Estabilidade numérica de fórmulas baricêntricas para interpolação

❊st❛❜✐❧✐❞❛❞❡ ♥✉♠ér✐❝❛ ❞❡ ❢ór♠✉❧❛s

❜❛r✐❝ê♥tr✐❝❛s ♣❛r❛ ✐♥t❡r♣♦❧❛çã♦

❆♥❞ré P✐❡rr♦ ❞❡ ❈❛♠❛r❣♦

❚❡s❡ ❛♣r❡s❡♥t❛❞❛

❛♦

■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛

❞❛

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦

♣❛r❛

♦❜t❡♥çã♦ ❞♦ tít✉❧♦

❞❡

❉♦✉t♦r ❡♠ ❈✐ê♥❝✐❛s

Pr♦❣r❛♠❛✿ ❉♦✉t♦r❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛

❖r✐❡♥t❛❞♦r✿ ❲❛❧t❡r ❋✐❣✉❡✐r❡❞♦ ▼❛s❝❛r❡♥❤❛s

❉✉r❛♥t❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦ ♦ ❛✉t♦r r❡❝❡❜❡✉ ❛✉①í❧✐♦ ✜♥❛♥❝❡✐r♦ ❞♦ ❈◆Pq

❊st❛❜✐❧✐❞❛❞❡ ♥✉♠ér✐❝❛ ❞❡ ❢ór♠✉❧❛s ❜❛r✐❝ê♥tr✐❝❛s ♣❛r❛ ✐♥t❡r♣♦❧❛çã♦

❊st❛ ✈❡rsã♦ ❞❛ t❡s❡ ❝♦♥té♠ ❛s ❝♦rr❡çõ❡s ❡ ❛❧t❡r❛çõ❡s s✉❣❡r✐❞❛s ♣❡❧❛ ❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛ ❞✉r❛♥t❡ ❛ ❞❡❢❡s❛ ❞❛ ✈❡rsã♦ ♦r✐❣✐♥❛❧ ❞♦ tr❛❜❛❧❤♦✱ r❡❛❧✐③❛❞❛ ❡♠ ✶✺✴✶✷✴✷✵✶✺✳ ❯♠❛ ❝ó♣✐❛ ❞❛ ✈❡rsã♦ ♦r✐❣✐♥❛❧ ❡stá ❞✐s♣♦♥í✈❡❧ ♥♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✳

❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛✿

• Pr♦❢✳ ❉r✳ ❲❛❧t❡r ❋✐❣✉❡✐r❡❞♦ ▼❛s❝❛r❡♥❤❛s ✭♦r✐❡♥t❛❞♦r✮ ✲ ■▼❊✲❯❙P • Pr♦❢✳ ❉r✳ ❙❛✉❧♦ ❘❛❜❡❧❧♦ ▼❛❝✐❡❧ ❞❡ ❇❛rr♦s ✲ ■▼❊✲❯❙P

• Pr♦❢✳ ❉r✳ ➪❧✈❛r♦ ❘♦❞♦❧❢♦ ❉❡ P✐❡rr♦ ✲ ❯◆■❈❆▼P

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉ ♣❛✐s✱ q✉❡ s❡♠♣r❡ ♠❡ ❛♣♦✐❛r❛♠ ❡ ♠❡ ❞❡r❛♠ s✉♣♦rt❡ ❡♠ t♦❞❛s ❛s ♠✐♥❤❛s ❡s❝♦❧❤❛s ♣❡ss♦❛✐s ❡ ♣r♦✜ss✐♦♥❛✐s✳ ❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ❡s♣♦s❛ ❘♦❜❡rt❛ ❡ à ♠✐♥❤❛ ✜❧❤❛ ▼❛r✐♥❛ ♣♦r t♦❞♦ ♦ ❛♣♦✐♦ ❡ ❝❛r✐♥❤♦ r❡❝❡❜✐❞♦s ♥❡ss❡ ♣❡rí♦❞♦✳ ❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ♠❡✉s ♠❡♥t♦r❡s q✉❡ t♦r♥❛r❛♠ ♦ ♣r♦❝❡ss♦ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ ♠❛✐s ✐♥t❡r❡ss❛♥t❡ ❡ ♠❡ ❡st✐♠✉❧❛r❛♠✱ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡✱ ❛ ♣r♦ss❡❣✉✐r ♥❡ss❛ ❥♦r♥❛❞❛ s✉♣❡r❛♥❞♦ ♦s ♦❜stá❝✉❧♦s✳ ❆❣r❛❞❡ç♦✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ❛♦s ♠❡✉s ❡①✲♦r✐❡♥t❛❞♦r❡s ❞❡ ✐♥✐❝✐❛çã♦ ❝✐❡♥tí✜❝❛✿ ❊❞✉❛r❞♦ ❞♦ ◆❛s❝✐♠❡♥t♦ ▼❛r❝♦s ❡ P❛✉❧♦ ❆❣♦③③✐♥✐ ▼❛rt✐♥✱ ❛♦ ♣r♦❢❡ss♦r ❆♥tô♥✐♦ ❞❡ P❛❞✉❛ ❋r❛♥❝♦ ❋✐❧❤♦ ❡ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❞❡ ❞♦✉t♦r❛❞♦ ❲❛❧t❡r ❋✐❣✉❡✐r❡❞♦ ▼❛s❝❛r❡♥❤❛s✳ ❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ♠❡✉s ❛♠✐❣♦s ♣❡❧❛s ❞✐✈❡rs❛s ❢♦r♠❛s ❞❡ ❛♣♦✐♦✳ ❊♠ ❡s♣❡❝✐❛❧ ❛❣r❛❞❡ç♦ ❛♦ ❛♠✐❣♦ ▲❡❛♥❞r♦ ❈â♥❞✐❞♦ ❇❛t✐st❛ ❡ ❛♦s ❛♠✐❣♦s P❡❞r♦ ❞❛ ❙✐❧✈❛ P❡✐①♦t♦ ❡ ❚✐❛❣♦ ❉❡ ▼♦r❛✐s ▼♦♥t❛♥❤❡r ♣❡❧❛ ❛❥✉❞❛ ❝♦♠ ❧✐♥✉① ❡ ❈✰✰✱ ❞❡♥tr❡ ♦✉tr❛s ❝♦✐s❛s✳ ❆❣r❛❞❡ç♦✱ t❛♠❜é♠✱ ❛♦ ❈◆Pq ♣❡❧♦ ❛✉①í❧✐♦ ✜♥❛♥❝❡✐r♦✳ P♦r ✜♠✱ ❛❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s q✉❡ ♥ã♦ ❢♦r❛♠ ❝✐t❛❞♦s ❡①♣❧✐❝✐t❛♠❡♥t❡ ♠❛s q✉❡ ❝♦♥tr✐❜✉ír❛♠✱ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛✱ ♣❛r❛ q✉❡ ❡✉ ♣✉❞❡ss❡ ❝♦♠♣❧❡t❛r ♠❛✐s ❡ss❛ ❡t❛♣❛ ❞❛ ♠✐♥❤❛ ❝❛rr❡✐r❛✳

❆ t♦❞♦s ✈♦❝ês✱ ♠✉✐t♦ ♦❜r✐❣❛❞♦✦

❘❡s✉♠♦

❈❛♠❛r❣♦✱ ❆✳ P✳ ❊st❛❜✐❧✐❞❛❞❡ ♥✉♠ér✐❝❛ ❞❡ ❢ór♠✉❧❛s ❜❛r✐❝ê♥tr✐❝❛s ♣❛r❛ ✐♥t❡r♣♦❧❛çã♦✳ ✷✵✶✺✳ ✶✷✵ ❢✳ ❚❡s❡ ✭❉♦✉t♦r❛❞♦✮ ✲ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛✱ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ❙ã♦ P❛✉❧♦✱ ✷✵✶✺✳

❖ ♣r♦❜❧❡♠❛ ❞❡ r❡❝♦♥str✉✐r ✉♠❛ ❢✉♥çã♦ f : [a, b]_{−→ R}❛ ♣❛rt✐r ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ✈❛❧♦r❡s

❝♦♥❤❡❝✐❞♦s f(x0), f(x1), . . . , f(xn) ❛♣❛r❡❝❡ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ❡♠ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛✳ ❊♠ ❣❡r❛❧✱ ♥ã♦ é ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛r f ❝♦♠♣❧❡t❛♠❡♥t❡ ❛ ♣❛rt✐r ❞❡ f(x0), f(x1), . . . , f(xn)✱ ♠❛s✱ ❡♠ ♠✉✐t♦s ❝❛s♦s ❞❡ ✐♥t❡r❡ss❡✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ❛♣r♦①✐♠❛çõ❡s r❛③♦á✈❡✐s ♣❛r❛ f ✉s❛♥❞♦ ✐♥t❡r♣♦❧❛çã♦✱ q✉❡

❝♦♥s✐st❡ ❡♠ ❞❡t❡r♠✐♥❛r ✉♠❛ ❢✉♥çã♦ ✭✉♠ ♣♦❧✐♥ô♠✐♦✱ ♦✉ ✉♠❛ ❢✉♥çã♦ r❛❝✐♦♥❛❧ ♦✉ tr✐❣♦♥♦♠étr✐❝❛✱ ❡t❝✮

g q✉❡ s❛t✐s❢❛ç❛

g(xi) =f(xi), i= 0,1, . . . , n.

◆❛ ♣rát✐❝❛✱ ❛ ❢✉♥çã♦ ✐♥t❡r♣♦❧❛❞♦r❛ g é ❛✈❛❧✐❛❞❛ ❡♠ ♣r❡❝✐sã♦ ✜♥✐t❛ ❡ ♦ ✈❛❧♦r ✜♥❛❧ ❝♦♠♣✉t❛❞♦ gd(x)

♣♦❞❡ ❞✐❢❡r✐r ❞♦ ✈❛❧♦r ❡①❛t♦g(x) ❞❡✈✐❞♦ ❛ ❡rr♦s ❞❡ ❛rr❡❞♦♥❞❛♠❡♥t♦✳ ❊ss❛ ❞✐❢❡r❡♥ç❛ ♣♦❞❡✱ ✐♥❝❧✉s✐✈❡✱

✉❧tr❛♣❛ss❛r ♦ ❡rr♦ ❞❡ ✐♥t❡r♣♦❧❛çã♦ E(x) = f(x)₋g(x) ❡♠ ✈ár✐❛s ♦r❞❡♥s ❞❡ ♠❛❣♥✐t✉❞❡✱ ❝♦♠♣r♦✲

♠❡t❡♥❞♦ t♦❞♦ ♦ ♣r♦❝❡ss♦ ❞❡ ❛♣r♦①✐♠❛çã♦✳ ❆ ❡st❛❜✐❧✐❞❛❞❡ ♥✉♠ér✐❝❛ ❞❡ ✉♠ ❛❧❣♦r✐t♠♦ r❡✢❡t❡ s✉❛ s❡♥s✐❜✐❧✐❞❛❞❡ ❡♠ r❡❧❛çã♦ ❛ ❡rr♦s ❞❡ ❛rr❡❞♦♥❞❛♠❡♥t♦✳ ◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❛♥á❧✐s❡ ❞❡t❛❧❤❛❞❛ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ♥✉♠ér✐❝❛ ❞❡ ❛❧❣✉♥s ❛❧❣♦r✐t♠♦s ✉t✐❧✐③❛❞♦s ♥♦ ❝á❧❝✉❧♦ ❞❡ ✐♥t❡r♣♦❧❛❞♦r❡s ♣♦❧✐♥♦♠✐❛✐s ♦✉ r❛❝✐♦♥❛✐s q✉❡ ♣♦❞❡♠ s❡r ♣♦st♦s ♥❛ ❢♦r♠❛ ❜❛r✐❝ê♥tr✐❝❛✳

❖s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❡st❡ tr❛❜❛❧❤♦ t❛♠❜é♠ ❡stã♦ ❞✐s♣♦♥í✈❡✐s ❡♠ ❧✐♥❣✉❛ ✐♥❣❧❡s❛ ♥♦s ❛rt✐❣♦s

• ▼❛s❝❛r❡♥❤❛s✱ ❲ ❡ ❈❛♠❛r❣♦✱ ❆✳ P✳✱ ❖♥ t❤❡ ❜❛❝❦✇❛r❞ st❛❜✐❧✐t② ♦❢ t❤❡ s❡❝♦♥❞ ❜❛r②❝❡♥tr✐❝ ❢♦r♠✉❧❛ ❢♦r ✐♥t❡r♣♦❧❛t✐♦♥✱ ❉♦❧♦♠✐t❡s r❡s❡❛r❝❤ ♥♦t❡s ♦♥ ❛♣♣r♦①✐♠❛t✐♦♥ ✈✳ ✼ ✭✷✵✶✹✮ ♣♣✳ ✶✕✶✷✳ • ❈❛♠❛r❣♦✱ ❆✳ P✳✱ ❖♥ t❤❡ ♥✉♠❡r✐❝❛❧ st❛❜✐❧✐t② ♦❢ ❋❧♦❛t❡r✲❍♦r♠❛♥♥✬s r❛t✐♦♥❛❧ ✐♥t❡r♣♦❧❛♥t✱

◆✉♠❡r✐❝❛❧ ❆❧❣♦r✐t❤♠s✱ ❉❖■ ✶✵✳✶✵✵✼✴s✶✶✵✼✺✲✵✶✺✲✵✵✸✼✲③✳

• ❈❛♠❛r❣♦✱ ❆✳ P✳✱ ❊rr❛t✉♠✿ ✏❖♥ t❤❡ ♥✉♠❡r✐❝❛❧ st❛❜✐❧✐t② ♦❢ ❋❧♦❛t❡r✲❍♦r♠❛♥♥✬s r❛t✐♦♥❛❧ ✐♥t❡r✲ ♣♦❧❛♥t✧✱ ◆✉♠❡r✐❝❛❧ ❆❧❣♦r✐t❤♠s✱ ❉❖■ ✶✵✳✶✵✵✼✴s✶✶✵✼✺✲✵✶✺✲✵✵✼✶✲①✳

• ❈❛♠❛r❣♦✱ ❆✳ P✳ ❡ ▼❛s❝❛r❡♥❤❛s✱ ❲✳✱ ❚❤❡ st❛❜✐❧✐t② ♦❢ ❡①t❡♥❞❡❞ ❋❧♦❛t❡r✲❍♦r♠❛♥♥ ✐♥t❡r♣♦✲ ❧❛♥ts✱ ◆✉♠❡r✐s❝❤❡ ▼❛t❤❡♠❛t✐❦✱ s✉❜♠❡t✐❞♦✳ ❛r❳✐✈✿✶✹✵✾✳✷✽✵✽✈✺

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ■♥t❡r♣♦❧❛çã♦✱ ❋ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛✱ ❊st❛❜✐❧✐❞❛❞❡ ♥✉♠ér✐❝❛✳

❆❜str❛❝t

❈❛♠❛r❣♦✱ ❆✳ P✳ ◆✉♠❡r✐❝❛❧ st❛❜✐❧✐t② ♦❢ ❜❛r②❝❡♥tr✐❝ ❢♦r♠✉❧❛❡ ❢♦r ✐♥t❡r♣♦❧❛t✐♦♥✳ ✷✵✶✺✳ ✶✷✵ ❢✳ ❚❡s❡ ✭❉♦✉t♦r❛❞♦✮ ✲ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛✱ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ❙ã♦ P❛✉❧♦✱ ✷✵✶✺✳

❚❤❡ ♣r♦❜❧❡♠ ♦❢ r❡❝♦♥str✉❝t✐♥❣ ❛ ❢✉♥❝t✐♦♥ f : [a, b] _−→ R ❢r♦♠ ❛ ✜♥✐t❡ s❡t ♦❢ ❦♥♦✇♥ ✈❛❧✉❡s

f(x0), f(x1), . . . , f(xn) ❛♣♣❡❛rs ❢r❡q✉❡♥t❧② ✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧✐♥❣✳ ■t ✐s ♥♦t ♣♦ss✐❜❧❡✱ ✐♥ ❣❡♥❡✲ r❛❧✱ t♦ ❝♦♠♣❧❡t❡❧② ❞❡t❡r♠✐♥❡ f ❢r♦♠ f(x0), f(x1), . . . , f(xn) ❜✉t✱ ✐♥ s❡✈❡r❛❧ ❝❛s❡s ♦❢ ✐♥t❡r❡st✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ✜♥❞ r❡❛s♦♥❛❜❧❡ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦rf ❜② ✐♥t❡r♣♦❧❛t✐♦♥✱ ✇❤✐❝❤ ❝♦♥s✐sts ✐♥ ✜♥❞✐♥❣ ❛ s✉✐t❛❜❧❡

❢✉♥❝t✐♦♥ ✭❛ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥✱ ❛ r❛t✐♦♥❛❧ ♦r tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥✱ ❡t❝✳✮ gs✉❝❤ t❤❛t

g(xi) =f(xi), i= 0,1, . . . , n.

■♥ ♣r❛❝t✐❝❡✱ t❤❡ ✐♥t❡r♣♦❧❛t✐♥❣ ❢✉♥❝t✐♦♥g✐s ❡✈❛❧✉❛t❡❞ ✐♥ ✜♥✐t❡ ♣r❡❝✐s✐♦♥ ❛♥❞ t❤❡ ✜♥❛❧ ❝♦♠♣✉t❡❞ ✈❛❧✉❡

d

g(x) ♠❛② ❞✐✛❡r ❢r♦♠ t❤❡ ❡①❛❝t ✈❛❧✉❡g(x)❞✉❡ t♦ r♦✉♥❞✐♥❣✳ ■♥ ❢❛❝t✱ s✉❝❤ ❞✐✛❡r❡♥❝❡ ❝❛♥ ❡✈❡♥ ❡①❝❡❡❞

t❤❡ ✐♥t❡r♣♦❧❛t✐♦♥ ❡rr♦rE(x) =f(x)₋g(x) ✐♥ s❡✈❡r❛❧ ♦r❞❡rs ♦❢ ♠❛❣♥✐t✉❞❡✱ ❝♦♠♣r♦♠✐s✐♥❣ t❤❡ ❡♥t✐r❡

❛♣♣r♦①✐♠❛t✐♦♥ ♣r♦❝❡ss✳ ❚❤❡ ♥✉♠❡r✐❝❛❧ st❛❜✐❧✐t② ♦❢ ❛♥ ❛❧❣♦r✐t❤♠ r❡✢❡❝t ✐s s❡♥s✐❜✐❧✐t② ✇✐t❤ r❡s♣❡❝t t♦ r♦✉♥❞✐♥❣✳ ■♥ t❤✐s ✇♦r❦ ✇❡ ♣r❡s❡♥t ❛ ❞❡t❛✐❧❡❞ ❛♥❛❧②s✐s ♦❢ t❤❡ ♥✉♠❡r✐❝❛❧ st❛❜✐❧✐t② ♦❢ s♦♠❡ ❛❧❣♦r✐t❤♠s ✉s❡❞ t♦ ❡✈❛❧✉❛t❡ ♣♦❧②♥♦♠✐❛❧ ♦r r❛t✐♦♥❛❧ ✐♥t❡r♣♦❧❛♥ts ✇❤✐❝❤ ❝❛♥ ❜❡ ♣✉t ✐♥ t❤❡ ❜❛r②❝❡♥tr✐❝ ❢♦r♠❛t✳

❚❤❡ ♠❛✐♥ r❡s✉❧ts ♦❢ t❤✐s ✇♦r❦ ❛r❡ ❛❧s♦ ❛✈❛✐❧❛❜❧❡ ✐♥ ❡♥❣❧✐s❤ ✐♥ t❤❡ ♣❛♣❡rs

• ▼❛s❝❛r❡♥❤❛s✱ ❲ ❡ ❈❛♠❛r❣♦✱ ❆✳ P✳✱ ❖♥ t❤❡ ❜❛❝❦✇❛r❞ st❛❜✐❧✐t② ♦❢ t❤❡ s❡❝♦♥❞ ❜❛r②❝❡♥tr✐❝ ❢♦r♠✉❧❛ ❢♦r ✐♥t❡r♣♦❧❛t✐♦♥✱ ❉♦❧♦♠✐t❡s r❡s❡❛r❝❤ ♥♦t❡s ♦♥ ❛♣♣r♦①✐♠❛t✐♦♥ ✈✳ ✼ ✭✷✵✶✹✮ ♣♣✳ ✶✕✶✷✳ • ❈❛♠❛r❣♦✱ ❆✳ P✳✱ ❖♥ t❤❡ ♥✉♠❡r✐❝❛❧ st❛❜✐❧✐t② ♦❢ ❋❧♦❛t❡r✲❍♦r♠❛♥♥✬s r❛t✐♦♥❛❧ ✐♥t❡r♣♦❧❛♥t✱

◆✉♠❡r✐❝❛❧ ❆❧❣♦r✐t❤♠s✱ ❉❖■ ✶✵✳✶✵✵✼✴s✶✶✵✼✺✲✵✶✺✲✵✵✸✼✲③✳

• ❈❛♠❛r❣♦✱ ❆✳ P✳✱ ❊rr❛t✉♠✿ ✏❖♥ t❤❡ ♥✉♠❡r✐❝❛❧ st❛❜✐❧✐t② ♦❢ ❋❧♦❛t❡r✲❍♦r♠❛♥♥✬s r❛t✐♦♥❛❧ ✐♥t❡r✲ ♣♦❧❛♥t✧✱ ◆✉♠❡r✐❝❛❧ ❆❧❣♦r✐t❤♠s✱ ❉❖■ ✶✵✳✶✵✵✼✴s✶✶✵✼✺✲✵✶✺✲✵✵✼✶✲①✳

• ❈❛♠❛r❣♦✱ ❆✳ P✳ ❡ ▼❛s❝❛r❡♥❤❛s✱ ❲✳✱ ❚❤❡ st❛❜✐❧✐t② ♦❢ ❡①t❡♥❞❡❞ ❋❧♦❛t❡r✲❍♦r♠❛♥♥ ✐♥t❡r♣♦✲ ❧❛♥ts✱ ◆✉♠❡r✐s❝❤❡ ▼❛t❤❡♠❛t✐❦✱ s✉❜♠✐t❡❞✳ ❛r❳✐✈✿✶✹✵✾✳✷✽✵✽✈✺

❑❡②✇♦r❞s✿ ■♥t❡r♣♦❧❛t✐♦♥✱ ❇❛r②❝❡♥tr✐❝ ❢♦r♠✉❧❛❡✱ ◆✉♠❡r✐❝❛❧ st❛❜✐❧✐t②✳

❙✉♠ár✐♦

▲✐st❛ ❞❡ ❙í♠❜♦❧♦s ✐①

▲✐st❛ ❞❡ ❋✐❣✉r❛s ①✐

▲✐st❛ ❞❡ ❚❛❜❡❧❛s ①✐✐✐

✶ ■♥tr♦❞✉çã♦ ✶

✷ Pr❡❧✐♠✐♥❛r❡s ✸

✷✳✶ ❆ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✷✳✷ ❉❡r✐✈❛❞❛s ❞❡ ✐♥t❡r♣♦❧❛❞♦r❡s ♥❛ ❢♦r♠❛ ❜❛r✐❝ê♥tr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✸ ❋✉♥❞❛♠❡♥t♦s ❞❛ ❛r✐t♠ét✐❝❛ ❞❡ ♣♦♥t♦ ✢✉t✉❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✸ ■♥t❡r♣♦❧❛❞♦r❡s ❡ ❛ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛✿ ❡①❡♠♣❧♦s ✾ ✸✳✶ ■♥t❡r♣♦❧❛❞♦r ♣♦❧✐♥♦♠✐❛❧ ❞❡ ▲❛❣r❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✸✳✶✳✶ ❊rr♦ ❞❡ ✐♥t❡r♣♦❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✸✳✶✳✷ ❆ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ♣❛r❛ ♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ▲❛❣r❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✸✳✶✳✸ ❋❛♠í❧✐❛s ❡s♣❡❝✐❛✐s ❞❡ ♥ós ♣❛r❛ ✐♥t❡r♣♦❧❛çã♦ ♣♦❧✐♥♦♠✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✸✳✶✳✹ ❈♦♥st❛♥t❡s ❞❡ ▲❡❜❡s❣✉❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✸✳✶✳✺ ❆ ✐♥✢✉ê♥❝✐❛ ❞❛ ❝♦♥st❛♥t❡ ❞❡ ▲❡❜❡s❣✉❡ ♥❛ ❡t❛♣❛ ♥✉♠ér✐❝❛ ❞❡ ❛♣r♦①✐♠❛çã♦ ✳ ✳ ✶✺ ✸✳✷ ■♥t❡r♣♦❧❛❞♦r ❞❡ ❋❧♦❛t❡r✲❍♦r♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✷✳✶ ❆ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ♣❛r❛ ♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ❋❧♦❛t❡r✲❍♦r♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✷✳✷ ❆✉sê♥❝✐❛ ❞❡ ♣ó❧♦s r❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✷✳✸ ❊rr♦ ❞❡ ✐♥t❡r♣♦❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✷✳✹ ❈♦♥st❛♥t❡s ❞❡ ▲❡❜❡s❣✉❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✷✳✺ ❙♦❜r❡ ❛ ♠❛❣♥✐t✉❞❡ ❞❛ ❢✉♥çã♦ ❞❡ ▲❡❜❡s❣✉❡ ♣❛r❛ ♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ❋❧♦❛t❡r✲

❍♦r♠❛♥♥ ♥♦ ✐♥t❡r✐♦r ❞♦ ✐♥t❡r✈❛❧♦ ❞❡ ✐♥t❡r♣♦❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✹ ❆ ❡st❛❜✐❧✐❞❛❞❡ ❜❛❝❦✇❛r❞ ❞❛ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ♣❛r❛ ✐♥t❡r♣♦❧❛çã♦ ✷✺ ✹✳✶ ❆ ❡st❛❜✐❧✐❞❛❞❡ ❜❛❝❦✇❛r❞ ❞❛ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ♣❛r❛ ✐♥t❡r♣♦❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✹✳✶✳✶ ❘❡s✉❧t❛❞♦s t❡ór✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✹✳✶✳✷ ❊①♣❡r✐♠❡♥t♦s ♥✉♠ér✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✺ ❆ ❡st❛❜✐❧✐❞❛❞❡ ♥✉♠ér✐❝❛ ❞♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ❋❧♦❛t❡r✲❍♦r♠❛♥♥ ✸✼ ✺✳✶ ❆❧❣♦r✐t♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✺✳✷ ❆♥á❧✐s❡ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ❜❛❝❦✇❛r❞ ❞♦s ❛❧❣♦r✐t♠♦s ❞♦ ❚✐♣♦ ■ ❡ ❞♦ ❚✐♣♦ ■■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

✈✐✐✐ ❙❯▼➪❘■❖

✺✳✷✳✶ ❊rr♦ ♥♦ P❛ss♦ ■■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✺✳✷✳✷ ❊rr♦ ♥♦ P❛ss♦ ■■■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✺✳✸ ❆♥á❧✐s❡ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ❢♦r✇❛r❞ ❞♦s ❛❧❣♦r✐t♠♦s ❞♦ ❚✐♣♦ ■ ❡ ❞♦ ❚✐♣♦ ■■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✺✳✹ ❊①♣❡r✐♠❡♥t♦s ♥✉♠ér✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✺✳✹✳✶ ■♥t❡r♣♦❧❛çã♦ ❞♦s ♣♦❧✐♥ô♠✐♦s ▲❛❣r❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✺✳✹✳✷ ❆✈❛❧✐❛çã♦ ❡stá✈❡❧ ❞❛ ❢✉♥çã♦✴❝♦♥st❛♥t❡ ❞❡ ▲❡❜❡s❣✉❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✺✳✹✳✸ ❋✉♥çõ❡s ♦r❞✐♥ár✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✺✳✹✳✹ ❉✐s❝✉ssã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✻ ❆ ❡st❛❜✐❧✐❞❛❞❡ ♥✉♠ér✐❝❛ ❞♦s ✐♥t❡r♣♦❧❛❞♦r❡s ❞❡ ❋❧♦❛t❡r✲❍♦r♠❛♥♥ ❡st❡♥❞✐❞♦s ✺✶ ✻✳✶ ❉❡✜♥✐çã♦ ❢♦r♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✻✳✷ ❈rít✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✻✳✷✳✶ ■♥t❡r♣r❡t❛çã♦ ✐♥❝♦♠✉♠ ❞❛ ❝♦♥st❛♥t❡ ❞❡ ▲❡❜❡s❣✉❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✻✳✷✳✷ ❊st❛❜✐❧✐❞❛❞❡ ✈❡rs✉s ❝♦♥✈❡r❣ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✻✳✸ ❊st❛❜✐❧✐❞❛❞❡ ♥✉♠ér✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✻✳✸✳✶ ❖ ❝❛s♦ ♠✐♥✐♠❛❧ _d˜_{= ˜n=d ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼}

✻✳✸✳✷ ❈❛s♦ ❣❡r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✻✳✸✳✸ ❯♠❛ ♣r♦♣♦st❛ ♣❛r❛ ♠❡❧❤♦r❛r ❛ ❡st❛❜✐❧✐❞❛❞❡ ♥✉♠ér✐❝❛ ❞♦s ✐♥t❡r♣♦❧❛❞♦r❡s ❞❡

❋❧♦❛t❡r✲❍♦r❛♠♥♥ ❡st❡♥❞✐❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷

✼ ❈♦♥❝❧✉sõ❡s ✻✺

✼✳✶ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✼✳✷ ❙✉❣❡stõ❡s ♣❛r❛ P❡sq✉✐s❛s ❋✉t✉r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✻✼

▲✐st❛ ❞❡ ❙í♠❜♦❧♦s

x ♥ós ❞❡ ✐♥t❡r♣♦❧❛çã♦

y ✈❛❧♦r❡s ✐♥t❡r♣♦❧❛❞♦s

w ♣❡s♦s ❞❡ ✐♥t❡r♣♦❧❛çã♦

n+ 1 ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ♣❛r❛ ✐♥t❡r♣♦❧❛çã♦ f(x) ✐♠❛❣❡♠ ❞♦ ❝♦♥❥✉♥t♦x s♦❜ ❛ ❢✉♥çã♦f

xcheb1 ♥ós ❞❡ ❈❤❡❜②s❤❡✈ ❞♦ ♣r✐♠❡✐r♦ t✐♣♦

xcheb2 ♥ós ❞❡ ❈❤❡❜②s❤❡✈ ❞♦ s❡❣✉♥❞♦ t✐♣♦

xeq ♥ós ✐❣✉❛❧♠❡♥t❡ ❡s♣❛ç❛❞♦s

γ(.) ♣❡s♦s ❞❡ ✐♥t❡r♣♦❧❛çã♦ ♣❛r❛ ♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ▲❛❣r❛♥❣❡

d ♣❛râ♠❡tr♦ q✉❡ ❞❡✜♥❡ ❛ ♦r❞❡♠ ❞❡ ❛♣r♦①✐♠❛çã♦ ❞♦s ✐♥t❡r♣♦❧❛❞♦r❡s ❞❡ ❋❧♦❛t❡r✲❍♦r♠❛♥♥ µd(.) ♣❡s♦s ❞❡ ✐♥t❡r♣♦❧❛çã♦ ♣❛r❛ ♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ❋❧♦❛t❡r✲❍♦r♠❛♥♥

rd(x,x,y) ✐♥t❡r♣♦❧❛❞♦r ❞❡ ❋❧♦❛t❡r✲❍♦r♠❛♥♥

˜

r_d,n,˜d,κ˜ (x,x,y) ✐♥t❡r♣♦❧❛❞♦r ❞❡ ❋❧♦❛t❡r✲❍♦r♠❛♥♥ ❡st❡♥❞✐❞♦ p(x,x,y,w) ♣r✐♠❡✐r❛ ❋ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛

q(x,x,y,w) ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛✴s❡❣✉♥❞❛ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ Leb(x,x,y,w)) ❢✉♥çã♦ ❞❡ ▲❡❜❡s❣✉❡

Λ(x,w) ❝♦♥st❛♥t❡ ❞❡ ▲❡❜❡s❣✉❡

C[a, b] ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ♥♦ ✐♥t❡r✈❛❧♦ [a, b]

||._||∞ ♥♦r♠❛ ❞♦ ❙✉♣r❡♠♦

⌊ ._⌋ ♣❛rt❡ ✐♥t❡✐r❛

χ ❤♦♠❡♦♠♦r✜s♠♦ ❛✜♠ ♣♦r ♣❛rt❡s

ζζζ ✈❡t♦r ❞❡ ❡rr♦s r❡❧❛t✐✈♦s ♥♦s ♣❡s♦s f l(.) ♦♣❡r❛❞♦r ❞❡ ❛rr❡❞♦♥❞❛♠❡♥t♦ cond(x,x,y_b) ♥ú♠❡r♦ ❞❡ ❝♦♥❞✐çã♦

h._i ❝♦♥t❛❞♦r ❞❡ ❙t❡✇❛rt

ǫ ♣r❡❝✐sã♦ ❞❛ ♠áq✉✐♥❛

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✸✳✶ ❋✉♥çõ❡s ❞❡ ▲❡❜❡s❣✉❡ ❛ss♦❝✐❛❞❛s ❛♦s ♥ós (xd7₎∗_,(xd8₎∗ _{❡ (x}d9₎∗_{✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺} ✸✳✷ ❋✉♥çõ❡s ❞❡ ▲❡❜❡s❣✉❡ ✭n= 9✮ ♣❛r❛✿ ♥ós ❞❡ ❈❤❡❜②s❤❡✈ ❞♦ ♣r✐♠❡✐r♦ t✐♣♦ ✭❛✮✱ ♥ós ❞❡

❈❤❡❜②s❤❡✈ ❞♦ s❡❣✉♥❞♦ t✐♣♦ ✭❜✮ ❡ ♥ós ✐❣✉❛❧♠❡♥t❡ ❡s♣❛ç❛❞♦s ✭❝✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✸✳✸ ❊rr♦ t♦t❛❧ ❞❡ ✐♥t❡r♣♦❧❛çã♦ log₁₀ max

x ∈[−5,5]|f(x)−f l(p(x,x,y, γ(x)))| ✭❛✮ ❡ ❡rr♦ ♥❛

s❡❣✉♥❞❛ ❡t❛♣❛ log₁₀ max

x ∈[−5,5]|p(x,ˆx,yˆ, γ(xˆ))−f l(p(x,ˆx,yˆ, γ(xˆ)))| ✭❜✮ ♣❛r❛ ❛ ❢✉♥✲

çã♦ f(x) = sin(x)✱ ♣❛r❛ ♦s ♥ós ❞❡ ❈❤❡❜②s❤❡✈ ❞♦ s❡❣✉♥❞♦ t✐♣♦ x=x(_cheb2n) ❡ ♥ós

✐❣✉❛❧♠❡♥t❡ ❡s♣❛ç❛❞♦s x=x(eqn)

✱ ❛♠❜♦s ❞❡✜♥✐❞♦s ❡♠ [₋5,5]✳ ❆s ❧✐♥❤❛s tr❛❝❡❥❛❞❛s

✐♥❞✐❝❛♠ ♦s ✈❛❧♦r❡s ❞♦ ♣r♦❞✉t♦ nǫΛ (ˆx, γ(xˆ))✱ ❡♠ ❝❛❞❛ ❝❛s♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✸✳✹ ❆s ❝♦♥st❛♥t❡s ❞❡ ▲❡❜❡s❣✉❡ Λ xn_eq, µd(xneq)

❡ Λ (xn_cheb2, µd(xncheb2))✱ ❡♠ ❡s❝❛❧❛ ❧♦✲

❣❛r✐t♠✐❝❛ ✭❜❛s❡= 10✮ ♣❛r❛ n= 50❡1_≤d_≤50✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✹✳✶ ❉❡❝♦♠♣♦s✐çã♦ ❞♦ ❡rr♦ ❞❡ ✐♥t❡r♣♦❧❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✹✳✷ ❊rr♦ r❡❧❛t✐✈♦ ♥♦s ♣❡s♦s ❞❡ ✐♥t❡r♣♦❧❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✹✳✸ ❖ ❡rr♦ ❜❛❝❦✇❛r❞ max

1≤k≤n/2log10(|β0|) ♣❛r❛ ♦s ♣❡s♦swb

num _❡wbsal_{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹}

✹✳✹ ❖ ❡rr♦ ❜❛❝❦✇❛r❞ ♠á①✐♠♦max log₂(_|β0|) s♦❜10002♣♦♥t♦s ✐❣✉❛❧♠❡♥t❡ ❡s♣❛ç❛❞♦s ❡♠

[0, n]✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✺✳✶ ❱❛❧♦r ❛❜s♦❧✉t♦ ❞♦ ❡rr♦ ❜❛❝❦✇❛r❞ ✭❡♠ ❡s❝❛❧❛ ❧♦❣❛r✐t♠✐❝❛✮ max

j ∈ Ilog10(|βj|)♣❛r❛ ✐♥t❡r✲

♣♦❧❛çã♦ ✭❛✮ ❡ ♦ ❡rr♦ ❢♦r✇❛r❞ r❡✈❡rs♦ max

j ∈ Ilog10

|β∗

j|

♣❛r❛ ❡①tr❛♣♦❧❛çã♦ ✭❜✮✱ ✭❝✮ ❡ ✭❞✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✺✳✷ log₁₀L(x,xb, µ(bx)) ❡♠ [₋5,5]✱ ♣❛r❛ d = 5,10,15 ❛♥❞ 20✳ bx ❝♦rr❡s♣♦♥❞❡ à ✈❡rsã♦

❛rr❡❞♦♥❞❛❞❛ ✭❡♠ ♣r❡❝✐sã♦ ❞✉♣❧❛✮ ❞❡ n+ 1 = 52 ♣♦♥t♦s ✐❣✉❛❧♠❡♥t❡ ❡s♣❛ç❛❞♦s ❡♠ [₋1,1]✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽

✺✳✸ max

x∈ [−5,5]|f l(u(x,xb, f(ˆx), µ(bx)))−u(x,bx, f(xˆ), µ(xb))|✱u=p❡u=q✱ ❝♦♠n+1 = 201

♣♦♥t♦s ✐❣✉❛❧♠❡♥t❡ ❡s♣❛ç❛❞♦s ❡♠ [a=₋5, b= 5]✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

✺✳✹ max

x∈ [tk−1,tk]

f lu(u(x,(x,xbxb,f,f(ˆx(ˆx),µ),µ(bx(bx))))) −1

, k = 0,1, . . . ,99 ✭u=p ❡u=q✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✻✳✶ ❆ ❢✉♥çã♦ ❞❡ ▲❡❜❡s❣✉❡ Leb_d,˜n,d,κ˜ (x,x)✱ ♣❛r❛n+ 1 = 51♥ós ✐❣✉❛❧♠❡♥t❡ ❡s♣❛ç❛❞♦s ❡♠

[₋1,1]✱ ❝♦♠ d=κ= 18❡ ˜n= 20✜①♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✻✳✷ ❖ ❡rr♦ ❧♦❣❛r✐t♠✐❝♦log₁₀

max

x ∈ [−5,5]|■(x)−f(x)|

♣❛r❛ ■=❋❍ ❡ ■=❊❋❍ ♣❛r❛n+1 =

103+ 1 ♥ós ✐❣✉❛❧♠❡♥t❡ ❡s♣❛ç❛❞♦s ❡♠ [₋5,5] ✭d˜= ˜n= 3✱κ = d✮✱ ❝♦♠ ♣❡rt✉r❜❛çã♦ ❞❡ 10−10 _{♥♦s ✈❛❧♦r❡s ✐♥t❡r♣♦❧❛❞♦s ✭❝♦♠ s✐♥❛✐s ❛❧t❡r♥❛❞♦s✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻}

①✐✐ ▲■❙❚❆ ❉❊ ❋■●❯❘❆❙

✻✳✸ ❖ ❡rr♦ ❧♦❣❛r✐t♠✐❝♦log₁₀

max

x ∈[−5,5]|■(x)−f(x)|

♣❛r❛ ■=❋❍ ❡ ■=❊❋❍ ♣❛r❛n+1 =

103+ 1♥ós ✐❣✉❛❧♠❡♥t❡ ❡s♣❛ç❛❞♦s ❡♠[₋5,5]✭d˜= ˜n= 3✱κ=d✮✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✻✳✹ ❖ ❡rr♦ ❧♦❣❛r✐t♠✐❝♦log₁₀

max

x ∈[−5,5]|■(x)−f(x)|

♣❛r❛ ■=❋❍ ❡ ■=❊❋❍ ♣❛r❛n+1 =

103+ 1♥ós ✐❣✉❛❧♠❡♥t❡ ❡s♣❛ç❛❞♦s ❡♠[₋5,5]✭κ= ˜d= ˜n=d✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✻✳✺ ❖ ❡rr♦ ❧♦❣❛r✐t♠✐❝♦ log₁₀

max

x ∈ [−5,5]|■(x)−f(x)|

♣❛r❛ ■ =❋❍ ❡ ■ =❊❋❍ ✭κ= ˜d=

˜

n=d✮ ❡n= 103✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

✻✳✻ ❈♦♥st❛♥t❡s ❞❡ ▲❡❜❡s❣✉❡ ♣❛r❛ ♦s ✐♥t❡r♣♦❧❛❞♦r❡s ❞❡ ❋❧♦❛t❡r✲❍♦r♠❛♥♥ ✭❋❍✮✱ ❋❧♦❛t❡r✲ ❍♦r♠❛♥♥ ❡st❡♥❞✐❞♦s ✭❊❋❍✯✮ ❡ ♣❛r❛ ♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ▲❛❣r❛♥❣❡ ✭▲■✮ ❝♦♠d+ 1 ♥ós

✐❣✉❛❧♠❡♥t❡ ❡s♣❛ç❛❞♦s ✭κ= ˜d= ˜n=d✮ ❡n= 103✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷

✻✳✼ ❈♦♥st❛♥t❡s ❞❡ ▲❡❜❡s❣✉❡ ♣❛r❛ ♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ❋❧♦❛t❡r✕❍♦r♠❛♥♥ ❡①t❡♥❞✐❞♦ ❡♠ ❢✉♥✲ çã♦ ❞♦ ♣❛r❛♠❡tr♦κ✱ ❝♦♠ n= 2000 ❡d˜= ˜n=d✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✻✳✽ log₁₀Λ˜_n,d,˜n,d,d˜ (x)

❡♠ ❢✉♥çã♦ ❞❛ ❞✐❢❡r❡♥ç❛n˜₋d˜✱ ❝♦♠ n= 100✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷

✻✳✾ ❖ ❡rr♦ ❧♦❣❛r✐t♠✐❝♦ log₁₀

max

x ∈ [−5,5]|■(x)−f(x)|

♣❛r❛ ■ =❋❍ ❡ ■ =❊❋❍ ✭κ= ˜d=

˜

n=d✮ ❡n= 103✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹

✻✳✶✵ ❖ ❡rr♦ ❧♦❣❛r✐t♠✐❝♦ ♥♦s ✈❛❧♦r❡s ❡①tr❛♣♦❧❛❞♦s ♣❛r❛ d= 43✿log_10|f l(˜y−τ)−y−˜ τ|✭❛✮❀ ❡

log_10|f l(˜yn+τ)−y˜n+τ|✭❜✮✳ ❊♠ ❛♠❜♦s ♦s ♣r♦❝❡ss♦s ❞❡ ❡①tr❛♣♦❧❛çã♦✱ ♦s ♥♦✈♦s ♣♦♥t♦s

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

✸✳✶ ▼❛❧❤❛s ♦t✐♠✐③❛❞❛s ❡♠ [₋1,1]s❡♠ ❡①tr❡♠♦s ✜①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✺✳✶ ❱❛❧♦r❡s ❞❡ max

0≤i≤n

|θi|

n||x−xb||∞(1+log(d))♣❛r❛ ♥ós ✐❣✉❛❧♠❡♥t❡ ❡s♣❛ç❛❞♦s ❞❡✜♥✐❞♦s ❡♠[−1,1]❀ ||x₋bx_||∞:= 10−15✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

✺✳✷ ❱❛❧♦r❡s ❞❡ max

0≤i≤n

|θi|

||x−xb||∞

2

n2 ♣❛r❛ ♦s ♥ós ❞❡ ❈❤❡❜②s❤❡✈ ❞♦ s❡❣✉♥❞♦ t✐♣♦❀||x−xb||∞:=

10−15✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❖ ♣r♦❜❧❡♠❛ ❞❡ r❡❝♦♥str✉✐r ✉♠❛ ❢✉♥çã♦ f : [a, b]_−→ R❛ ♣❛rt✐r ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ✈❛❧♦r❡s ❝♦♥❤❡❝✐❞♦s f(x0), f(x1), . . . , f(xn) ❛♣❛r❡❝❡ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ❡♠ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛✳ ❊♠ ❣❡r❛❧✱ ♥ã♦ é ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛rf ❝♦♠♣❧❡t❛♠❡♥t❡ ❛ ♣❛rt✐r ❞❡f(x0), f(x1), . . . , f(xn)♠❛s✱ ❡♠ ♠✉✐t♦s ❝❛s♦s ❞❡ ✐♥t❡r❡ss❡✱ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ❛♣r♦①✐♠❛çõ❡s r❛③♦á✈❡✐s ♣❛r❛f ✉s❛♥❞♦ ✐♥t❡r♣♦❧❛çã♦✱ q✉❡ ❝♦♥s✐st❡

❡♠ ❞❡t❡r♠✐♥❛r ✉♠❛ ❢✉♥çã♦ ✭✉♠ ♣♦❧✐♥ô♠✐♦✱ ♦✉ ✉♠❛ ❢✉♥çã♦ r❛❝✐♦♥❛❧ ♦✉ tr✐❣♦♥♦♠étr✐❝❛✱ ❡t❝✮ g q✉❡

s❛t✐s❢❛ç❛

g(xi) =f(xi), i= 0,1, . . . , n.

◆❛ ♣rát✐❝❛✱ ❛ ❢✉♥çã♦ ✐♥t❡r♣♦❧❛❞♦r❛ g é ❛✈❛❧✐❛❞❛ ❡♠ ♣r❡❝✐sã♦ ✜♥✐t❛ ❡ ♦ ✈❛❧♦r ✜♥❛❧ ❝♦♠♣✉t❛❞♦ gd(x)

♣♦❞❡ ❞✐❢❡r✐r ❞♦ ✈❛❧♦r ❡①❛t♦g(x) ❞❡✈✐❞♦ ❛ ❡rr♦s ❞❡ ❛rr❡❞♦♥❞❛♠❡♥t♦✳ ❊ss❛ ❞✐❢❡r❡♥ç❛ ♣♦❞❡✱ ✐♥❝❧✉s✐✈❡✱

✉❧tr❛♣❛ss❛r ♦ ❡rr♦ ❞❡ ✐♥t❡r♣♦❧❛çã♦ E(x) = f(x)₋g(x) ❡♠ ✈ár✐❛s ♦r❞❡♥s ❞❡ ♠❛❣♥✐t✉❞❡✱ ❝♦♠♣r♦✲

♠❡t❡♥❞♦ t♦❞♦ ♦ ♣r♦❝❡ss♦ ❞❡ ❛♣r♦①✐♠❛çã♦✳ ❆ ❡st❛❜✐❧✐❞❛❞❡ ♥✉♠ér✐❝❛ ❞❡ ✉♠ ❛❧❣♦r✐t♠♦ r❡✢❡t❡ ❛ s✉❛ s❡♥s✐❜✐❧✐❞❛❞❡ ❡♠ r❡❧❛çã♦ ❛ ❡rr♦s ❞❡ ❛rr❡❞♦♥❞❛♠❡♥t♦✳ ◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❛♥á❧✐s❡ ❞❡t❛❧❤❛❞❛ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ♥✉♠ér✐❝❛ ❞❡ ❛❧❣✉♥s ❛❧❣♦r✐t♠♦s ✉t✐❧✐③❛❞♦s ♥♦ ❝á❧❝✉❧♦ ❞❡ ✐♥t❡r♣♦❧❛❞♦r❡s ♣♦❧✐♥♦♠✐❛✐s ❡ r❛❝✐♦♥❛✐s q✉❡ ♣♦❞❡♠ s❡r ♣♦st♦s ♥❛ ❢♦r♠❛ ❜❛r✐❝ê♥tr✐❝❛

g(x) = Pn

i=0 wi

x−xif(xi)

_n P i=0

wi

x−xi, x /∈ {x0, x1, . . . , xn},

❝♦♠♦ ♦s ✐♥t❡r♣♦❧❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡ ❡ ❞❡ ❋❧♦❛t❡r✲❍♦r♠❛♥♥✳

◆♦ ❈❛♣ít✉❧♦ ✷ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ r❡✈✐sã♦ s♦❜r❡ ♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s r❡❧❛❝✐♦♥❛❞♦s à ✐♥t❡r♣♦✲ ❧❛çã♦ ❜❛r✐❝ê♥tr✐❝❛ ❡ t❛♠❜é♠ s♦❜r❡ ♦s ❢✉♥❞❛♠❡♥t♦s ❞❛ ❛r✐t♠ét✐❝❛ ❞❡ ♣♦♥t♦ ✢✉t✉❛♥t❡ ❡ ♥♦ ❈❛♣ít✉❧♦ ✸ r❡✉♥✐♠♦s ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❝♦♥❤❡❝✐❞♦s s♦❜r❡ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❡ ❡st❛❜✐❧✐❞❛❞❡ ♥✉♠ér✐❝❛ ❞♦s ■♥t❡r♣♦❧❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡ ✭♣♦❧✐♥♦♠✐❛❧✮ ❡ ❞❡ ❋❧♦❛t❡r✲❍♦r♠❛♥♥ ✭r❛❝✐♦♥❛❧✮✳ ❆s ♥♦ss❛s ❝♦♥tr✐❜✉✐çõ❡s sã♦ ❛♣r❡s❡♥t❛❞❛s ♥♦s ❈❛♣ít✉❧♦s✹✱✺ ❡✻✳

◆♦ ❈❛♣ít✉❧♦ ✹❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❛♥á❧✐s❡ ❞❛ s❡♥s✐❜✐❧✐❞❛❞❡ ❞❛ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ❣❡♥ér✐❝❛ ❝♦♠ r❡❧❛çã♦ à ♣❡rt✉r❜❛çõ❡s ❞♦s s❡✉s ♣❛râ♠❡tr♦s✿ ♥ós ❞❡ ✐♥t❡r♣♦❧❛çã♦✱ ✈❛❧♦r❡s ✐♥t❡r♣♦❧❛❞♦s ❡ ♣❡s♦s ❞❡ ✐♥t❡r♣♦❧❛çã♦✳ ▼♦str❛♠♦s✱ t❛♠❜é♠✱ q✉❡ ❛ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ♣♦ss✉✐ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ❜❛❝❦✇❛r❞ q✉❛♥❞♦ ❛ ❝♦♥st❛♥t❡ ❞❡ ▲❡❜❡s❣✉❡ ❛ss♦❝✐❛❞❛ ❛♦s ♥ós ❞❡ ✐♥t❡r♣♦❧❛çã♦ é ♣❡q✉❡♥❛✱ ✐st♦ é✿ q✉❡ ♦ ✈❛❧♦r ❝♦♠♣✉t❛❞♦ ❞❛ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ✭❡♠ ❛r✐t♠ét✐❝❛ ❞❡ ♣r❡❝✐sã♦ ✜♥✐t❛✮ ❝♦rr❡s♣♦♥❞❡ ❛♦ ✈❛❧♦r ❡①❛t♦ ❞❛ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ❝♦♠ ♣❛râ♠❡tr♦s ✭♥ós✱ ✈❛❧♦r❡s ❡ ♣❡s♦s✮ ♣❡rt✉r❜❛❞♦s✳ ❊ss❡ r❡s✉❧t❛❞♦ ❣❡♥❡r❛❧✐③❛ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❡♠ ❬▼❛s✶✹❪ ♣❛r❛ ✐♥t❡r♣♦❧❛çã♦ ♥♦s ♥ós ❞❡ ❈❤❡❜②s❤❡✈ ❞♦ s❡❣✉♥❞♦ t✐♣♦✳

◆♦ ❈❛♣ít✉❧♦ ✺ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❝❛❧❝✉❧❛r ♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ❋❧♦❛t❡r✲❍♦r♠❛♥♥ q✉❡ ♣♦ss✉✐ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ❜❛❝❦✇❛r❞ s♦❜r❡ t♦❞❛ ❛ r❡t❛ r❡❛❧✳ ❊ss❡ ❛❧❣♦r✐t♠♦ é ❜❛s❡❛❞♦ ❡♠ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦✱ ♣❛r❛ ♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ❋❧♦❛t❡r✲❍♦r♠❛♥♥✱ ❞❛ ♣r✐♠❡✐r❛ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ♣❛r❛ ♦ ✐♥t❡r♣♦❧❛❞♦r ♣♦❧✐♥♦♠✐❛❧ ❞❡ ▲❛❣r❛♥❣❡✳ ❆ ♥♦ss❛ ❛♥á❧✐s❡ ❣❡♥❡r❛❧✐③❛ ♦s r❡s✉❧t❛❞♦s ❞❡ ❬❍✐❣✵✹❪ s♦❜r❡ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❛ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ♣❛r❛ ♦ ✐♥t❡r♣♦❧❛❞♦r ♣♦❧✐♥♦♠✐❛❧ ❞❡ ▲❛❣r❛♥❣❡✳ ❚❛♠❜é♠ ✉t✐❧✐③❛♠♦s ♦s r❡s✉❧t❛❞♦s ❞♦ ❈❛♣ít✉❧♦ ✹ ♣❛r❛ ❢❛③❡r ✉♠❛ ❛♥á❧✐s❡ ❞❡t❛❧❤❛❞❛ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ♥✉♠ér✐❝❛

✷ ■◆❚❘❖❉❯➬➹❖ ✶✳✵

❞❛ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ♣❛r❛ ♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ❋❧♦❛t❡r✲❍♦r♠❛♥♥✳ ❉✐✈❡rs♦s ❡①♣❡r✐♠❡♥t♦s ♥✉♠ér✐❝♦s sã♦ ❛♣r❡s❡♥t❛❞♦s ♣❛r❛ ❝♦♥s♦❧✐❞❛r ♦s r❡s✉❧t❛❞♦s t❡ór✐❝♦s✳

❈❛♣ít✉❧♦ ✷

Pr❡❧✐♠✐♥❛r❡s

✷✳✶ ❆ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛

❆ ❢ó♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ♣❛r❛ ✐♥t❡r♣♦❧❛çã♦ ❞♦ ✈❡t♦r r❡❛❧ y= (y0, y1, . . . , yn)✱ ❛ss♦❝✐❛❞❛ ❛♦s ♥ós ❞❡ ✐♥t❡r♣♦❧❛çã♦ x = (x0, x1, . . . , xn), a ≤ x0 < x1 < . . . xn ≤ b✱ ❡ ❛♦s ♣❡s♦s w= (w0, w1, . . . , wn) é ❞❛❞❛ ♣♦r

q(x,x,y,w) =

    

n P i=0

wiyi

x−xi

_n P i=0

wi

x−xi , ♣❛r❛x ∈ [a, b]/{x0, x1, . . . , xn}. yi, ♣❛r❛x = xi.

✭✷✳✶✮

❊✈❡♥t✉❛❧♠❡♥t❡ t❛♠❜é♠ ❡s❝r❡✈❡r❡♠♦sx=xn= (xn

0, xn1, . . . , xnn)♣❛r❛ ❞❡♥♦t❛r q✉❡ x❞❡♣❡♥❞❡ ❞❡ n✳ ❉✉r❛♥t❡ ♦ t❡①t♦✱ ❛ss✉♠✐r❡♠♦s q✉❡ ♦s ♣❡s♦s ❞❡ ✐♥t❡r♣♦❧❛çã♦ sã♦ ♥ã♦ ♥✉❧♦s✱ ✐st♦ é

wi 6= 0, i= 0,1, . . . , n. ❆ss✉♠✐r❡♠♦s✱ t❛♠❜é♠✱ q✉❡

n X

i=0 wi

x₋xi 6

= 0, _∀x _∈ [a, b]/_{x0, x1, . . . , xn}. ✭✷✳✷✮ ❉❡ss❛ ❢♦r♠❛✱ ❛ ❡①♣r❡ssã♦ ✭✷✳✶✮ ❞❡✜♥❡ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡♠ [a, b] q✉❡ ✐♥t❡r♣♦❧❛ y ❡♠ x ❡ ♦

♦♣❡r❛❞♦r ❧✐♥❡❛r

y_7−→Φ q(.,x,y,w)_∈C[a, b], ✭✷✳✸✮

❡♥tr❡ ♦s ❡s♣❛ç♦s ♥♦r♠❛❞♦s(Rn+1_,||.||

∞) ❡(C[a, b],||.||∞)✱ ❡stá ❜❡♠ ❞❡✜♥✐❞♦✳

◗✉❛♥❞♦ ♦s ✈❛❧♦r❡s ✐♥t❡r♣♦❧❛❞♦s y ♣r♦✈é♠ ❞❡ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛f _∈ C[a, b]✱ ✐st♦ é✱ yi=f(xi), i= 0,1, . . . , n,

❛ tr❛♥s❢♦r♠❛çã♦ Φt❛♠❜é♠ ♣♦❞❡ s❡r ✐♥t❡r♣r❡t❛❞❛ ❝♦♠♦ ♦ ♦♣❡r❛❞♦r ❧✐♥❡❛r

f _7−→q(.,x, f(x),w)_∈C[a, b], f(x) := (f(x0), f(x1), . . . , f(xn)). ✭✷✳✹✮ ➱ ❢á❝✐❧ ✈❡r q✉❡ ❛♠❜♦s ♦s ♦♣❡r❛❞♦r❡s ❞❡✜♥✐❞♦s ♣♦r ✭✷✳✸✮ ❡ ✭✷✳✹✮ ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ♥♦r♠❛

Λ(x,w) := max

||y||∞≤1||

q(.,x,y,w)_||∞ = max ||f||∞≤1||

q(.,x, f(x),w)_||∞ ✭✷✳✺✮ ❝♦♠ r❡❧❛çã♦ à ♥♦r♠❛ ❞♦ s✉♣r❡♠♦||._||∞❞❡✜♥✐❞❛ ❡♠C[a, b]❡ ❡♠Rn+1✳ ❖ ♥ú♠❡r♦Λ(x,w)é ❤✐st♦r✐❝❛✲ ♠❡♥t❡ ❞❡♥♦♠✐♥❛❞♦ ❛ ❝♦♥st❛♥t❡ ❞❡ ▲❡❜❡s❣✉❡ ❞♦ ✐♥t❡r♣♦❧❛❞♦r ✭✷✳✶✮ ❬❇▼❍❑✶✷❪✱ ❬❇▼❍❙✶✸❪✱ ❬▼❞❈✶✹❪✳ ❊✈❡♥t✉❛❧♠❡♥t❡ t❛♠❜é♠ ❡s❝r❡✈❡r❡♠♦s Λ(x,w)_|b_a ♣❛r❛ ❡①♣❧✐❝✐t❛r ♦ ✐♥t❡r✈❛❧♦ [a, b] ❞❡ r❡❢❡rê♥❝✐❛ ❡♠

✹ P❘❊▲■▼■◆❆❘❊❙ ✷✳✷

q✉❡stã♦✳ ◆♦t❡ q✉❡ ❛ ❝♦♥st❛♥t❡ ❞❡ ▲❡❜❡s❣✉❡ t❛♠❜é♠ ❝♦rr❡s♣♦♥❞❡ ❛♦ ✈❛❧♦r ♠á①✐♠♦ ❞❛ ❢✉♥çã♦ ❞❡ ▲❡❜❡s❣✉❡Leb(x,x,y,w) =

max

||y||∞≤1|

q(x,x,y,w)_| =

    

1, s❡x _{∈ {}x0, x1, . . . , xn}, n P i=0 wi

x−xi

n P i=0 wi

x−xi

, ❝❛s♦ ❝♦♥trár✐♦,

✭✷✳✻✮

♣♦✐s

max

||y||∞≤1

max

x ∈ [a,b]|q(x,x,y,w)|

= max

x ∈ [a,b]

max

||y||∞≤1|

q(x,x,y,w)_|

.

❆ ❝♦♥st❛♥t❡ ❞❡ ▲❡❜❡s❣✉❡ ♣♦ss✉✐ ✉♠ ♣❛♣❡❧ ❢✉♥❞❛♠❡♥t❛❧ ♥♦ ❡st✉❞♦ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ♥✉♠ér✐❝❛ ❞❛ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ✭✷✳✶✮✱ ♣♦✐s ❡❧❛ ♠❡❞❡ ❛ s❡♥s✐❜✐❧✐❞❛❞❡ ❞♦ ✐♥t❡r♣♦❧❛❞♦rq❝♦♠ r❡❧❛çã♦ à ♣❡rt✉r❜❛çõ❡s

♥♦s ✈❛❧♦r❡s ✐♥t❡r♣♦❧❛❞♦sy✳ ❉❡ ❢❛t♦✱ ♣❛r❛ y1,y2 ∈ Rn+1✱ t❡♠♦s

||q(.,x,y2,w)−q(.,x,y1,w)||∞≤Λ(x,w) ||y2−y1||∞. ✭✷✳✼✮ ❆ ❝♦♥st❛♥t❡ ❞❡ ▲❡❜❡s❣✉❡ t❛♠❜é♠ é út✐❧ ♣❛r❛ ♠❡❞✐r ❛ t❛①❛ ❞❡ ❛♣r♦①✐♠❛çã♦ ❞❡ q(.,x, f(x),w)✱

♣❛r❛ ✉♠❛ ❢✉♥çã♦ ♣❛rt✐❝✉❧❛r f✱ ❡♠ r❡❧❛çã♦ à ♠❡❧❤♦r ❛♣♣r♦①✐♠❛çã♦ q(.,x,v∗,w) ♣❛r❛ f ❞❡♥tr♦ ❞♦

s✉❜❡s♣❛ç♦ Φ(Rn+1_{)✳ ❉❡ ❢❛t♦✱ s❡❣✉❡ ❞✐r❡t♦ ❞❡ ✭✷✳✼✮ q✉❡}

||f ₋q(.,x, f(x),w)_||∞ ≤ ||f−q(.,x,v∗,w)||∞ + ||q(.,x,v∗,w)−q(.,x, f(x),w)||∞ ≤ ||f₋q(.,x,v∗,w)_||∞ + Λ(x,w)||v∗−f(x)||∞

≤ (1 + Λ(x,w)) _||f₋q(.,x,v∗,w)_||∞. ✭✷✳✽✮ ❖❜s❡r✈❛çã♦ ✶✳ ❆ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✽✮ ✈❛❧❡✱ ❞❡ ❢❛t♦✱ ♣❛r❛ q✉❛❧q✉❡r ❛♣r♦①✐♠❛çã♦ q(.,x,v,w) _∈ Φ(Rn+1_)✳

P❛r❛ ♦s ♣r✐♥❝✐♣❛✐s ✐♥t❡r♣♦❧❛❞♦r❡s ❝♦♥s✐❞❡r❛❞♦s ♥❡ss❡ tr❛❜❛❧❤♦ ✭✐♥t❡r♣♦❧❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡ ❡ ❞❡ ❋❧♦❛t❡r✲❍♦r♠❛♥♥✮✱ ♦s ♣❡s♦s ❞❡ ✐♥t❡r♣♦❧❛çã♦w(x) sã♦ ❢✉♥çõ❡s ❤♦♠♦❣ê♥❡❛s ❞❡x✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡

w(ϕ(x)) =αkw(x), ϕ(x) := (ϕ(x0), ϕ(x1), . . . , ϕ(xn)), ✭✷✳✾✮ ♣❛r❛ t♦❞❛ ❢✉♥çã♦ ❛✜♠ ✐♥✈❡rtí✈❡❧ ϕ: [a, b]_→[c, d], x _7−→ϕ αx+β, ♦♥❞❡ké ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦

♥ã♦ ♥❡❣❛t✐✈♦ q✉❡ ❞❡♣❡♥❞❡ ❞❡ w✱ ♠❛s ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ ϕ✳ ❙♦❜ ❡ss❛ ❝♦♥❞✐çã♦✱ ♣❛r❛ g: [c, d]_−→R ❡

x′ =ϕ(x)✱ t❡♠♦s

q(u,x′, g(x′),w(x′)) =

n P i=0

w′ ig(x′i)

u−x′ i _n P i=0 w′ i

u−x′

i =

n P i=0

αk_w ig(x′i)

ϕ(ϕ−1_{(u))−ϕ(xi) n}

P i=0

αk_w i

ϕ(ϕ−1_{(u))−ϕ(xi)}

= Pn

i=0 αk_w

ig(x′i)

α(ϕ−1_{(u)−xi) n}

P i=0

αk_w i

α(ϕ−1_(u)−xi) = n P i=0

wig(x′i)

ϕ−1_{(u)−xi n}

P i=0

wi

ϕ−1_(u)−xi

= q(ϕ−1_{(u),x, g(ϕ(x)),w(x)). ✭✷✳✶✵✮} P♦rt❛♥t♦✱ s♦❜ ✭✷✳✾✮✱ ❛ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ✭✷✳✶✮ ♣❛r❛ x′ ❡ g é ❢❛❝✐❧♠❡♥t❡ ♦❜t✐❞❛ ❝♦♠♣♦♥❞♦✲s❡ ❛

❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ♣❛r❛ x =ϕ−1(x′) ❡ ✭g_◦ϕ✮ ❝♦♠ ❛ tr❛♥s❢♦r♠❛çã♦ ❛✜♠ ϕ−1✳ P♦r ❡ss❡ ♠♦t✐✈♦✱

♠✉✐t❛s ✈❡③❡s ♦s r❡s✉❧t❛❞♦s r❡❢❡r❡♥t❡s à ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ✭✷✳✶✮ sã♦ ❡♥✉♥❝✐❛❞♦s ♣❛r❛ ♦s ✐♥t❡r✈❛❧♦s ❞❡ ✐♥t❡r♣♦❧❛çã♦ ♣❛❞rã♦[₋1,1]♦✉ [0,1]✳ P♦r ❡①❡♠♣❧♦✱ s❡❣✉❡ ❞❡ ✭✷✳✶✵✮ q✉❡ Λ(ϕ(x),w(ϕ(x)))_|d_c =

max

||g||∞≤1||

q(.,x′, g(x′),w(x′))_||∞ = max ||g◦ϕ||∞≤1||

✷✳✸ ❉❊❘■❱❆❉❆❙ ❉❊ ■◆❚❊❘P❖▲❆❉❖❘❊❙ ◆❆ ❋❖❘▼❆ ❇❆❘■❈✃◆❚❘■❈❆ ✺

✷✳✷ ❉❡r✐✈❛❞❛s ❞❡ ✐♥t❡r♣♦❧❛❞♦r❡s ♥❛ ❢♦r♠❛ ❜❛r✐❝ê♥tr✐❝❛

❙♦❜ ❛ ❤✐♣ót❡s❡ ✭✷✳✷✮✱ ❛s s✐♥❣✉❧❛r✐❞❛❞❡sx0, x1, . . . , xn❞♦ ✐♥t❡r♣♦❧❛❞♦r ✭✷✳✶✮ sã♦ r❡♠♦✈í✈❡✐s ❡ t❡♠♦s ✉♠❛ ❢✉♥çã♦ ❛♥❛❧ít✐❝❛ ❡♠ ✉♠ ❡♥t♦r♥♦ ❞❛ r❡t❛ r❡❛❧✶_{✳ ❆s ❞❡r✐✈❛❞❛s ❞❡q(x,x,y,w) sã♦ ❧✐♥❡❛r❡s ❡♠y✿}

∂kq(x,x,y,w)

∂xk =

n X

i=0

∂kq x,x,e(i),w

∂xk

!

yi, ✭✷✳✶✶✮

❝♦♠e(i)= (0, . . . ,0,

♣♦s✐çã♦ ✐ z}|{

1 ,0. . . ,0), i= 0,1, . . . , n.

◆❡st❡ tr❛❜❛❧❤♦ ♥ã♦ ❡st✉❞❛♠♦s ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ss❛s ❞❡r✐✈❛❞❛s✱ ♣♦ré♠ ♣r❡❝✐s❛♠♦s ❞❡ s❡✉s ✈❛❧♦r❡s ♥♦s ♥ós ❞❡ ✐♥t❡r♣♦❧❛çã♦ ♣❛r❛ ❞❡✜♥✐r ♦s ✐♥t❡r♣♦❧❛❞♦r❡s ❞❡ ❋❧♦❛t❡r✲❍♦r❛♠♥♥ ❡st❡♥❞✐❞♦s ♥♦ ❈❛♣ít✉❧♦✻✳ ❇❛s❡❛❞♦s ♥❛ Pr♦♣♦s✐çã♦ ✶✷ ❞❡ ❬❙❲✽✻❪✱ ❑❧❡✐♥ ❡ ❇❡rr✉t ❬❑❇✶✷❪ ❛♣r❡s❡♥t❛r❛♠ ✉♠❛ ❢ór♠✉❧❛ ❞❡ r❡❝♦rrê♥❝✐❛ s✐♠♣❧❡s✷ _{♣❛r❛ ❝❛❧❝✉❧❛r ♦s ❝♦❡✜❝✐❡♥t❡s D}(k)

j,i := ∂k_q(x

j,x,e(i),w)

∂xk ❡♠ ✭✷✳✶✶✮✿

D(0)_j,i =

1, s❡i=j

0, s❡i₆=j ❡✱ ♣❛r❛ k≥1, D (k) j,i =

  

wi

wjD

(k−1) j,j −D

(k−1)

j,i , s❡i6=j −P

ℓ6=j

D_j,ℓ(k), s❡i=j. ✭✷✳✶✷✮

❚❡♠♦s✱ ♣♦rt❛♥t♦✱ q✉❡

∂k_q(x

j,x,y,w)

∂xk =

n X

i=0

D_j,i(k) yi. ✭✷✳✶✸✮

✷✳✸ ❋✉♥❞❛♠❡♥t♦s ❞❛ ❛r✐t♠ét✐❝❛ ❞❡ ♣♦♥t♦ ✢✉t✉❛♥t❡

P❛r❛ ❝♦♠♣r❡❡♥❞❡r ♦s ❡❢❡✐t♦s ❞♦s ❡rr♦s ❞❡ ❛rr❡❞♦♥❞❛♠❡♥t♦ ♥❛ ❛✈❛❧✐❛çã♦ ♥✉♠ér✐❝❛ ❞❡ ❡①♣r❡ssõ❡s ♠❛t❡♠át✐❝❛s✱ é ♥❡❝❡ssár✐♦ ❝♦♥❤❡❝❡r ❛s ♣r♦♣r✐❡❞❛❞❡s ❢✉♥❞❛♠❡♥t❛✐s ❞♦ s✐st❡♠❛ ❞❡ ♥ú♠❡r♦s ❞❡ ♣♦♥t♦ ✢✉t✉❛♥t❡✳ ❖ ♥♦ss♦ tr❛❜❛❧❤♦ ❜❛s❡✐❛✲s❡ ♥♦ ♠♦❞❡❧♦ ♣❛❞rã♦ ❛♣r❡s❡♥t❛❞♦ ❡♠ ❬❍✐❣✵✷❪✱ s♦❜ ♦ q✉❛❧ ❡stã♦ ❛❧✐❝❡rç❛❞❛s ❛s ❛r✐t♠ét✐❝❛s ❞❡ ♣♦♥t♦ ✢✉t✉❛♥t❡ ✉t✐❧✐③❛❞❛s ♣♦r ❞✐✈❡rs❛s ❧✐♥❣✉❛❣❡♥s ❞❡ ♣r♦❣r❛çã♦ ❝♦♠♦ ❈✱ ❈✰✰✱ ❏❛✈❛ ❡ ❋♦rtr❛♥✱ ❡ t❛♠❜é♠ ❛ ♠❛✐♦r✐❛ ❞♦s ♠✐❝r♦♣r♦❝❡ss❛❞♦r❡s ❛t✉❛✐s✳ ◆❡ss❡ ♠♦❞❡❧♦✱ ♦ s✐st❡♠❛ ❞❡ ♥ú♠❡r♦s ❞❡ ♣♦♥t♦ ✢✉t✉❛♥t❡F é ❞❡✜♥✐❞♦ ❝♦♠♦ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❛ r❡t❛ r❡❛❧ ❝✉❥♦s ❡❧❡♠❡♥t♦s

y=_±m_×βe−t

sã♦ ❝❛r❛❝t❡r✐③❛❞♦s ♣♦r ✹ ♣❛râ♠❡tr♦s ✐♥t❡✐r♦s • ❛ ❜❛s❡ β✱

• ❛ ♣r❡❝✐sã♦t

• ❡ ♦ ✐♥t❡r✈❛❧♦emin≤e≤emax ♣❛r❛ ♦ ❡①♣♦❡♥t❡✳

❆ ♠❛♥t✐ss❛ m é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ t❛❧ q✉❡ 0 _≤m < βt _{❡ m ≥ β}t−1 _{♣❛r❛ y 6= 0 ❡ e > emin✳ ❊ss❛} ú❧t✐♠❛ ❝♦♥❞✐çã♦ ❣❛r❛♥t❡ ❛ ✉♥✐❝✐❞❛❞❡ ❞❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❝❛❞❛ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦ ✢✉t✉❛♥t❡ ❡♠F✳ ❖ t✐♣♦ ❞♦✉❜❧❡ ✭■❊❊❊ ✼✺✹✮ ❞❡✜♥✐❞♦ ❡♠ ❈✱❈✰✰ ❡ ❏❛✈❛ ❝♦rr❡s♣♦♥❞❡ ❛ β = 2✱t = 53✱ emin =−1021 ❡

emax= 1024 ✭❬❍✐❣✵✷❪✱ ♣✳ ✸✼✳✮

P❛r❛ ✉♠❛ ❡①♣r❡ssã♦ ♠❛t❡♠át✐❝❛ expr = expr(α1, α2, . . . , αk)✱ ❢✉♥çã♦ ❞♦s ♣❛râ♠❡tr♦s r❡❛✐s

α1, α2, . . . , αk✱ ❞❡♥♦t❛r❡♠♦s ♣♦rf l(expr)✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡expr[✱ ♦ ❡❧❡♠❡♥t♦ ❞❡F ♦❜t✐❞♦ ♣❡❧❛ ❛✈❛✲ ❧✐❛çã♦ ♥✉♠ér✐❝❛ ❞❡ expr s❡❣✉♥❞♦ ❛s r❡❣r❛s ❞❡ ❛rr❡❞♦♥❞❛♠❡♥t♦ q✉❡ ❞❡✜♥❡♠ ❛ ❛r✐t♠ét✐❝❛ ❞❡ ♣♦♥t♦

✢✉t✉❛♥t❡ ❡♠ q✉❡stã♦✳ P♦r ❡①❡♠♣❧♦✱

✶_{❈♦♠♦ ❛ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ r❡♣r❡s❡♥t❛ s❡♠♣r❡ ✉♠ ✐♥t❡r♣♦❧❛❞♦r r❛❝✐♦♥❛❧✱ ❡♥tã♦ ❡❧❛ ♣♦ss✉✐ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡}

♣ó❧♦s✳

✻ P❘❊▲■▼■◆❆❘❊❙ ✷✳✸

f l(α1+ (α2+α3)), f l(f l(α1) +f l(α2+α3)), ❡ f l(f l(α1) +f l(f l(α2) +f l(α3)))

❞❡♥♦t❛♠ ♦ ♠❡s♠♦ ❡❧❡♠❡♥t♦ ❞❡ F✳ ◆❛ t❡r❝❡✐r❛ ❡①♣r❡ssã♦ ❛❝✐♠❛✱ t♦❞♦s ♦s ❛rr❡❞♦♥❞❛♠❡♥t♦s ♥❛ ❛✈❛✲ ❧✐❛çã♦ ♥✉♠ér✐❝❛ ❞❡ α1+ (α2+α3) ❡stã♦ ❡①♣❧í❝✐t♦s ❡✱ ♥❛s ❞✉❛s ♣r✐♠❡✐r❛s✱ ❛❧❣✉♥s ❡stã♦ ✐♠♣❧í❝✐t♦s✳

◆❛ ♥♦ss❛ ❛♥á❧✐s❡ ✐r❡♠♦s ❝♦♥s✐❞❡r❛r ♦ ♠♦❞❡❧♦ ❞❡ ❛r✐t♠ét✐❝❛ ❞❡ ♣♦♥t♦ ✢✉t✉❛♥t❡ ♣❛❞rã♦✸ _{❞❡s❝r✐t♦}

♥❛ ♣á❣✐♥❛ ✹✵ ❞❡ ❬❍✐❣✵✷❪

f l(w op z) = (w op z)(1 +δ), _|δ_{| ≤}ǫ, op= +,₋,_∗, /, _∀w, z _{∈ F}, ✭✷✳✶✹✮

♦♥❞❡ ǫ = β1−t _{❞❡♥♦t❛ ❛ ♣r❡❝✐sã♦ ❞♦ s✐st❡♠❛ F✱ ♦✉ ♣r❡❝✐sã♦ ❞❛ ♠áq✉✐♥❛ ✭♥ã♦ ❝♦♥❢✉♥❞✐r ❝♦♠ ♦} ♣❛râ♠❡tr♦ ♣r❡❝✐sã♦ t✳✮ P❛r❛ ❛ ❛r✐t♠ét✐❝❛ ❞❡ ♣r❡❝✐sã♦ ❞✉♣❧❛ ✭t✐♣♦ ❞♦✉❜❧❡✮✱ ♣♦r ❡①❡♠♣❧♦✱ t❡♠♦s ǫ= 2−52_≈2.22_×10−16✳

❆ ♣r✐♥❝✐♣❛❧ ❢❡rr❛♠❡♥t❛ ♣❛r❛ ❛♥❛❧✐s❛r ❛ ♣r♦♣❛❣❛çã♦ ❞❡ ❡rr♦s ❞❡ ❛rr❡❞♦♥❞❛♠❡♥t♦ é ♦ ❝♦♥t❛❞♦r ❞❡ ❡rr♦ ❞❡ ❙t❡✇❛rt ❬❍✐❣✵✷❪

hk_i =

k Y

i=1

(1 +ξi)ρi, ❝♦♠ ρi =±1 ❡ |ξi| ≤ǫ. ✭✷✳✶✺✮ ❊♠ ✈✐st❛ ❞❡ ✭✷✳✶✺✮✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ✭✷✳✶✹✮ ❝♦♠♦

f l(w op z) = (w op z)_h1_i, op= +,₋,_∗, /. ✭✷✳✶✻✮

➱ ❝♦♥✈❡♥✐❡♥t❡✱ t❛♠❜é♠✱ ❡s❝r❡✈❡rhk_iu q✉❛♥❞♦ ❞❡s❡❥❛♠♦s ✐♥❞❡①❛r ♣♦ru✱k❡rr♦s ❞❡ ❛rr❡❞♦♥❞❛♠❡♥t♦

❡s♣❡❝í✜❝♦s✳ ❆s ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❞♦ ❝♦♥t❛❞♦r ❞❡ ❡rr♦ ❞❡ ❙t❡✇❛rt ♣♦❞❡♠ s❡r r❡s✉♠✐❞❛s ♣♦r

1

hk_iu =hkiw, hk1iu1hk2iu2 =hk1+k2iv, ✭✷✳✶✼✮

s❡ k1 ≤k2 ❡♥tã♦ hk1iu =hk2iv ✭✷✳✶✽✮

❡ ✭❧❡♠❛ ✸✳✶ ❞❡ ❬❍✐❣✵✷❪✮

s❡ kǫ <0.001 ❡♥tã♦ _|hk_{i −}1_{| ≤} kǫ

1₋kǫ ≤1.01kǫ. ✭✷✳✶✾✮

P♦r ❡①❡♠♣❧♦✱ s✉♣♦♥❤❛ q✉❡ ❞❡s❡❥❛♠♦s ❛✈❛❧✐❛r ♦ ♣r♦❞✉t♦(w₋z)(r₋s)♥✉♠❡r✐❝❛♠❡♥t❡✱ ❝♦♠w, z, r, s _∈ F✳ P♦r ✭✷✳✶✻✮✱ t❡♠♦s q✉❡

f l(w₋z) = (w₋z)_h1_i1 f l(r₋s) = (r₋s)_h1_i2 ❡✱ ♣♦rt❛♥t♦✱

f l((w₋z)(r₋s)) = f l((w₋z)(r₋s)_h1_i1h1i2)

✭✷✳✶✻✮

= (w₋z)(r₋s)_h1_i1h1i2h1i3

✭✷✳✶✼✮

= (w₋z)(r₋s)_h3_i1,

❝♦♠ |h3_i1−1| ≤ 3.03ǫ✱ s❡ 3ǫ < 0.001✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♦ ❡rr♦ r❡❧❛t✐✈♦

_{f l((w−z)(r−s))} (w−z)(r−s) −1

❡♥tr❡ ♦ ✈❛❧♦r ❝♦♠♣✉t❛❞♦ ❡ ♦ ✈❛❧♦r ❡①❛t♦ ❞❡(w₋z)(r₋s) ♣♦ss✉✐ ✈❛❧♦r ❛❜s♦❧✉t♦ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ 3.03ǫ✳ ◆♦ ❝❛s♦ ❣❡r❛❧✱ é ♣♦ssí✈❡❧ ♠♦str❛r q✉❡✱ s❡ wi, zi, rj, sj ∈ F, i≤p, j≤q✱ ❡♥tã♦

✸_{❆ ❡q✉❛çã♦ ✭✷✳✹✮ ❞❛ ♣á❣✐♥❛ ✹✵ ❞❡ ❬❍✐❣✵✷❪ ✉t✐❧✐③❛u=} 1

✷✳✸ ❋❯◆❉❆▼❊◆❚❖❙ ❉❆ ❆❘■❚▼➱❚■❈❆ ❉❊ P❖◆❚❖ ❋▲❯❚❯❆◆❚❊ ✼ f l      p Q i=1

(wi±zi) q

Q j=1

(rj±sj)     =      p Q i=1

(wi±zi) q

Q j=1

(rj±sj)    

α, ❝♦♠ α=   

h2p+ 2q₋1_i s❡p_≥1 ❡q _≥1,

h2p₋1_i s❡p_≥1 ❡q = 0,

h2q_i s❡p= 0 ❡q _≥1.

✭✷✳✷✵✮

P♦r ✭✷✳✶✼✮ ❡ ✭✷✳✶✽✮✱ t❛♠❜é♠ é ❢á❝✐❧ ❝♦♥st❛t❛r q✉❡✿ s❡ ✉♠❛ s♦♠❛ ❞❡ m+ 1 ♥ú♠❡r♦s ❞❡ ♣♦♥t♦

✢✉t✉❛♥t❡a0, a1, . . . , amé ❛✈❛❧✐❛❞❛ ❞❡ ❢♦r♠❛ r❡❝✉rs✐✈❛ _r+1

P i=0

ai = _r

P i=0

ai

+ar+1

✱ ❡♥tã♦ ♦ ✈❛❧♦r ✜♥❛❧ ❝♦♠♣✉t❛❞♦ ❞❛ s♦♠❛ s❛t✐s❢❛③

f l m X i=0 ai ! = m X i=0

a_ihm_ii. ✭✷✳✷✶✮

P♦r ✜♠✱ ✈❛❧❡ q✉❡

f l

m X

i=0 a_ihk_ii

! = m X i=0 ai !

hm+k_i ✭✷✳✷✷✮

s❡♠♣r❡ q✉❡ a0hki0, a1hki1, . . . , amhkim ∈ F ♣♦ss✉ír❡♠ ♦ ♠❡s♠♦ s✐♥❛❧ ✭k ≥0✮✳ ◆❡ss❡ ❝❛s♦✱ ✭✷✳✷✷✮ ♠♦str❛ q✉❡ ❛ s♦♠❛ Pm

i=0

ai ♣♦❞❡ s❡r ❝♦♠♣✉t❛❞❛ ❝♦♠ ❡rr♦ r❡❧❛t✐✈♦ ♣❡q✉❡♥♦✳ ❆ ❡q✉❛çã♦ ✭✷✳✷✷✮ s❡❣✉❡ ❞❡ ✭✷✳✷✶✮ ❡

1 (1+ǫ)m+k ≤

min

i hm+kii

≤

m

P

i=0

aihm+kii m P i=0 ai ≤ max

i hm+kii

❈❛♣ít✉❧♦ ✸

■♥t❡r♣♦❧❛❞♦r❡s ❡ ❛ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛✿

❡①❡♠♣❧♦s

✸✳✶ ■♥t❡r♣♦❧❛❞♦r ♣♦❧✐♥♦♠✐❛❧ ❞❡ ▲❛❣r❛♥❣❡

❉❛❞♦s x ❡ y✱ ♦ ✐♥t❡r♣♦❧❛❞♦r ♣♦❧✐♥♦♠✐❛❧ ❞❡ ▲❛❣r❛♥❣❡ é ❞❡✜♥✐❞♦ ❝♦♠♦ ♦ ú♥✐❝♦ ♣♦❧✐♥ô♠✐♦ pn(x)✱ ❞❡ ❣r❛✉ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛n✱ q✉❡ s❛t✐s❢❛③

pn(xi) =yi, i= 0,1, . . . , n. ✭✸✳✶✮ ❊①♣❧✐❝✐t❛♠❡♥t❡✱ t❡♠♦s

pn(x) = n X

i=0

yiℓi(x,x), ℓi(x,x) = n Y

j= 0 j6=i

x₋xj

xi−xj

, i= 0,1, . . . , n. ✭✸✳✷✮

❖s ♣♦❧✐♥ô♠✐♦s ℓi(x,x) sã♦ ❝❤❛♠❛❞♦s ♦s ♣♦❧✐♥ô♠✐♦s ❞❡ ▲❛❣r❛♥❣❡ ❛ss♦❝✐❛❞♦s ❛♦s ♥ós x ❡ sã♦ ❝❛r❛❝✲ t❡r✐③❛❞♦s ♣♦r

ℓi(xk,x) =δi,k ✭❞❡❧t❛ ❞❡ ❑r♦♥❡❝❦❡r✳✮ ✭✸✳✸✮ ❙❡❣✉❡ ❞✐r❡t♦ ❞❡ ✭✸✳✸✮ q✉❡ ♦s ♣♦❧✐♥ô♠✐♦s ❞❡ ▲❛❣r❛♥❣❡ ❢♦r♠❛♠ ✉♠ ❝♦♥❥✉♥t♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡✱ ❡✱ ♣♦rt❛♥t♦✱ ✉♠❛ ❜❛s❡ ❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧Pn❞♦s ♣♦❧✐♥ô♠✐♦s ❞❡ ❣r❛✉ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛n✳ ■ss♦ ❞❡st❛❝❛ ❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡r✐③❛❞♦ ♣♦r ✭✸✳✶✮✱ ♣♦✐sy❢♦r♥❡❝❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞♦ ♣♦❧✐♥ô♠✐♦pn(x) ❝♦♠ r❡❧❛çã♦ à ❜❛s❡{ℓ0(x,x), ℓ1(x,x), . . . , ℓn(x,x)}✳

✸✳✶✳✶ ❊rr♦ ❞❡ ✐♥t❡r♣♦❧❛çã♦

P❛r❛ ❢✉♥çõ❡s s✉❛✈❡s✱ ♦ ❡rr♦ ❞❡ ✐♥t❡r♣♦❧❛çã♦ ♣♦❞❡ s❡r ❡st✐♠❛❞♦ ❝♦♠ ❜❛s❡ ♥♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ❝❧áss✐❝♦ ✭❬❍❡♥✽✷❪✱ ♣✳ ✷✸✶ ♦✉ ❬■❑✾✹❪✱ ♣✳ ✶✾✵✮✿

❚❡♦r❡♠❛ ✶✳ ❙❡❥❛♠ f : [a, b]_{−→ R} ✉♠❛ ❢✉♥çã♦ ❞❡ ❝❧❛ss❡ Cn+1 ❡ pn(x) ♦ ♣♦❧✐♥ô♠✐♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ▲❛❣r❛♥❣❡ ♣❛r❛ f ❞❡ ♠♦❞♦ q✉❡

pn(xi) =f(xi), i= 0,1, . . . , n.

❊♥tã♦✱ ❞❛❞♦ x _∈ [a, b] ❡①✐st❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ξ ✭❝♦♠ min_{x0, x}< ξ <max{x, xn}✮ t❛❧ q✉❡

f(x)₋pn(x) =

f(n+1)_(ξ)

(n+ 1)! ω(x), ω(x) :=

n Y

i=0

(x₋xi). ✭✸✳✹✮

❈♦♠♦ ❝♦r♦❧ár✐♦ ❞✐r❡t♦✱ s❡❣✉❡ q✉❡✱ s❡ f é ❞❡ ❝❧❛ss❡ C∞ ❡ ♣♦ss✉✐ t♦❞❛s ❛s s✉❛s ❞❡r✐✈❛❞❛s ❧✐♠✐t❛❞❛s

❡♠[a, b]♣♦r ✉♠❛ ♠❡s♠❛ ❝♦♥st❛♥t❡ ✭❝♦♠♦ ❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s s❡♥♦ ❡ ❝♦ss❡♥♦✱ ♣♦r ❡①❡♠♣❧♦✮✱

✶✵ ■◆❚❊❘P❖▲❆❉❖❘❊❙ ❊ ❆ ❋Ó❘▼❯▲❆ ❇❆❘■❈✃◆❚❘■❈❆✿ ❊❳❊▼P▲❖❙ ✸✳✶

❡♥tã♦ ♣❛r❛ q✉❛❧q✉❡r s❡q✉ê♥❝✐❛ ❞❡ ♠❛❧❤❛s ❞❡ ✐♥t❡r♣♦❧❛çã♦x1,x2. . . . ,xk, . . . ✭xk❝♦♠k+1♣♦♥t♦s✮✱ ❛

s❡q✉ê♥❝✐❛ ❞❡ ✐♥t❡r♣♦❧❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡ ❛ss♦❝✐❛❞♦sp1(x), p2(x), . . . , pk(x), . . . ,❝♦♥✈❡r❣❡ ✉♥✐❢♦r♠❡✲ ♠❡♥t❡ ♣❛r❛f✳ P♦ré♠✱ ♣❛r❛ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s s❡♠ ♥❡♥❤✉♠ ❣r❛✉ ❞❡ ❞✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡✱ ❡ss❡ r❡s✉❧t❛❞♦

é ❢❛❧s♦✳ ❉❡ ❢❛t♦✱ ❋❛❜❡r ♠♦str♦✉✱ ❡♠ ✶✾✶✹✱ q✉❡✱ ♣❛r❛ q✉❛❧q✉❡r s❡q✉ê♥❝✐❛ ❞❡ ♠❛❧❤❛sx1,x2. . . . ,xk, . . .

❝♦♥t✐❞❛s ❡♠[a, b]✱ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛f∗ : [a, b]_−→Rt❛❧ q✉❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ✐♥t❡r♣♦❧❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡ ❛ss♦❝✐❛❞♦sp1(x), p2(x), . . . , pk(x), . . . ♥ã♦ ❝♦♥✈❡r❣❡ ✉♥✐❢♦r♠❡♠❡♥t❡✶ ♣❛r❛f∗ ❬❙♠✐✵✻❪✳ ✸✳✶✳✷ ❆ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ♣❛r❛ ♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ▲❛❣r❛♥❣❡

❙❡ x /_{∈ {}x0, x1, . . . , xn}✱ s❡❣✉❡✱ ♣♦r ✭✸✳✷✮✱ q✉❡

pn(x) = n P i=0 yi     n Q

j= 1 j6=i

x−xj

xi−xj

  

xx−−xxii = ω(x)

n P i=0

yiγ(x)i

x−xi = p(x,x,y, γ(x)), ✭✸✳✺✮

♣❛r❛ω(x)❞❡✜♥✐❞♦ ♣♦r ✭✸✳✹✮ ❡

γ(x)i := n Y

j= 0 j6=i

1 xi−xj

= 1

ω′_(x i)

, i= 0,1, . . . , n. ✭✸✳✻✮

❆ ❡①♣r❡ssã♦ ✭✸✳✺✮ ♣❛r❛ ♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ▲❛❣r❛♥❣❡ é ❝♦♠✉♠❡♥t❡ ❝❤❛♠❛❞❛ ❞❡ ❢ór♠✉❧❛ ❞❡ ▲❛✲ ❣r❛♥❣❡ ♠♦❞✐✜❝❛❞❛✱ ♦✉ ♣r✐♠❡✐r❛ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛✳ ❆♣❧✐❝❛♥❞♦ ✭✸✳✺✮ ♣❛r❛ ♦ ♣♦❧✐♥ô♠✐♦ ❝♦♥st❛♥t❡ ✏1✧✱ ♦❜t❡♠♦s

1 = ω(x)Pn

i=0 γ(x)i

x−xi. ✭✸✳✼✮

▲♦❣♦✱ ♣♦r ✭✸✳✺✮ ❡ ✭✸✳✼✮✱ s❡❣✉❡ q✉❡

pn(x) = n P i=0

γ(x)iyi

x−xi

_n P i=0

γ(x)i

x−xi = q(x,x,y, γ(x)). ✭✸✳✽✮

❚❡♠♦s✱ ❡♥tã♦✱ q✉❡ ♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ▲❛❣r❛♥❣❡ ♣♦❞❡ s❡r ❡①♣r❡ss♦ ❡♠ ❢✉♥çã♦ ❞❛ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ✭✷✳✶✮✱ ♣❛r❛ ♦s ♣❡s♦s ❞❡✜♥✐❞♦s ♣♦r ✭✸✳✻✮✳ ❆ ❡①♣r❡ssã♦ ✭✸✳✽✮ ♣❛r❛ ♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ▲❛❣r❛♥❣❡ é t❛♠❜é♠ ❝❤❛♠❛❞❛ ❞❡ s❡❣✉♥❞❛ ❢ór♠✉❧❛ ❜❛r✐❝ê♥tr✐❝❛ ✭❬❚r❡✶✷❪✱ ❝❛♣✳ ✺✱ ♣✳ ✸✹✳✮

✸✳✶✳✸ ❋❛♠í❧✐❛s ❡s♣❡❝✐❛✐s ❞❡ ♥ós ♣❛r❛ ✐♥t❡r♣♦❧❛çã♦ ♣♦❧✐♥♦♠✐❛❧ ◆ós ✐❣✉❛❧♠❡♥t❡ ❡s♣❛ç❛❞♦s

❖s ❝♦♥❥✉♥t♦ ❞❡ ✭n+ 1✮ ♥ós ✐❣✉❛❧♠❡♥t❡ ❡s♣❛ç❛❞♦s ❡♠[a, b]é ❞❡✜♥✐❞♦ ♣♦r

xi = (xeq)i =a+ih, i= 0,1, . . . , n, h:=

b₋a

n . ✭✸✳✾✮

P♦r ♠❡✐♦ ❞❡ ♠❛♥✐♣✉❧❛çõ❡s ❛❧❣é❜r✐❝❛s s✐♠♣❧❡s ✭❬❍❡♥✽✷❪✱ ♣✳ ✷✸✾✮✱ ♣♦❞❡✲s❡ ♠♦str❛r q✉❡✱ ♣❛r❛ ❡ss❛ ❢❛♠✐❧✐❛ ❞❡ ♥ós✱ ♦s ♣❡s♦s ❞❡ ✐♥t❡r♣♦❧❛çã♦ ✭✸✳✻✮ s❛t✐s❢❛③❡♠

γ(xeq)i=

(₋1)n−i n!hn

n i

. ✭✸✳✶✵✮

P♦r ✉♠ ❧❛❞♦✱ ♥ós ✐❣✉❛❧♠❡♥t❡ ❡s♣❛ç❛❞♦s sã♦ ❝♦♥✈❡♥✐❡♥t❡s✱ ♣♦✐s ♣♦ss✐❜✐❧✐t❛♠ ❛ ❞✐s❝r❡t✐③❛çã♦ ❞❡ ♦♣❡r❛❞♦r❡s ❞❡ ❛❧t❛s ♦r❞❡♥s ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s ♣♦r ❞✐❢❡r❡♥ç❛s ✜♥✐t❛s✱ ♣♦r ❡①❡♠♣❧♦ ✭❬▲❡✈✵✼❪✱ ❝❛♣✳ ✶✳✮ P♦r ♦✉tr♦ ❧❛❞♦✱ ❛♦s ♥ós ✐❣✉❛❧♠❡♥t❡ ❡s♣❛ç❛❞♦s ❡stá ❛ss♦❝✐❛❞♦ ♦ ❝❤❛♠❛❞♦ ❢❡♥ô♠❡♥♦ ❞❡ ❘✉♥❣❡✱ ♥♦ q✉❛❧ ❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦❧✐♥ô♠✐♦s q✉❡ ✐♥t❡r♣♦❧❛♠ ❛ ❢✉♥çã♦

✶_{❉❡ ❢❛t♦✱ ✈❛❧❡ q✉❡lim}

k→||f ∗_−p