt; 0✱ ❛ ♠❛tr✐③ ❞♦ s✐st❡♠❛✱ Bx = y. 2

              ❖ ❛❧❣♦r✐t♠♦ ❞❡ ♠á①✐♠❛ ❞❡s❝✐❞❛ ♣❛r❛ ♠✐♥✐♠✐③❛r kBx − yk ✱ ❢♦r♥❡❝❡ ❛

              ✺✷ ✐t❡r❛çã♦ ❞❡ ▲❛♥❞✇❡❜❡r ❝♦♠ ❜✉s❝❛ ❧✐♥❡❛r x = x d , n+1 n n n

              − α ♦♥❞❡ T d d n n

              α n = n = B(Bx n T 2 ❡ d − y). d B d n n

              ❋✐♥❛❧♠❡♥t❡✱ ♦ ú❧t✐♠♦ ♠ét♦❞♦ ✉s❛❞♦ ♣❛r❛ ❛ ❝♦♠♣❛r❛çã♦ é ♦ ♠ét♦❞♦

              N ×N

              ❙t❡❡♣❡st ❉❡s❝❡♥t✱ ♣❛r❛ ♠❛tr✐③❡s B ∈ R ❝♦♠ B &gt; 0✱ q✉❡ é ❞❛❞♦ ♣❡❧❛ ✐t❡r❛çã♦ x n+1 = x n n d n ,

              − α ♦♥❞❡ T d d n n

              α = = (Bx n ❡ d n n T − y). d Bd n n

              ❉❡t❛❧❤❡s s♦❜r❡ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ss❡s ♠ét♦❞♦s ♣♦❞❡♠ s❡r ❡♥✲ ❝♦♥tr❛❞♦s ♥❛ ❧✐t❡r❛t✉r❛ ❝❧áss✐❝❛ ❡ ❡♠ ❬✽✱ ✶✻❪ ♥♦ ❝❛s♦ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✳

              ❈♦♠♦ ❝r✐tér✐♦ ❞❡ ♣❛r❛❞❛ ♣❛r❛ ♦s ♠ét♦❞♦s ❞❡ ▲❛♥❞✇❡❜❡r ❡ ▲❛♥❞✇❡✲ ❜❡r ❝♦♠ ❜✉s❝❛ ❧✐♥❡❛r ♥♦ ❝❛s♦ ❞❡ ❞❛❞♦s ❝♦♠ r✉í❞♦✱ s❡rá ✉t✐❧✐③❛❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ❞✐s❝r❡♣â♥❝✐❛✱ n ✭✹✳✶✮ δ kBx − y k ≤ τδ, ✐st♦ é✱ ♦ ♠ét♦❞♦ ♣❛r❛ q✉❛♥❞♦ ♣♦r ♣r✐♠❡✐r❛ ✈❡③ é s❛t✐s❢❡✐t❛ ❛ ❝♦♥❞✐✲ n = x δ çã♦ ✭✹✳✶✮✱ ❡ ♥❡ss❡ ❝❛s♦ x n ✳ ❖ ♠ét♦❞♦ ❙t❡❡♣❡st ❉❡s❝❡♥t ♥ã♦ é ❝♦♥s✐❞❡r❛❞♦ ♥♦ ❝❛s♦ ❞❡ ❞❛❞♦s ❝♦♠ r✉í❞♦✳

              ✹✳✷ ❉❡s❝r✐çã♦ ❞♦ Pr♦❜❧❡♠❛ ❚❡st❡

              P❛r❛ ❝♦♠♣❛r❛r ♦s ♠ét♦❞♦s ❝✐t❛❞♦s ❛♥t❡r✐♦r♠❡♥t❡✱ s❡rá r❡s♦❧✈✐❞♦ ♦ s❡❣✉✐♥t❡ s✐st❡♠❛ ❧✐♥❡❛r✱ Bx = y,

              ✭✹✳✷✮ ♦♥❞❡

              1 N ×N B := H , N N ∈ R N kH k N

              ❝♦♠ H ❛ ♠❛tr✐③ ❞❡ ❍✐❧❜❡rt ❞❡ ♦r❞❡♠ N ❡ x, y ∈ R ✳ ❆ ♠❛tr✐③ B é s✐♠étr✐❝❛ ❞❡✜♥✐❞❛ ♣♦s✐t✐✈❛✱ ❝♦♠ ♥♦r♠❛ ✐❣✉❛❧ ❛ ✶✱ ♠❛s é ♠❛❧ ❝♦♥❞✐❝✐♦✲ ♥❛❞❛✱ ❞❡ ❢❛t♦✱ ♣❛r❛ N &gt; 15 é q✉❛s❡ s✐♥❣✉❧❛r✳ ❈♦♠ ❛ ♠❛tr✐③ B ❞❡✜♥✐❞❛ ❞❡ss❛ ❢♦r♠❛✱ sã♦ ❣❛r❛♥t✐❞❛s ❛s ❝♦♥❞✐çõ❡s ✭✷✳✶✮ ❡ ✭✷✳✷✮ ✉t✐❧✐③❛❞❛s ♥❛ ❛♥á✲ ❧✐s❡ ❞♦s ♠ét♦❞♦s ❞❡ ▲❛♥❞✇❡❜❡r ❡ ▲❛♥❞✇❡❜❡r✲ ❑❛❝③♠❛r③ ♥♦s ❝❛♣ít✉❧♦s ❛♥t❡r✐♦r❡s✳

              ✺✸ P❛r❛ r❡s♦❧✈❡r ♦ s✐st❡♠❛ ✭✹✳✷✮ ♠❡❞✐❛♥t❡ ♦s ♠ét♦❞♦s LK ❡ sLK ❞❡s✲

              ❝r✐t♦s ♥♦ ❈❛♣ít✉❧♦ ✸✱ ❛ ♠❛tr✐③ B s❡rá ❞✐✈✐❞✐❞❛ ❡♠ ❜❧♦❝♦s ❞❡ ♠❛✐s ❞❡ ✉♠❛ ❧✐♥❤❛✱ ❥á q✉❡ s❡ ❝❛❞❛ ❧✐♥❤❛ é ❝♦♥s✐❞❡r❛❞❛ ❝♦♠♦ ✉♠ ❜❧♦❝♦✱ ❞❛❞❛ ❛ ❡str✉t✉r❛ ❝í❝❧✐❝❛ ❞♦s ♠ét♦❞♦s✱ t❡r✐❛♠✲s❡ N ❜❧♦❝♦s✱ ❡ ♣❛r❛ N ❣r❛♥❞❡ ✐ss♦ ❝♦♠♣❧✐❝❛ ❛ ✈✐s✉❛❧✐③❛çã♦ ❞♦s r❡s✉❧t❛❞♦s✱ ❛❧é♠ ❞❡ ❞❡s❛♣r♦✈❡✐t❛r ❛s ❜♦♥❞❛❞❡s ❞❛ ❡str✉t✉r❛✳

              ❆❣♦r❛✱ ♣❛r❛ r❡s♦❧✈❡r ♦ s✐st❡♠❛ ✭✹✳✷✮ ♠❡❞✐❛♥t❡ ♦s ♠ét♦❞♦s ❞❡s❝r✐t♦s ♥❛ ❙❡❝çã♦ ✹✳✶✱ s❡rá ❝♦♥s✐❞❡r❛❞♦ ♦ s✐st❡♠❛ ❞❡ ❢♦r♠❛ ✐♥t❡❣r❛✳

              ✹✳✷✳✶ ●❡r❛♥❞♦ ❉❛❞♦s ❊①❛t♦s ❡ ❝♦♠ ❘✉í❞♦

              P❛r❛ ❣❡r❛r ♦s ✈❡t♦r ❞❡ ❞❛❞♦s ❡①❛t♦s y✱ ♦ ♣r♦❝❡❞✐♠❡♥t♦ é ♦ s❡❣✉✐♥t❡✳ N

              † †

              = 1 = (1, 1, . . . , 1) ❋✐①❡ x ✱ ❡ ❝❛❧❝✉❧❡ y = Bx ✳

              ∈ R δ ❆❣♦r❛✱ ♣❛r❛ ❝❛❧❝✉❧❛r ♦ ✈❡t♦r ❞❡ ❞❛❞♦s ♣❡rt✉r❜❛❞♦s y ✱ ❝♦♠ δ ❛

              †

              ♣♦r❝❡♥t❛❣❡♠ ❞❡ r✉í❞♦✱ ♦ ♣r♦❝❡❞✐♠❡♥t♦ é ♦ s❡❣✉✐♥t❡✳ ❉❛❞♦ y = Bx ✱ 1 := 2δ(θ ), i = 1, . . . , N

              ❢❛ç❛ eδ i i ✱ ♦♥❞❡ θ i é ✉♠ ♥ú♠❡r♦ ❛❧❡❛tór✐♦ ❝♦♠ − 2

              ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ✉♥✐❢♦r♠❡ ♥♦ ✐♥t❡r✈❛❧♦ ✭✵✱✶✮✳ ❆❣♦r❛✱ ❞❡✜♥❛ δ y := e δ y , i = 1, . . . , N := y y , i = e i i i ✱ ♣❛r❛ ✜♥❛❧♠❡♥t❡ ❝❛❧❝✉❧❛r y i + e i i 1, . . . , N

              ✳ ❆ss✐♠✱ ♦ ♥í✈❡❧ ❞❡ r✉í❞♦ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ t❛♠❛♥❤♦ ❞❡ y✳ P❛r❛ ♦s ♠ét♦❞♦s ♥ã♦ ❝í❝❧✐❝♦s✱ ✐st♦ é✱ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✱ ▲❛♥❞✇❡✲

              ❜❡r ❝♦♠ ❜✉s❝❛ ❧✐♥❡❛r✱ ♦ ✈❛❧♦r ❞♦ ♥í✈❡❧ ❞❡ r✉í❞♦ ✉t✐❧✐③❛❞♦ ♣❛r❛ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ❞✐s❝r❡♣â♥❝✐❛ s❡rá keyk✱ ❡ ♣❛r❛ ♦s ♠ét♦❞♦s t✐♣♦ ❑❛❝③♠❛r③ ♦ k✲és✐♠♦

              := y] ♥í✈❡❧ ❞❡ r✉í❞♦ s❡rá δ k k k é ♦ k✲és✐♠♦ ❜❧♦❝♦ ❞♦ ✈❡t♦r b✳ k[e k✱ ♦♥❞❡ [b]

              ✹✳✸ ❘❡s✉❧t❛❞♦s

              ❋♦r❛♠ r❡❛❧✐③❛❞♦s t❡st❡s ❝♦♠♣❛r❛t✐✈♦s ♣❛r❛ ❞❛❞♦s ❡①❛t♦s ❡ ❝♦♠ r✉í❞♦✱ ♥♦ ú❧t✐♠♦ ❝❛s♦ ♣❛r❛ ❞♦✐s ✈❛❧♦r❡s ❞❡ δ✱ ❛ s❛❜❡r✱ δ = 0.05 ❡ δ = 0.1✳

              ❖ ✈❡t♦r ✐♥✐❝✐❛❧ ❡♠ ❝❛❞❛ ❡①❡❝✉çã♦ ❞❛ ❜❛t❡r✐❛ ❞❡ ♠ét♦❞♦s é ❛❧❡❛tór✐♦ = 0.1θ

              ❣❡r❛❞♦ ♣❡❧❛ ❡①♣r❡ssã♦ x ✱ ♦♥❞❡ θ é ✉♠ ✈❡t♦r ❛❧❡❛tór✐♦ ❝♦♠ ❞✐s✲ tr✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ✉♥✐❢♦r♠❡ ♥♦ ✐♥t❡r✈❛❧♦ ✭✵✱✶✮✳ P♦rt❛♥t♦ ♣❛r❛ ❝❛❞❛ ✈❡t♦r ✐♥✐❝✐❛❧ ♦s ♠ét♦❞♦s t❡♠ t❡♠♣♦s ❞❡ ❡①❡❝✉çã♦ ❞✐❢❡r❡♥t❡s✳ ❆s✲ s✐♠✱ ♣❛r❛ ❛❝❤❛r ♦ t❡♠♣♦ ❞❡ ❡①❡❝✉çã♦ ❞❡ ❝❛❞❛ ♠ét♦❞♦✭t❡♠♣♦ ♥❡❝❡ssár✐♦ ♣❛r❛ ❝♦♥✈❡r❣✐r✮✱ s❡ ❝❛❧❝✉❧♦✉ ♦ t❡♠♣♦ ♠é❞✐♦ ❡♠ ✶✵✵✵ r❡♣❡t✐çõ❡s✱ ♦♥❞❡ ♥♦ ❝❛s♦ ❞❡ ❞❛❞♦s ❝♦♠ r✉í❞♦✱ ♠❛♥t❡✈❡✲s❡ ♦ ✈❡t♦r ❞❡ ❞❛❞♦s ♣❡rt✉r❜❛❞♦s✱ δ y

              ✱ ✜①♦✳ ❊♠ t♦❞♦s ♦s t❡st❡s ♦s ♣❛râ♠❡tr♦s N = 120 ❡ τ = 5 sã♦ ✜✲ ①♦s✳ P❛r❛ ❡ss❡ ✈❛❧♦r ❞❡ N✱ ♦ ♥ú♠❡r♦ ❞❡ ❝♦♥❞✐❝✐♦♥❛♠❡♥t♦ ❞❛ ▼❛tr✐③ B 19

              é K(B) = 2.8438 × 10 ✱ ♦♥❞❡ ♦ ♠❛❧ ❝♦♥❞✐❝✐♦♥❛♠❡♥t♦ ❞❛ ♠❛tr✐③ B é ❡✈✐❞❡♥t❡✳

              ✺✹ ◆❛ s❡q✉ê♥❝✐❛✱ ❞❡✜♥✐♥❞♦ t i ✿❂ ❚❡♠♣♦ ❞❡ ❡①❡❝✉çã♦ ❞♦ ♠ét♦❞♦ ♥❛ r❡✲

              ♣❡t✐çã♦ i ❡ x n ✿❂ s♦❧✉çã♦ ❢♦r♥❡❝✐❞❛ ♣❡❧♦ r❡s♣❡❝t✐✈♦ ♠ét♦❞♦✱ ❛ ♥♦♠❡♥✲ ❝❧❛t✉r❛ ♥❛s t❛❜❡❧❛s ❡ ♥♦s ❣rá✜❝♦s é ❛ s❡❣✉✐♥t❡✿

              ◆♦♠❡♥❝❧❛t✉r❛ ❙✐❣♥✐✜❝❛❞♦ ▲❲ ▲❛♥❞✇❡❜❡r ▲❲ ❇▲ ▲❛♥❞✇❡❜❡r ❝♦♠ ❜✉s❝❛ ❧✐♥❡❛r ❙❉ ❙t❡❡♣❡st✲❉❡s❝❡♥t ▲❲✲❑❛❝③ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ▲❲✲❑❛❝③ ❙ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ s✐♠étr✐❝♦ ❚❡♠♣♦ ♠é❞✐♦

              P 1000 i =1 t i 1000

              ❊rr♦ ♥❛ s♦❧✉çã♦ kx n − x

              †

              k ❊rr♦ r❡❧❛t✐✈♦✭♣♦r❝❡♥t✉❛❧✮

              kx n∗ −x k kx k

              × 100% ❚❛❜❡❧❛ ✹✳✶✿ ◆♦♠❡♥❝❧❛t✉r❛ ♥❛s t❛❜❡❧❛s ❡ ✜❣✉r❛s✳

              ✹✳✸✳✶ ❉❛❞♦s ❊①❛t♦s

              P❛r❛ ❞❛❞♦s ❡①❛t♦s ❢♦✐ r❡❛❧✐③❛❞♦ ✉♠ ú♥✐❝♦ t❡st❡✳ ❈♦♠♦ ❝r✐tér✐♦ ❞❡ ♣❛r❛❞❛ ❢♦✐ ❛❞♦t❛❞♦ ♦ ♥ú♠❡r♦ ✜①♦ ❞❡ ❝✐❝❧♦s✱ C = 150✱ ♦♥❞❡ ♣❛r❛ ♦s ♠ét♦❞♦s ▲❲✲❑❛❝③ ❡ ▲❲✲❑❛❝③ ❙ ✉♠ ❝✐❝❧♦ ❡♥t❡♥❞❡✲s❡ ♣♦r ✉♠ p−❝✐❝❧♦ ❡ 2p−❝✐❝❧♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ ♣❛r❛ ♦s ♠ét♦❞♦ ♥ã♦ ❝í❝❧✐❝♦s ✉♠ ❝✐❝❧♦ é ✉♠❛ ✐t❡r❛çã♦✳ ❖s ❞❡t❛❧❤❡s ❞♦s r❡s✉❧t❛❞♦s ♥❛ ❚❛❜❡❧❛ ✹✳✷ ❡ ♥❛ ❋✐❣✉r❛ ✹✳✶✳

              ▼ét♦❞♦ ❚❡♠♣♦ ♠é❞✐♦ ❊rr♦ ♥❛ s♦❧✉çã♦ ❊rr♦ r❡❧❛t✐✈♦ ▲❲ ✵✳✵✵✸✷ ✭s✮ ✷✳✵✵✼✸ ✶✽✳✸✷✸✽% ▲❲ ❇▲ ✵✳✵✵✹✺ ✭s✮ ✶✳✻✺✺✾ ✶✺✳✶✶✻✹% ❙❉ ✵✳✵✵✺✶ ✭s✮ ✵✳✹✷✺✻ ✸✳✽✽✺✹% ▲❲✲❑❛❝③ ✵✳✵✷✽✻ ✭s✮ ✷✳✵✵✾✶ ✶✽✳✸✹✵✷% ▲❲✲❑❛❝③ ❙ ✵✳✵✺✷✵ ✭s✮ ✶✳✼✽✺✻ ✶✻✳✸✵✵✹%

              ❚❛❜❡❧❛ ✹✳✷✿ ❚❡♠♣♦ ♠é❞✐♦ ❞❡ ❡①❡❝✉çã♦ ❡ ❡rr♦s ❛❜s♦❧✉t♦ ❡ r❡❧❛t✐✈♦ ♥❛ s♦❧✉çã♦ ♣❛r❛ ❞❛❞♦s ❡①❛t♦s✳

              10 Métodos Landweber e Landweber−Kaczmarz − Dados exatos LW−Kacz SD LW BL LW ✺✺ − x + || / ||x || n LW−Kacz S + Erro relativo na iteração || x 10 −1 20 40 60 Número de ciclos 80 100 120 140

              ❋✐❣✉r❛ ✹✳✶✿ ❊✈♦❧✉çã♦ ❞♦ ❡rr♦ r❡❧❛t✐✈♦ ♥❛ ✐t❡r❛çã♦ ♣❛r❛ ❞❛❞♦s ❡①❛t♦s✳

              ✹✳✸✳✷ ❉❛❞♦s ❝♦♠ ❘✉í❞♦

              ◆❡st❛ ❡t❛♣❛✱ ♦ ♣r✐♠❡✐r♦ ❞♦s t❡st❡s ❢♦✐ ❢❡✐t♦ ♣❛r❛ ❞❛❞♦s ❝♦♠ 5% ❞❡ r✉í❞♦✱ ✐st♦ é✱ δ = 0.05. ❖s ❞❡t❛❧❤❡s ❞♦s r❡s✉❧t❛❞♦s ❡stã♦ ♥❛ ❚❛❜❡❧❛ ✹✳✸ ❡ ♥❛ ❋✐❣✉r❛ ✹✳✷✱ ♦♥❞❡ t❛♠❜é♠ ♣♦❞❡✲s❡ ❛♣r❡❝✐❛r ♦ ❣rá✜❝♦ ❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ n = 1 ❛t✉❛❧✐③❛çõ❡s ❞❛ ❛♣r♦①✐♠❛çã♦ ❡❢❡t✉❛❞❛s ♣♦r ❝✐❝❧♦✱ ✐✳é✱ q✉❛♥❞♦ ω ♥♦s ♠ét♦❞♦s t✐♣♦ ❑❛❝③♠❛r③✳ ◆♦ ❝❛s♦ s✐♠étr✐❝♦✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❛✈❛❧✐❛çõ❡s ❞♦ ♣r✐♥❝í♣✐♦ ❞❛ ❞✐s❝r❡♣â♥❝✐❛ é ♦ ❞♦❜r♦✳

              ▼ét♦❞♦ ❚❡♠♣♦ ♠é❞✐♦ ❊rr♦ ♥❛ s♦❧✉çã♦ ❊rr♦ r❡❧❛t✐✈♦ ▲❲ ✵✳✵✵✵✶ ✭s✮ ✹✳✾✹✻✶ ✹✺✳✶✺✶✼% ▲❲ ❇▲ ✵✳✵✵✵✶ ✭s✮ ✹✳✵✵✶✽ ✸✻✳✺✸✶✷% ▲❲✲❑❛❝③ ✵✳✵✵✸✹ ✭s✮ ✸✳✾✺✵✺ ✸✻✳✵✻✸✵% ▲❲✲❑❛❝③ ❙ ✵✳✵✵✸✸ ✭s✮ ✹✳✵✽✵✵ ✸✼✳✷✹✺✸%

              ❚❛❜❡❧❛ ✹✳✸✿ ❚❡♠♣♦ ♠é❞✐♦ ❞❡ ❡①❡❝✉çã♦ ❡ ❡rr♦s ❛❜s♦❧✉t♦ ❡ r❡❧❛t✐✈♦ ♥❛ s♦❧✉çã♦ ♣❛r❛ ❞❛❞♦s ❝♦♠ ✺✪ ❞❡ r✉í❞♦✳

              ✺✻

              5 10

            15

            20 25 30 Erro relativo na iteração || x 10 Número de ciclos n − x + || / ||x + || Métodos Landweber e Landweber−Kaczmarz − Dados com ruído LW LW BL LW−Kacz LW−Kacz S

              5 10

            15

            20 25 30 2 4 6 8 10 12 14 16 18 Atualizações efetuadas por ciclo 20 Número de ciclos

            Método Landweber−Kaczmarz − Atual. efetuadas / ciclo − Dados com ruído

            LW−Kacz LW−Kacz S

              ❋✐❣✉r❛ ✹✳✷✿ ❊✈♦❧✉çã♦ ❞♦ ❡rr♦ r❡❧❛t✐✈♦ ♥❛ ✐t❡r❛çã♦ ❡ ❛t✉❛❧✐③❛çõ❡s ❡❢❡t✉✲ ❛❞❛s ♣♦r ❝✐❝❧♦ ♣❛r❛ ❞❛❞♦s ❝♦♠ ✺✪ ❞❡ r✉í❞♦✳

              ❋✐♥❛❧♠❡♥t❡✱ ♦ t❡st❡ ♣❛r❛ ❞❛❞♦s ❝♦♠ 10% ❞❡ r✉í❞♦✱ ❝✉❥♦s r❡s✉❧t❛❞♦s s❡ ❛♣r❡s❡♥t❛♠ ♥❛ ❚❛❜❡❧❛ ✹✳✹ ❡ ♥♦s ❣rá✜❝♦s ❞❛ ❋✐❣✉r❛ ✹✳✸✳ ▼ét♦❞♦ ❚❡♠♣♦ ♠é❞✐♦ ❊rr♦ ♥❛ s♦❧✉çã♦ ❊rr♦ r❡❧❛t✐✈♦ ▲❲ ✵✳✵✵✵✶ ✭s✮ ✻✳✽✷✻✽ ✻✷✳✸✶✾✹% ▲❲ ❇▲ ✵✳✵✵✵✶ ✭s✮ ✻✳✽✶✾✹ ✻✷✳✷✺✷✶% ▲❲✲❑❛❝③ ✵✳✵✵✸✷ ✭s✮ ✺✳✺✾✵✻ ✺✶✳✵✸✹✾% ▲❲✲❑❛❝③ ❙ ✵✳✵✵✸✸ ✭s✮ ✻✳✶✻✹✷ ✺✻✳✷✼✶✵%

              ❚❛❜❡❧❛ ✹✳✹✿ ❚❡♠♣♦ ♠é❞✐♦ ❞❡ ❡①❡❝✉çã♦ ❡ ❡rr♦s ❛❜s♦❧✉t♦ ❡ r❡❧❛t✐✈♦ ♥❛ s♦❧✉çã♦ ♣❛r❛ ❞❛❞♦s ❝♦♠ ✶✵✪ ❞❡ r✉í❞♦✳

              10 Métodos Landweber e Landweber−Kaczmarz − Dados com ruído LW−Kacz S LW−Kacz LW BL LW ✺✼

            • + || / ||x +

              || − x n Erro relativo na iteração || x 18 2 Método Landweber−Kaczmarz − Atual. efetuadas / ciclo − Dados com ruído 4 6 Número de ciclos 8 10 12 14 10 12 14 16 LW−Kacz S LW−Kacz Atualizações efetuadas por ciclo 2 4

              8 6

              10 15

              ❋✐❣✉r❛ ✹✳✸✿ ❊✈♦❧✉çã♦ ❞♦ ❡rr♦ r❡❧❛t✐✈♦ ♥❛ ✐t❡r❛çã♦ ❡ ❛t✉❛❧✐③❛çõ❡s ❡❢❡t✉✲ ❛❞❛s ♣♦r ❝✐❝❧♦ ♣❛r❛ ❞❛❞♦s ❝♦♠ ✶✵✪ ❞❡ r✉í❞♦✳

              ✺✽

              ✹✳✹ ❈♦♥❝❧✉sõ❡s

              P❛r❛ ❞❛❞♦s ❡①❛t♦s✱ ❛♣r❡❝✐❛✲s❡ q✉❡ ♦ ♠ét♦❞♦ ❝♦♠ ♠❡❧❤♦r ❞❡s❡♠♣❡✲ ♥❤♦ é ♦ ♠ét♦❞♦ ❙t❡❡♣❡st✲❉❡s❝❡♥t✳ ❆❣♦r❛✱ ♥❛ ❝❧❛ss❡ ❞♦s ♠ét♦❞♦s ❝í❝❧✐❝♦s ♦ ♠ét♦❞♦ sLK ♦❜t❡✈❡ ♦ ♠❡♥♦r ❡rr♦ ♥❛ ú❧t✐♠❛ ❛♣r♦①✐♠❛çã♦ ❞♦ q✉❡ ♦ ♠ét♦❞♦ LK✱ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ 11% ♠❡♥♦r✱ ♠❛s ♦ t❡♠♣♦ ❞❡ ❡①❡❝✉çã♦ ❞♦ ♠ét♦❞♦ sLK é q✉❛s❡ ♦ ❞♦❜r♦✳

              ❆❣♦r❛✱ ♣❛r❛ ❞❛❞♦s ❝♦♠ 5% ❞❡ r✉í❞♦✱ ❛♣r❡❝✐❛✲s❡ q✉❡ ♥❛ ❝❧❛ss❡ ❞❡ ♠é✲ t♦❞♦s ❝í❝❧✐❝♦s ♦ ♠❡❧❤♦r ❞❡s❡♠♣❡♥❤♦ é ♦❜t✐❞♦ ♣❡❧♦ ♠ét♦❞♦ LK✱ ♦❜t❡✈❡ ♦ ♠❡♥♦r ❡rr♦✱ ♠❛s só ❝♦♥s❡❣✉✐✉ ❞✐♠✐♥✉✐r ♦ ❡rr♦ ♥❛ s♦❧✉çã♦ ❛♣r♦①✐♠❛✲ ❞❛♠❡♥t❡ ❡♠ ❛♣❡♥❛s 3% ❝♦♠ r❡❧❛çã♦ ❛♦ ♠ét♦❞♦ sLK✱ ❛✉♠❡♥t❛♥❞♦ ♦ t❡♠♣♦ ♠é❞✐♦ ❞❡ ❝♦♠♣✉t♦ ✭❝♦♠ r❡❧❛çã♦ ❛♦ ♠ét♦❞♦ sLK✮ ❡♠ ❛♣r♦①✐♠❛✲ ❞❛♠❡♥t❡ 3%✳

              ◆♦ ❝❛s♦ ❞❡ ❞❛❞♦s ❝♦♠ 10% ❞❡ r✉í❞♦✱ ♦ ♠❡♥♦r ❡rr♦ ❢♦✐ ♦❜t✐❞♦ ♣❡❧♦ ♠ét♦❞♦ LK✱ ❞❡ss❛ ✈❡③ ❝♦♥s❡❣✉✐✉ ❞✐♠✐♥✉✐r ♦ ❡rr♦ ♥❛ s♦❧✉çã♦ ❝♦♠ r❡❧❛çã♦ ❛♦ ♠ét♦❞♦ sLK ❡♠ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❡♠ 9%✱ ❡ ❞✐♠✐♥✉✐♥❞♦ ♦ t❡♠♣♦ ♠é❞✐♦ ❞❡ ❝♦♠♣✉t♦ ❡♠ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ 3%✳

              ❋✐♥❛❧♠❡♥t❡✱ ♦s ❡①♣❡r✐♠❡♥t♦s ♠♦str❛♠ q✉❡ ♥♦ ❝❛s♦ ❞❡ ❞❛❞♦s ❝♦♠ r✉í❞♦ ❞❡ 5% ♦s ♠ét♦❞♦s t✐♣♦ ❑❛❝③♠❛r③ t❡♠ ✉♠ ♠❡❧❤♦r ❞❡s❡♠♣❡✲ ♥❤♦✭❝♦♠ r❡❧❛çã♦ ❛♦ ❡rr♦ ♥❛ s♦❧✉çã♦✮ q✉❡ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ❡ ♣❛r❛ ❞❛❞♦s ❝♦♠ r✉í❞♦ ❞❡ 10% ♦s ♠ét♦❞♦s t✐♣♦ ❑❛❝③♠❛r③ t❡♠ ♠❡❧❤♦r ❞❡s❡♠♣❡♥❤♦ q✉❡ ♦s ♠ét♦❞♦s ▲❛♥❞✇❡❜❡r ❡ ▲❛♥❞✇❡❜❡r ❝♦♠ ❜✉s❝❛ ❧✐♥❡❛r✳

              ❈♦♥❝❧✉sõ❡s ●❡r❛✐s

              ◆❡st❡ tr❛❜❛❧❤♦ ❢♦✐ ♣r♦✈❛❞❛ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ ✉♠❛ ✈❛r✐❛çã♦ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③✱ ❝♦♠ ✉♠❛ ♥♦✈❛ r❡❣r❛ ❞❡ ♣❛r❛❞❛ ❡ ✉♠ ♣❛râ♠❡tr♦ q✉❡ ♣❡r♠✐t❡ ♦♠✐t✐r ❛❧❣✉♠❛s ✐t❡r❛çõ❡s ❞❡♥tr♦ ❞❡ ✉♠ ❝✐❝❧♦✱ s❡ ♦s r❡sí❞✉♦s ❝♦rr❡s♣♦♥❞❡♥t❡s sã♦ ♦ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡✲ ♥♦s✳ ❯♠❛ ✈❡♥t❛❣❡♠ ❞♦s ♠ét♦❞♦s t✐♣♦ ❑❛❝③♠❛r③✱ é q✉❡ ♦ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ r❡s✉❧t❛♥t❡ ❛♣r♦✈❡✐t❛ ❞❡ ♠❡❧❤♦r ♠❛♥❡✐r❛ ❛ ❡str✉t✉r❛ ❡s♣❡✲ i δ i i

              ❝✐❛❧ ❞♦ ♠♦❞❡❧♦ ❡ ❛ ❡st✐♠❛t✐✈❛ ♣♦♥t✉❛❧ ❞♦ ♥í✈❡❧ ❞❡ r✉í❞♦ ky ✳ − y k ≤ δ

              ❆ ❝❤❛✈❡ ♣❛r❛ ♣r♦✈❛r ❝♦♥✈❡r❣ê♥❝✐❛✱ q✉❛♥❞♦ ♦ ♥í✈❡❧ ❞❡ r✉í❞♦ t❡♥❞❡ ❛ ③❡r♦✱ ❢♦✐ ❛ ✐♥tr♦❞✉çã♦ ❞♦ ♣❛râ♠❡tr♦ ❞❡ r❡❧❛①❛çã♦ ❜❛♥❣✲❜❛♥❣✱ ω n ♥❛ ✐t❡r❛çã♦ ✭✶✳✾✮✳

              ❈♦♠ r❡❧❛çã♦ às t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛✱ ♥♦s ❝❛s♦ s✐♠étr✐❝♦ ❡ ♥ã♦ s✐✲ ♠étr✐❝♦✱ ❞❛❞♦ q✉❡ ❛ ú♥✐❝❛ ❝♦♥❞✐çã♦ ♣❛r❛ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ♥♦s t❡♦r❡♠❛s

              ∗

              M A é q✉❡ ♦s ♦♣❡r❛❞♦r❡s A i i s❡❥❛♠ ❧✐♠✐t❛❞♦s✱ ✐♥❢❡r❡✲s❡ q✉❡ ♣❛r❛ t❛✲ i ♠❛♥❤♦s ❞❡ ♣❛ss♦ τ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦s ✭♦✉ ♦♣❡r❛❞♦r❡s ❡s❝❛❧❛❞♦s ❛♣r♦♣r✐❛❞❛♠❡♥t❡✮✱ s❡♠♣r❡ ♣♦❞❡♠ s❡r ❡st❛❜❡❧❡❝✐❞❛s t❛①❛s ❞❡ ❝♦♥✈❡r✲ ❣ê♥❝✐❛ ✉s✉❛✐s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♥♦s t❡♦r❡♠❛s ✉s❛♠✲s❡ ❝♦t❛s ✭✈❡❥❛ ✭✸✳✸✹✮✱ ✭✸✳✸✼✮✮ q✉❡ sã♦ ❝♦♠♣✉tá✈❡✐s ❡ ♣♦❞❡♠ s❡r ✉s❛❞❛s ❡♠ ✐♠♣❧❡♠❡♥t❛çõ❡s ♥✉♠ér✐❝❛s✳

              ❆ ✐t❡r❛çã♦ s✐♠étr✐❝❛✱ t❡♠ ❛ ✈❡♥t❛❣❡♠ q✉❡ ♣♦❞❡ s❡r ✉s❛❞❛ ❛✐♥❞❛ ❡♠ ❝❛s♦s ♦♥❞❡ ❛s ❝♦♥❞✐çõ❡s ♣❛r❛ ❛ ✐t❡r❛çã♦ ♥ã♦ s✐♠étr✐❝❛ ♥ã♦ sã♦ s❛t✐s❢❡✐t❛s✳ ◆♦ ❡♥t❛♥t♦✱ ❞❡✈❡✲s❡ ♣❛❣❛r ♦ ♣r❡ç♦ ❞❡ ❛✉♠❡♥t❛r ❛♦ ❞♦❜r♦ ♦ ❡s❢♦rç♦ ❝♦♠♣✉t❛❝✐♦♥❛❧✳

              ❋✐♥❛❧♠❡♥t❡✱ ♥❛s ❡①♣❡r✐ê♥❝✐❛s ♥✉♠ér✐❝❛s ❞♦ ❈❛♣ít✉❧♦ ✹ ❡ ♥❛ ❝❧❛ss❡ ❞❡ ♠ét♦❞♦s ❝í❝❧✐❝♦s✱ ♦❜s❡r✈❛✲s❡ ♦ ❜♦♠ ❞❡s❡♠♣❡♥❤♦ ❞♦ ♠ét♦❞♦ sLK ♣❛r❛ ❞❛❞♦s ❡①❛t♦s ❝♦♠ r❡❧❛çã♦ ❛♦ ❡rr♦ ♥❛ s♦❧✉çã♦✱ ❡♠❜♦r❛ t❡♠ ✉♠ ❝✉st♦ ♥♦ t❡♠♣♦ ♠é❞✐♦ ❞❡ ❡①❡❝✉çã♦ ❝♦♠ r❡❧❛çã♦ ❛♦ ♠ét♦❞♦ LK ❞❡ q✉❛s❡ ♦ ❞♦❜r♦✳ ❆❣♦r❛ ♣❛r❛ ❞❛❞♦s ❝♦♠ r✉í❞♦ ❛ s✐t✉❛çã♦ ♠✉❞❛✱ q✉❡♠ t❡♠ ♠❡❧❤♦r ❞❡s❡♠♣❡♥❤♦ ❝♦♠ r❡❧❛çã♦ ❛♦ ❡rr♦ ♥❛ s♦❧✉çã♦ é ♦ ♠ét♦❞♦ LK✱ ♠❛s q✉❡♠ t❡♠ ❛ ♠❡❧❤♦r r❡❧❛çã♦ ❝✉st♦✴❜❡♥❡✜❝✐♦✭❝♦♠ r❡❧❛çã♦ ❛♦ ♠ét♦❞♦ LK✮✱ é ♦ ♠ét♦❞♦ sLK✳

              ✻✵

              ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

              ❬✶❪ ❆◆❍✱ P✳ ❑✳❀ ❈❍❯◆●✱ ❈✳ ❱✳✳ P❛r❛❧❧❡❧ r❡❣✉❧❛r✐③❡❞ ◆❡✇t♦♥ ♠❡t❤♦❞ ❢♦r ♥♦♥❧✐♥❡❛r ✐❧❧✲♣♦s❡❞ ❡q✉❛t✐♦♥s✳ ◆✉♠❡r✳ ❆❧❣♦r✐t❤♠s✳ ✈✳ ✺✽✱ ♣✳ ✸✼✾✲✸✾✽✱ ✷✵✶✶✳

              ❬✷❪ ❇❆❯▼❊■❙❚❊❘✱ ❏✳❀ ❑❆▲❚❊◆❇❆❈❍❊❘✱ ❇✳❀ ▲❊■❚➹❖✱ ❆✳✳ ❖♥ ▲❡✈❡♥❜❡r❣✲▼❛rq✉❛r❞t✲❑❛❝③♠❛r③ ✐t❡r❛t✐✈❡ ♠❡t❤♦❞s ❢♦r s♦❧✈✐♥❣ s②s✲ t❡♠s ♦❢ ♥♦♥❧✐♥❡❛r ✐❧❧✲♣♦s❡❞ ❡q✉❛t✐♦♥s✳ ■♥✈❡rs❡ Pr♦❜❧❡♠s ❛♥❞ ■♠❛❣✐♥❣✳ ✈✳ ✹✱ ♣✳ ✸✸✺✲✸✺✵✱ ✷✵✶✵✳

              ❬✸❪ ❇❯❘●❊❘✱ ▼✳❀ ❑❆▲❚❊◆❇❆❈❍❊❘✱ ❇✳✳ ❘❡❣✉❧❛r✐③✐♥❣ ◆❡✇t♦♥✲ ❑❛❝③♠❛r③ ♠❡t❤♦❞s ❢♦r ♥♦♥❧✐♥❡❛r ✐❧❧✲♣♦s❡❞ ♣r♦❜❧❡♠s✳ ❙■❆▼ ❏✳ ◆✉♠❡r✳ ❆♥❛❧✳ ✈✳ ✹✹✱ ♣✳ ✶✺✸✲✶✽✷✱ ✷✵✵✻✳

              ❬✹❪ ❈❘❖❯❩❊■❳✱ ▼✳✳ ◆✉♠❡r✐❝❛❧ r❛♥❣❡ ❛♥❞ ❢✉♥❝t✐♦♥❛❧ ❝❛❧❝✉❧✉s ✐♥ ❍✐❧✲ ❜❡rt s♣❛❝❡✳ ❏♦✉r♥❛❧ ♦❢ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s✱ ✈✳ ✷✹✹✱ ♣✳ ✻✻✽✲✻✾✵✱ ✷✵✵✼✳

              ❬✺❪ ❉❊ ❈❊❩❆❘❖✱ ❆✳❀ ❇❆❯▼❊■❙❚❊❘✱ ❏✳❀ ▲❊■❚➹❖✱ ❆✳✳ ▼♦❞✐✜❡❞ ✐t❡✲ r❛t❡❞ ❚✐❦❤♦♥♦✈ ♠❡t❤♦❞s ❢♦r s♦❧✈✐♥❣ s②st❡♠s ♦❢ ♥♦♥❧✐♥❡❛r ✐❧❧✲♣♦s❡❞ ❡q✉❛t✐♦♥s✳ ■♥✈❡rs❡ Pr♦❜❧❡♠s ❛♥❞ ■♠❛❣✐♥❣✳ ✈✳ ✺✱ ♣✳ ✶✲✶✼✱ ✷✵✶✶✳

              ❬✻❪ ❉❊ ❈❊❩❆❘❖✱ ❆✳❀ ❍❆▲❚▼❊■❊❘✱ ▼✳❀ ▲❊■❚➹❖✱ ❆✳❀ ❙❈❍❊❘❩❊❘✱ ❖✳✳ ❖♥ st❡❡♣❡st✲❞❡s❝❡♥t✲❑❛❝③♠❛r③ ♠❡t❤♦❞s ❢♦r r❡❣✉❧❛r✐③✐♥❣ s②s✲ t❡♠s ♦❢ ♥♦♥❧✐♥❡❛r ✐❧❧✲♣♦s❡❞ ❡q✉❛t✐♦♥s✳ ❆♣♣❧✳ ▼❛t❤✳ ❈♦♠♣✉t✳ ✈✳ ✷✵✷✱ ♣✳ ✺✾✻✲✻✵✼✱ ✷✵✵✽✳

              ❬✼❪ ❊▲❋❱■◆●✱ ❚✳❀ ◆■❑❆❩❆❉✱ ❚✳✳ Pr♦♣❡rt✐❡s ♦❢ ❛ ❝❧❛ss ♦❢ ❜❧♦❝❦✲ ✐t❡r❛t✐✈❡ ♠❡t❤♦❞s✳ ■♥✈❡rs❡ Pr♦❜❧❡♠s✱ ✈✳ ✷✺✱ ♣✳ ✶✸✱ ✷✵✵✾✳

              ❬✽❪ ❊◆●▲✱ ❍✳ ❲✳❀ ❍❆◆❑❊✱ ▼✳❀ ◆❊❯❇❆❯❊❘✱ ❆✳✳ ❘❡❣✉❧❛r✐③❛t✐♦♥ ♦❢ ■♥✈❡rs❡ Pr♦❜❧❡♠s✳ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✱ ✈✳ ✸✼✺✱ ❑❧✉✇❡r ❆❝❛❞❡♠✐❝ P✉❜❧✐s❤❡rs✱ ❉♦r❞r❡❝❤t✱ ✶✾✾✻✳

              ✻✷ ❬✾❪ ●❘❖❊❚❙❈❍✱ ❈✳ ❲✳✳ ❚❤❡ ❚❤❡♦r② ♦❢ ❚✐❦❤♦♥♦✈ ❘❡❣✉r❛❧✐③❛✲ t✐♦♥ ❢♦r ❋r❡❞❤♦❧♠ ❊q✉❛t✐♦♥s ♦❢ t❤❡ ❋✐rst ❑✐♥❞✳ P✐t♠❛♥ ❆❞✲

              ✈❛♥❝❡❞ P✉❜❧✐s❤✐♥❣ Pr♦❣r❛♠✱ ❇♦st♦♥✱ ▲♦♥❞♦♥✱ ▼❡❧❜♦✉r♥❡✱ ✶✾✽✹✳ ❬✶✵❪ ❍❆▲❚▼❊■❊❘✱ ▼✳✳ ❈♦♥✈❡r❣❡♥❝❡ ❛♥❛❧②s✐s ♦❢ ❛ ❜❧♦❝❦ ✐t❡r❛t✐✈❡ ✈❡r✲ s✐♦♥ ♦❢ ❛ ❧♦♣✐♥❣ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ✐t❡r❛t✐♦♥✳ ◆♦♥❧✐♥❡❛r ❆♥❛❧✱

              ✈✳ ✼✶✱ ♣✳ ❡✷✾✶✷✲❡✷✾✶✾✱ ✷✵✵✾✳ ❬✶✶❪ ❍❆▲❚▼❊■❊❘✱ ▼✳❀ ❑❖❲❆❘✱ ❘✳❀ ▲❊■❚➹❖✱ ❆✳❀ ❙❈❍❊❘❩❊❘✱ ❖✳✳

              ❑❛❝③♠❛r③ ♠❡t❤♦❞s ❢♦r r❡❣✉❧❛r✐③✐♥❣ ♥♦♥❧✐♥❡❛r ✐❧❧✲♣♦s❡❞ ❡q✉❛t✐♦♥s ■■✿ ❆♣♣❧✐❝❛t✐♦♥s✳ ■♥✈❡rs❡ Pr♦❜❧❡♠s ❛♥❞ ■♠❛❣✐♥❣✱ ✈✳ ✶✱ ♣✳ ✺✵✼✲ ✺✷✸✱ ✷✵✵✼✳

              ❬✶✷❪ ❍❆▲❚▼❊■❊❘✱ ▼✳❀ ▲❊■❚➹❖✱ ❆✳❀ ❘❊❙▼❊❘■❚❆✱ ❊✳✳ ❖♥ r❡❣✉❧❛r✐✲ ③❛t✐♦♥ ♠❡t❤♦❞s ♦❢ ❊▼✲❑❛❝③♠❛r③ t②♣❡✳ ■♥✈❡rs❡ Pr♦❜❧❡♠s✱ ✈✳ ✷✺✱ ♣✳ ✶✼✱ ✷✵✵✾✳

              ❬✶✸❪ ❍❆▲❚▼❊■❊❘✱ ▼✳❀ ▲❊■❚➹❖✱ ❆✳❀ ❙❈❍❊❘❩❊❘✱ ❖✳✳ ❑❛❝③♠❛r③ ♠❡t❤♦❞s ❢♦r r❡❣✉❧❛r✐③✐♥❣ ♥♦♥❧✐♥❡❛r ✐❧❧✲♣♦s❡❞ ❡q✉❛t✐♦♥s ■✿ ❝♦♥✈❡r✲ ❣❡♥❝❡ ❛♥❛❧②s✐s✳ ■♥✈❡rs❡ Pr♦❜❧❡♠s ❛♥❞ ■♠❛❣✐♥❣✱ ✈✳ ✶✱ ♣✳ ✷✽✾✲ ✷✾✽✱ ✷✵✵✼✳

              ❬✶✹❪ ❍❆◆❑❊✱ ▼✳❀ ◆❊❯❇❆❯❊❘✱ ❆✳❀ ❙❈❍❊❘❩❊❘✱ ❖✳✳ ❆ ❝♦♥✈❡r❣❡♥❝❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ▲❛♥❞✇❡❜❡r ✐t❡r❛t✐♦♥ ❢♦r ♥♦♥❧✐♥❡❛r ✐❧❧✲♣♦s❡❞ ♣r♦✲ ❜❧❡♠s✳ ◆✉♠❡r✐s❝❤❡ ▼❛t❤❡♠❛t✐❦✱ ✈✳ ✼✷✱ ♣✳ ✷✶✲✸✼✱ ✶✾✾✺✳

              ❬✶✺❪ ❑❆❈❩▼❆❘❩✱ ❙✳✳ ❆♥❣❡♥ä❤❡rt❡ ❆✉✢ös✉♥❣ ✈♦♥ ❙②st❡♠❡♠ ❧✐♥❡❛r❡r ●❧❡✐❝❤✉♥❣❡♥✳ ❇✉❧❧✳ ■♥t❡r♥❛t✳ ❆❝❛❞✳ P♦❧♦♥✳ ❙❝✐✳ ❆✳ ✈✳ ✸✺✱ ♣✳ ✸✺✺✲✸✺✼✱ ✶✾✸✼✳

              ❬✶✻❪ ❑❆▲❚❊◆❇❆❈❍❊❘✱ ❇✳❀ ◆❊❯❇❆❯❊❘✱ ❆✳❀ ❙❈❍❊❘❩❊❘✱ ❖✳✳ ■t❡✲ r❛t✐✈❡ ❘❡❣✉❧❛r✐③❛t✐♦♥ ▼❡t❤♦❞s ❢♦r ◆♦♥❧✐♥❡❛r ■❧❧✲P♦s❡❞ Pr♦❜❧❡♠s✳ ❲❛❧t❡r ❞❡ ●r✉②t❡r ●♠❣❍ ✫ ❈♦✳ ❑●✱ ❇❡r❧✐♥✱ ✷✵✵✽✳

              ❬✶✼❪ ❑❆❚❖✱ ❚✳✳ ❆ ❣❡♥❡r❛t✐③❛t✐♦♥ ♦❢ t❤❡ ❍❡✐♥③ ✐♥❡q✉❛❧✐t②✳ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❏❛♣❛♥ ❆❝❛❞❡♠②✳ ✈✳ ✸✼✱ ♣✳ ✸✵✺✲✸✵✽✱ ✶✾✻✶✳

              ❬✶✽❪ ❑❆❚❖✱ ❚✳✳ P❡rt✉r❜❛t✐♦♥ ❚❤❡♦r② ❢♦r ▲✐♥❡❛r ❖♣❡r❛t♦rs✳ ❈❧❛s✲ s✐❝s ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ❙♣r✐♥❣❡r✲ ❱❡r❧❛❣✱ ❇❡r❧✐♥✱ ✶✾✾✺✱ ❘❡♣r✐♥t ♦❢ t❤❡ ✶✾✽✵ ❡❞✐t✐♦♥✳

              ❬✶✾❪ ❑■◆❉❊❘▼❆◆◆✱ ❙✳❀ ▲❊■❚➹❖✱ ❆✳✳ ❈♦♥✈❡r❣❡♥❝❡ r❛t❡s ❢♦r ❑❛❝③♠❛r③✲t②♣❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞s✳ ■♥✈❡rs❡ Pr♦❜❧❡♠s ❛♥❞ ■♠❛❣✐♥❣✱ ✈✳ ✽✱ ♣✳ ✶✹✾✲✶✼✷✱ ✷✵✶✹✳

              ✻✸ ❬✷✵❪ ❑❖❲❆❘✱ ❘✳❀ ❙❈❍❊❘❩❊❘✱ ❖✳✳ ❈♦♥✈❡r❣❡♥❝❡ ❆♥❛❧②s✐s ♦❢ ❛

              ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ▼❡t❤♦❞ ❢♦r ❙♦❧✈✐♥❣ ◆♦♥❧✐♥❡❛r ■❧❧✲P♦s❡❞ Pr♦✲ ❜❧❡♠s✳ ■❧❧✲P♦s❡❞ ❛♥❞ ✐♥✈❡rs❡ Pr♦❜❧❡♠s ✭❜♦♦❦ s❡r✐❡s✮✳ ✈✳ ✷✸✱ ♣✳ ✻✾✲✾✵✱ ✷✵✵✷✳

              ❬✷✶❪ ◆❆●❨✱ ❇✳❀ ❩❊▼➪◆❊❑✱ ❏✳✳ ❆ r❡s♦❧✈❡♥t ❝♦♥❞✐t✐♦♥ ✐♠♣❧②✐♥❣ ♣♦✇❡r ❜♦✉♥❞❡❞♥❡ss✳ ❙t✉❞✐❛ ▼❛t❤❡♠❛t✐❝❛✱ ✈✳ ✶✸✹✱ ♣✳ ✶✹✸✲✶✺✶✱ ✶✾✾✾✳

              ❬✷✷❪ P▲❆❚❖✱ ❘✳❀ ❍➘▼❆❘■❑✱ ❯✳✳ ❖♥ ♣s❡✉❞♦✲♦♣t✐♠❛❧ ♣❛r❛♠❡t❡r ❝❤♦✐❝❡ ❛♥❞ st♦♣♣✐♥❣ r✉❧❡s ❢♦r r❡❣✉r❛❧✐③❛t✐♦♥s ♠❡t❤♦❞s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✳ ◆✉♠❡r✐❝❛❧ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥✱ ✈✳ ✶✼✱ ♣✳ ✶✽✶✲✶✾✺✱ ✶✾✾✻✳

              ❬✷✸❪ P▲❆❚❖✱ ❘✳✳ ■t❡r❛t✐✈❡ ❛♥❞ ❖t❤❡r ▼❡t❤♦❞s ❢♦r ▲✐♥❡❛r ■❧❧✲ P♦s❡❞ ❊q✉❛t✐♦♥s✳ ❍❛❜✐❧✐t❛t✐♦♥ t❤❡s✐s✱ ❉❡♣❛rt❛♠❡♥t ♦❢ ▼❛t❤❡✲ ♠❛t✐❝s✱ ❚❯ ❇❡r❧✐♥✱ ✶✾✾✺✳

Novo documento