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❖♥ s❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ♦❢ ✐❧❧✲♣♦s❡❞ ♣r♦❜❧❡♠s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❜② ♣r♦❥❡❝t✐♦♥ ♠❡t❤♦❞s

❯r✈❡ ❑❛♥❣r♦✱ ❯♥♦ ❍ä♠❛r✐❦

❯♥✐✈❡rs✐t② ♦❢ ❚❛rt✉✱ ❊st♦♥✐❛

◆❡✇ ❚r❡♥❞s ✐♥ P❛r❛♠❡t❡r ■❞❡♥t✐✜❝❛t✐♦♥ ❘✐♦ ❞❡ ❏❛♥❡✐r♦ ✷✵✶✼

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✶ ✴ ✷✺

✶ ❚❤❡ ♣r♦❜❧❡♠ ✷ Pr♦❥❡❝t✐♦♥ ♠❡t❤♦❞s

❈♦♥✈❡r❣❡♥❝❡ ✇✐t❤ ❛ ♣r✐♦r✐ ❝❤♦✐❝❡ ♦❢ n = n(δ) ❈❤♦✐❝❡ ♦❢ n = n(δ) ❜② t❤❡ ❞✐s❝r❡♣❛♥❝② ♣r✐♥❝✐♣❧❡ ▼♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❞✐s❝r❡♣❛♥❝② ♣r✐♥❝✐♣❧❡

✸ ❆♣♣❧✐❝❛t✐♦♥s✿ ❈♦❧❧♦❝❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s

❈♦r❞✐❛❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s P♦❧②♥♦♠✐❛❧ ❝♦❧❧♦❝❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ❝♦r❞✐❛❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ❙♣❧✐♥❡✲❝♦❧❧♦❝❛t✐♦♥ ❢♦r ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✷ ✴ ✷✺

❚❤❡ ♣r♦❜❧❡♠

▲❡t E, F ❜❡ ❇❛♥❛❝❤ s♣❛❝❡s✱ A ∈ L(E, F)✳ ❈♦♥s✐❞❡r t❤❡ ❡q✉❛t✐♦♥ Au = f , ✭✶✮ ✇❤❡r❡ f ∈ R(A) ✐s ✉♥❦♥♦✇♥✱ ♦♥❧② ♥♦✐s② ❞❛t❛ f δ ✇✐t❤f δ − f F ≤ δ ❛r❡ ❛✈❛✐❧❛❜❧❡✳ ❚②♣✐❝❛❧❧② t❤❡ ♣r♦❜❧❡♠ ✐s ❢♦r♠✉❧❛t❡❞ ✐♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✳ ■❢ t❤❡ ♣r♦❜❧❡♠ ✐s ✐❧❧✲♣♦s❡❞✱ t❤❡♥ ✉s✉❛❧❧② ✐t ✐s ✜rst r❡❣✉❧❛r✐③❡❞ ❛♥❞ t❤❡♥ ❞✐s❝r❡t✐③❡❞✳ ❖❢t❡♥ t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ ✇♦r❦s ❛s r❡❣✉❧❛r✐③❛t✐♦♥ ❛s ✇❡❧❧✿ ✐❢ ❞❛t❛ ❛r❡ ♥♦✐s② ✇✐t❤ ❦♥♦✇♥ ♥♦✐s❡ ❧❡✈❡❧ ✱ t❤❡♥ ❜② ♣r♦♣❡r ❝❤♦✐❝❡ ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ ❞✐s❝r❡t✐③❡❞ ❡q✉❛t✐♦♥s ✇✐t❤ ♥♦✐s② ❞❛t❛ ❝♦♥✈❡r❣❡ t♦ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♣r♦❜❧❡♠ ✇✐t❤ ❡①❛❝t ❞❛t❛✳ ▼♦st r❡s✉❧ts ❛❜♦✉t s❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ❤❛✈❡ ❜❡❡♥ ♦❜t❛✐♥❡❞ ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✳ ❇✉t ❢♦r ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s t❤❡ ♠♦st ♥❛t✉r❛❧ s♣❛❝❡ ✐s ✱ ❡s♣❡❝✐❛❧❧② ❢♦r ❝♦❧❧♦❝❛t✐♦♥ ♦r q✉❛❞r❛t✉r❡ ♠❡t❤♦❞s✳ ❖❢t❡♥ ❛❧s♦

✶ ✐s ✉s❡❢✉❧ t♦ r❡❝♦✈❡r s♣❛rs❡

s♦❧✉t✐♦♥✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✸ ✴ ✷✺

❚❤❡ ♣r♦❜❧❡♠

▲❡t E, F ❜❡ ❇❛♥❛❝❤ s♣❛❝❡s✱ A ∈ L(E, F)✳ ❈♦♥s✐❞❡r t❤❡ ❡q✉❛t✐♦♥ Au = f , ✭✶✮ ✇❤❡r❡ f ∈ R(A) ✐s ✉♥❦♥♦✇♥✱ ♦♥❧② ♥♦✐s② ❞❛t❛ f δ ✇✐t❤f δ − f F ≤ δ ❛r❡ ❛✈❛✐❧❛❜❧❡✳ ❚②♣✐❝❛❧❧② t❤❡ ♣r♦❜❧❡♠ ✐s ❢♦r♠✉❧❛t❡❞ ✐♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✳ ■❢ t❤❡ ♣r♦❜❧❡♠ ✐s ✐❧❧✲♣♦s❡❞✱ t❤❡♥ ✉s✉❛❧❧② ✐t ✐s ✜rst r❡❣✉❧❛r✐③❡❞ ❛♥❞ t❤❡♥ ❞✐s❝r❡t✐③❡❞✳ ❖❢t❡♥ t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ ✇♦r❦s ❛s r❡❣✉❧❛r✐③❛t✐♦♥ ❛s ✇❡❧❧✿ ✐❢ ❞❛t❛ ❛r❡ ♥♦✐s② ✇✐t❤ ❦♥♦✇♥ ♥♦✐s❡ ❧❡✈❡❧ δ✱ t❤❡♥ ❜② ♣r♦♣❡r ❝❤♦✐❝❡ ♦❢ n = n(δ) t❤❡ s♦❧✉t✐♦♥s ♦❢ ❞✐s❝r❡t✐③❡❞ ❡q✉❛t✐♦♥s ✇✐t❤ ♥♦✐s② ❞❛t❛ ❝♦♥✈❡r❣❡ t♦ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♣r♦❜❧❡♠ ✇✐t❤ ❡①❛❝t ❞❛t❛✳ ▼♦st r❡s✉❧ts ❛❜♦✉t s❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ❤❛✈❡ ❜❡❡♥ ♦❜t❛✐♥❡❞ ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✳ ❇✉t ❢♦r ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s t❤❡ ♠♦st ♥❛t✉r❛❧ s♣❛❝❡ ✐s ✱ ❡s♣❡❝✐❛❧❧② ❢♦r ❝♦❧❧♦❝❛t✐♦♥ ♦r q✉❛❞r❛t✉r❡ ♠❡t❤♦❞s✳ ❖❢t❡♥ ❛❧s♦

✶ ✐s ✉s❡❢✉❧ t♦ r❡❝♦✈❡r s♣❛rs❡

s♦❧✉t✐♦♥✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✸ ✴ ✷✺

❚❤❡ ♣r♦❜❧❡♠

▲❡t E, F ❜❡ ❇❛♥❛❝❤ s♣❛❝❡s✱ A ∈ L(E, F)✳ ❈♦♥s✐❞❡r t❤❡ ❡q✉❛t✐♦♥ Au = f , ✭✶✮ ✇❤❡r❡ f ∈ R(A) ✐s ✉♥❦♥♦✇♥✱ ♦♥❧② ♥♦✐s② ❞❛t❛ f δ ✇✐t❤f δ − f F ≤ δ ❛r❡ ❛✈❛✐❧❛❜❧❡✳ ❚②♣✐❝❛❧❧② t❤❡ ♣r♦❜❧❡♠ ✐s ❢♦r♠✉❧❛t❡❞ ✐♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✳ ■❢ t❤❡ ♣r♦❜❧❡♠ ✐s ✐❧❧✲♣♦s❡❞✱ t❤❡♥ ✉s✉❛❧❧② ✐t ✐s ✜rst r❡❣✉❧❛r✐③❡❞ ❛♥❞ t❤❡♥ ❞✐s❝r❡t✐③❡❞✳ ❖❢t❡♥ t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ ✇♦r❦s ❛s r❡❣✉❧❛r✐③❛t✐♦♥ ❛s ✇❡❧❧✿ ✐❢ ❞❛t❛ ❛r❡ ♥♦✐s② ✇✐t❤ ❦♥♦✇♥ ♥♦✐s❡ ❧❡✈❡❧ δ✱ t❤❡♥ ❜② ♣r♦♣❡r ❝❤♦✐❝❡ ♦❢ n = n(δ) t❤❡ s♦❧✉t✐♦♥s ♦❢ ❞✐s❝r❡t✐③❡❞ ❡q✉❛t✐♦♥s ✇✐t❤ ♥♦✐s② ❞❛t❛ ❝♦♥✈❡r❣❡ t♦ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♣r♦❜❧❡♠ ✇✐t❤ ❡①❛❝t ❞❛t❛✳ ▼♦st r❡s✉❧ts ❛❜♦✉t s❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ❤❛✈❡ ❜❡❡♥ ♦❜t❛✐♥❡❞ ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✳ ❇✉t ❢♦r ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s t❤❡ ♠♦st ♥❛t✉r❛❧ s♣❛❝❡ ✐s C✱ ❡s♣❡❝✐❛❧❧② ❢♦r ❝♦❧❧♦❝❛t✐♦♥ ♦r q✉❛❞r❛t✉r❡ ♠❡t❤♦❞s✳ ❖❢t❡♥ ❛❧s♦ L✶ ✐s ✉s❡❢✉❧ t♦ r❡❝♦✈❡r s♣❛rs❡ s♦❧✉t✐♦♥✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✸ ✴ ✷✺

Pr♦❥❡❝t✐♦♥ ♠❡t❤♦❞s

❆ss✉♠❡ t❤❛t ✭✶✮ ✐s ✉♥✐q✉❡❧② s♦❧✈❛❜❧❡✳ ▲❡t u∗ ❜❡ t❤❡ s♦❧✉t✐♦♥ ❢♦r ❡①❛❝t f ✳ ▲❡t En ⊆ E✱ Zn ⊆ F ∗ ❜❡ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s✱ ❞✐♠ En = ❞✐♠ Zn✳

• ❡♥❡r❛❧ ❧✐♥❡❛r ♣r♦❥❡❝t✐♦♥ ♠❡t❤♦❞✿ ✜♥❞ un ∈ En s✉❝❤ t❤❛t

∀zn ∈ Zn zn, AunF ∗,F = zn, f δF ∗,F. ✭✷✮ ❉❡✜♥❡ ✿ ✳ ❚❤❡♥ ✭✷✮ ✳ s✉♣

✶ ✶

s✉♣

✶ ✶

✶ ❊①❛♠♣❧❡✳ ❈♦❧❧♦❝❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s✳ ✱ s♣❛♥ ✶ ✱ ✳ ✭✷✮ ✶ ✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✹ ✴ ✷✺

Pr♦❥❡❝t✐♦♥ ♠❡t❤♦❞s

❆ss✉♠❡ t❤❛t ✭✶✮ ✐s ✉♥✐q✉❡❧② s♦❧✈❛❜❧❡✳ ▲❡t u∗ ❜❡ t❤❡ s♦❧✉t✐♦♥ ❢♦r ❡①❛❝t f ✳ ▲❡t En ⊆ E✱ Zn ⊆ F ∗ ❜❡ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s✱ ❞✐♠ En = ❞✐♠ Zn✳

• ❡♥❡r❛❧ ❧✐♥❡❛r ♣r♦❥❡❝t✐♦♥ ♠❡t❤♦❞✿ ✜♥❞ un ∈ En s✉❝❤ t❤❛t

∀zn ∈ Zn zn, AunF ∗,F = zn, f δF ∗,F. ✭✷✮ ❉❡✜♥❡ Qn ∈ L(F, Z ∗

n )✿ ∀g ∈ F , zn ∈ Zn : Qng, znZ ∗

n ,Zn = zn, gF ∗,F✳

❚❤❡♥ ✭✷✮ ⇐ ⇒ QnAun = Qnf δ✳ Qn = s✉♣

g∈F,gF =✶ zn∈Zn,znF∗=✶

Qng, znZ ∗

n ,Zn =

s✉♣

g∈F,gF =✶ zn∈Zn,znF∗=✶

zn, gF ∗,F = ✶. ❊①❛♠♣❧❡✳ ❈♦❧❧♦❝❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s✳ ✱ s♣❛♥ ✶ ✱ ✳ ✭✷✮ ✶ ✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✹ ✴ ✷✺

Pr♦❥❡❝t✐♦♥ ♠❡t❤♦❞s

❆ss✉♠❡ t❤❛t ✭✶✮ ✐s ✉♥✐q✉❡❧② s♦❧✈❛❜❧❡✳ ▲❡t u∗ ❜❡ t❤❡ s♦❧✉t✐♦♥ ❢♦r ❡①❛❝t f ✳ ▲❡t En ⊆ E✱ Zn ⊆ F ∗ ❜❡ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s✱ ❞✐♠ En = ❞✐♠ Zn✳

• ❡♥❡r❛❧ ❧✐♥❡❛r ♣r♦❥❡❝t✐♦♥ ♠❡t❤♦❞✿ ✜♥❞ un ∈ En s✉❝❤ t❤❛t

∀zn ∈ Zn zn, AunF ∗,F = zn, f δF ∗,F. ✭✷✮ ❉❡✜♥❡ Qn ∈ L(F, Z ∗

n )✿ ∀g ∈ F , zn ∈ Zn : Qng, znZ ∗

n ,Zn = zn, gF ∗,F✳

❚❤❡♥ ✭✷✮ ⇐ ⇒ QnAun = Qnf δ✳ Qn = s✉♣

g∈F,gF =✶ zn∈Zn,znF∗=✶

Qng, znZ ∗

n ,Zn =

s✉♣

g∈F,gF =✶ zn∈Zn,znF∗=✶

zn, gF ∗,F = ✶. ❊①❛♠♣❧❡✳ ❈♦❧❧♦❝❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s✳ F = C[a, b]✱ Zn = s♣❛♥{δ(t − ti), i = ✶, . . . , n}✱ ti ∈ [a, b]✳ ✭✷✮ ⇐ ⇒ Aun(ti) = f δ(ti), i = ✶, . . . , n✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✹ ✴ ✷✺

˜ κn = s✉♣

wn∈En

wnE QnAwnZ ∗

n

= s✉♣

wn∈En

wnE s✉♣zn∈Zn,znF∗=✶zn, AwnF ∗,F

▲❡♠♠❛ ✶ ✭❯♥✐q✉❡♥❡ss ♦❢ t❤❡ ❞✐s❝r❡t❡ s♦❧✉t✐♦♥✮

▲❡t ❞✐♠(En) = ❞✐♠(Zn) ❛♥❞ N(QnA) ∩ En = {✵}✳ ❚❤❡♥ ✭✷✮ ✐s ✉♥✐q✉❡❧② s♦❧✈❛❜❧❡ ❢♦r ❡❛❝❤ f δ ∈ F✳ ❉❡♥♦t❡ An := QnA|En : En → Z ∗

n ✳ ❚❤❡♥ t❤❡ ❧❡♠♠❛ ♠❡❛♥s t❤❛t An ❤❛s ❛♥

✐♥✈❡rs❡ ❛♥❞ A−✶

n = ˜

κn✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✺ ✴ ✷✺

κn = s✉♣

vn∈En

vnE AvnF , ˘ κn = A−✶

n Qn,

˜ κn = A−✶

n = s✉♣ vn∈En

vnE QnAvnF , τn = s✉♣

vn∈En,vn=✵

AvnF QnAvnZ ∗

n

.

▲❡♠♠❛ ✷

▲❡t ❞✐♠(En) = ❞✐♠(Zn) ❛♥❞ N(QnA) ∩ En = {✵} ❤♦❧❞✳ ❚❤❡♥ κn ≤ ˘ κn ≤ ˜ κn ≤ τnκn. ■❢ t❤❡r❡ ❡①✐sts τ < ∞ s✉❝❤ t❤❛t τn ≤ τ ❢♦r ❛❧❧ n ∈ N t❤❡♥ ❛❧s♦ ˜ κn ≤ τκn, ✐✳❡✳ t❤❡ q✉❛♥t✐t✐❡s κn✱ ˘ κn ❛♥❞ ˜ κn ❛r❡ ❛❧❧ ❡q✉✐✈❛❧❡♥t ❛s n → ∞✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✻ ✴ ✷✺

❈♦♥✈❡r❣❡♥❝❡ ✇✐t❤ ❛ ♣r✐♦r✐ ❝❤♦✐❝❡ ♦❢ n = n(δ)

❚❤❡♦r❡♠ ✸

▲❡t ✭✶✮ ❛♥❞ ✭✷✮ ❜❡ ✉♥✐q✉❡❧② s♦❧✈❛❜❧❡ ❢♦r ❡❛❝❤ n ≥ n✵ ❛♥❞ u∗✱ un ❜❡ t❤❡✐r s♦❧✉t✐♦♥s✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡rr♦r ❡st✐♠❛t❡ ❤♦❧❞s✿ un − u∗E ≤ ♠✐♥

vn∈En[u∗ − vnE + A−✶ n QnA(u∗ − vn)E] + ˘

κnδ ≤ (✶ + A−✶

n QnA) ❞✐st(u∗, En) + ˘

κnδ. ■♥ ❝❛s❡ ♦❢ ❡①❛❝t ❞❛t❛ ✭δ = ✵✮ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ un − u∗E → ✵ ❛s n → ∞ ❤♦❧❞s ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts ❛ (ˆ un)n∈N✱ ˆ un ∈ En✱ s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ❝♦♥❞✐t✐♦♥s u∗ − ˆ unE → ✵ ❛s n → ∞ ❛♥❞ A−✶

n QnA(u∗ − ˆ

un) → ✵ ❛s n → ∞. ■❢ t❤❡s❡ ❝♦♥❞✐t✐♦♥s ❤♦❧❞ ❛♥❞ t❤❡ ❞❛t❛ ❛r❡ ♥♦✐s②✱ t❤❡♥ ❝❤♦♦s✐♥❣ n = n(δ) ❛❝❝♦r❞✐♥❣ t♦ ❛ ♣r✐♦r✐ r✉❧❡ n(δ) → ∞ ❛♥❞ ˘ κn(δ)δ → ✵ ❛s δ → ✵ ✇❡ ❤❛✈❡ ❝♦♥✈❡r❣❡♥❝❡ un(δ) − u∗E → ✵ ❛s δ → ✵.

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✼ ✴ ✷✺

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♠❛② ❤♦❧❞ ❞✉❡ t♦ s✉✣❝✐❡♥t s♠♦♦t❤♥❡ss ♦❢ t❤❡ s♦❧✉t✐♦♥✳ ❚❤❡ ♥❡①t t❤❡♦r❡♠ ❣✐✈❡s ❝♦♥❞✐t✐♦♥s ❢♦r ❝♦♥✈❡r❣❡♥❝❡ ❢♦r ❡✈❡r② f ∈ R(A) ✭✐✳❡✳ ❢♦r ❡✈❡r② u∗ ∈ E ✇✐t❤♦✉t ❛❞❞✐t✐♦♥❛❧ s♠♦♦t❤♥❡ss r❡q✉✐r❡♠❡♥ts✮✳

❚❤❡♦r❡♠ ✹ ✭❈♦♥✈❡r❣❡♥❝❡ ❢♦r ❡✈❡r② f ✮

▲❡t ✭✶✮ ❛♥❞ ✭✷✮ ❜❡ ✉♥✐q✉❡❧② s♦❧✈❛❜❧❡ ❢♦r n ≥ n✵✳ ❚❤❡♥ ✐♥ ❝❛s❡ ♦❢ ❡①❛❝t ❞❛t❛ ✭δ = ✵✮ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ un − u∗E → ✵ ❛s n → ∞ ❤♦❧❞s ❢♦r ❡✈❡r② f ∈ R(A) ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ s✉❜s♣❛❝❡s En s❛t✐s❢② ❝♦♥❞✐t✐♦♥ ✐♥❢vn∈En vn − v → ✵ ∀v ∈ E ❛s n → ∞✱ ❛♥❞ t❤❡ ♣r♦❥❡❝t♦rs A−✶

n QnA : E → En ❛r❡ ✉♥✐❢♦r♠❧② ❜♦✉♥❞❡❞✱ ✐✳❡✳✱

A−✶

n QnA ≤ M

❢♦r ❛❧❧ n ≥ n✵ ❛♥❞ s♦♠❡ ❝♦♥st❛♥t M✳ ❚❤❡ ❧❛st t✇♦ ❝♦♥❞✐t✐♦♥s ❛r❡ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❢♦r ❡①✐st❡♥❝❡ ♦❢ r❡❧❛t✐♦♥s n = n(δ) ❢♦r ❝♦♥✈❡r❣❡♥❝❡ un(δ) − u∗E → ✵ ❛s δ → ✵ ❢♦r ❡✈❡r② f ∈ R(A) ❣✐✈❡♥ ❛♣♣r♦①✐♠❛t❡❧② ❛s ❛r❜✐tr❛r② f δ ✇✐t❤ f δ − f ≤ δ✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✽ ✴ ✷✺

❋♦r t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ❛♥❛❧②s✐s ✐♥ ❝❛s❡ ♦❢ ❡①❛❝t ❞❛t❛ ✇❡ ❝❛♥ ❝❤♦♦s❡ ❞✐✛❡r❡♥t ✐♠❛❣❡ s♣❛❝❡s✱ ♣❛rt✐❝✉❧❛r❧② s✉❝❤ t❤❛t t❤❡ ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s ✇❡❧❧✲♣♦s❡❞✳ ❇✉t ❢♦r ♥♦✐s② ❞❛t❛ t❤❡ ✐♠❛❣❡ s♣❛❝❡ ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❞❛t❛✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠ s❤♦✇s t❤❛t ❝♦♥✈❡r❣❡♥❝❡ ❢♦r t❤❡ ✏♠❛✐♥ ♣❛rt✑ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✐♠♣❧✐❡s ❝♦♥✈❡r❣❡♥❝❡ ❢♦r t❤❡ ✇❤♦❧❡ ❡q✉❛t✐♦♥✳

❚❤❡♦r❡♠ ✺

▲❡t ✭✶✮ ❛♥❞ ✭✷✮ ❜❡ ✉♥✐q✉❡❧② s♦❧✈❛❜❧❡ ❢♦r n ≥ n✵✳ ❆ss✉♠❡ t❤❛t t❤❡r❡ ❡①✐sts ❛ (ˆ un)n∈N✱ ˆ un ∈ En✱ s❛t✐s❢②✐♥❣ u∗ − ˆ unE → ✵ ❛s n → ∞✳ ▲❡t t❤❡ ♦♣❡r❛t♦r A : E → F ❤❛✈❡ t❤❡ ❢♦r♠ A = S + K✱ ✇❤❡r❡ S : E → W ⊂ F ✐s ✐♥✈❡rt✐❜❧❡✱ W ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❝♦♥t✐♥✉♦✉s ✐♠❜❡❞❞✐♥❣ ❛♥❞ K : E → W ✐s ❝♦♠♣❛❝t✳ ▲❡t t❤❡ ♦♣❡r❛t♦r Sn := QnS|En : En → Z ∗

n ❜❡ ✐♥✈❡rt✐❜❧❡ ❛♥❞

S−✶

n QnS ≤ M ❢♦r s♦♠❡ ❝♦♥st❛♥t M✳ ❚❤❡♥ t❤❡ ♣r♦❥❡❝t✐♦♥ ❡q✉❛t✐♦♥

QnAun = Qnf ❤❛s ❢♦r n ❧❛r❣❡ ❡♥♦✉❣❤ ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ un ∈ En ✱ ❛♥❞ un → u∗ ❛s n → ∞✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✾ ✴ ✷✺

❋♦r ❝♦♥s✐❞❡r✐♥❣ t❤❡ ✐♥✢✉❡♥❝❡ ♦❢ t❤❡ ♥♦✐s② ❞❛t❛✱ t❤❡ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ q✉❛♥t✐t✐❡s ˘ κn ✐s ❡ss❡♥t✐❛❧✳ ❋♦r ❡st✐♠❛t✐♥❣ t❤❡s❡ q✉❛♥t✐t✐❡s ✇❡ ✐♥tr♦❞✉❝❡ ♦♣❡r❛t♦rs Πn : Z ∗

n → F s✉❝❤ t❤❛t t❤❡ ❡q✉❛❧✐t② QnΠnQn = Qn ❤♦❧❞s✳ ❚❤❡♥

t❤❡ ♦♣❡r❛t♦r ΠnQn ✐s ❛ ♣r♦❥❡❝t♦r ✐♥ F✳ ▲❡t Fn = R(Πn)✳ ❲❡ ❛ss✉♠❡ t❤❛t Fn ⊂ W ❛♥❞ ❧❡t Wn = Fn✱ ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ♥♦r♠ ♦❢ W ✳ ▲❡t In ❜❡ t❤❡ ✐❞❡♥t✐t② ♦♣❡r❛t♦r✱ ❝♦♥s✐❞❡r❡❞ ❛s ❛❝t✐♥❣ ❢r♦♠ Fn t♦ Wn✳

❚❤❡♦r❡♠ ✻ ✭❊st✐♠❛t❡ ♦❢ ˘ κn✮

▲❡t ✭✶✮ ❛♥❞ ✭✷✮ ❜❡ ✉♥✐q✉❡❧② s♦❧✈❛❜❧❡ ❢♦r n ≥ n✵✳ ▲❡t t❤❡ ♦♣❡r❛t♦r A : E → W ❜❡ ✐♥✈❡rt✐❜❧❡✳ ❚❤❡♥ ˘ κn ≤ CInFn→Wn, n ≥ n✵.

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✶✵ ✴ ✷✺

❈❤♦✐❝❡ ♦❢ n = n(δ) ❜② t❤❡ ❞✐s❝r❡♣❛♥❝② ♣r✐♥❝✐♣❧❡

❚❤❡♦r❡♠ ✼

▲❡t ✭✶✮ ❛♥❞ ✭✷✮ ❜❡ ✉♥✐q✉❡❧② s♦❧✈❛❜❧❡ ❢♦r n ≥ n✵✳ ▲❡t t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ˘ κn+✶❞✐st(f , AEn) → ✵ ❛s n → ∞ ❤♦❧❞✳ ❲❡ ❛❧s♦ ❛ss✉♠❡ t❤❛t t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s (ˆ un)n∈N✱ ˆ un ∈ En✱ s❛t✐s❢②✐♥❣ u∗ − ˆ unE → ✵ ❛s n → ∞ ❛♥❞ A−✶

n QnA(u∗ − ˆ

un) → ✵ ❛s n → ∞. ❆ss✉♠❡ t❤❡r❡ ❡①✐sts τ < ∞ s✉❝❤ t❤❛t τn ≤ τ ❢♦r ❛❧❧ n ∈ N✳ ▲❡t b > τ + ✶ ❜❡ ✜①❡❞ ❛♥❞ ❢♦r δ > ✵✱ ❧❡t n = nDP(δ) ❜❡ t❤❡ ✜rst ✐♥❞❡① s✉❝❤ t❤❛t Aun − f δF ≤ bδ✳ ❚❤❡♥ nDP(δ) ✐s ✜♥✐t❡ ❛♥❞ unDP(δ) − u∗E → ✵ ❛s δ → ✵✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✶✶ ✴ ✷✺

▼♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❞✐s❝r❡♣❛♥❝② ♣r✐♥❝✐♣❧❡

■♥ ❚❤❡♦r❡♠ ✼ t❤❡ ❛ss✉♠♣t✐♦♥ τn ≤ τ ❢♦r ❛❧❧ n ∈ N ✇❛s r❡q✉✐r❡❞✳ ❋♦r ❝♦❧❧♦❝❛t✐♦♥ ♠❡t❤♦❞s t❤✐s ✐s t❤❡ ✉♥✐❢♦r♠ ❜♦✉♥❞❡❞♥❡ss ♦❢ t❤❡ ✐♥t❡r♣♦❧❛t✐♦♥ ♣r♦❥❡❝t♦r ♦♥t♦ t❤❡ s✉❜s♣❛❝❡ AEn ⊂ F✳ ■❢ F = C m✱ t❤✐s ❛ss✉♠♣t✐♦♥ ❞♦❡s ♥♦t ❤♦❧❞ ✐♥ ❣❡♥❡r❛❧✳

❚❤❡♦r❡♠ ✽

▲❡t t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ✼ ❜❡ s❛t✐s✜❡❞ ✇✐t❤♦✉t ✉♥✐❢♦r♠ ❜♦✉♥❞❡❞♥❡ss ♦❢ τn✳ ▲❡t t❤❡ s❡q✉❡♥❝❡ bn > (✶ + τn)(✶ + ε) ❜❡ ✜①❡❞ ✇✐t❤ s♦♠❡ ✜①❡❞ ε > ✵ ❛♥❞ n = nDP(δ) ❜❡ ❝❤♦s❡♥ ❛s t❤❡ ✜rst ✐♥❞❡① s✉❝❤ t❤❛t Aun − f δF ≤ bnδ✳ ❚❤❡♥ nDP(δ) ✐s ✜♥✐t❡ ❛♥❞ unDP(δ) − u∗E → ✵ ❛s δ → ✵✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✶✷ ✴ ✷✺

❈♦r❞✐❛❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s

❈♦♥s✐❞❡r ❝♦r❞✐❛❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ♦❢ ✜rst ❦✐♥❞ t

✵

✶ t a(t, s)ϕ(s t )u(s)ds = f (t), ✵ ≤ t ≤ T, ✭✸✮ ✇❤❡r❡ ϕ ∈ L✶(✵, ✶) ✐s ❝❛❧❧❡❞ t❤❡ ❝♦r❡ ♦❢ t❤❡ ❝♦r❞✐❛❧ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r✱ a, f ❛r❡ ❣✐✈❡♥ s♠♦♦t❤ ❡♥♦✉❣❤ ❢✉♥❝t✐♦♥s✳ ❉❡✜♥❡ t❤❡ ❝♦r❞✐❛❧ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦rs (Vϕu)(t) = t

✵

✶ t ϕ(s t )u(s)ds, (Vϕ,au)(t) = t

✵

✶ t a(t, s)ϕ(s t )u(s)ds. ▼❛❦❡ ❛ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ✱ ❞❡♥♦t❡ ❛♥❞ ❞❡✜♥❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r

✶ ✵

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✶✸ ✴ ✷✺

❈♦r❞✐❛❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s

❈♦♥s✐❞❡r ❝♦r❞✐❛❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ♦❢ ✜rst ❦✐♥❞ t

✵

✶ t a(t, s)ϕ(s t )u(s)ds = f (t), ✵ ≤ t ≤ T, ✭✸✮ ✇❤❡r❡ ϕ ∈ L✶(✵, ✶) ✐s ❝❛❧❧❡❞ t❤❡ ❝♦r❡ ♦❢ t❤❡ ❝♦r❞✐❛❧ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r✱ a, f ❛r❡ ❣✐✈❡♥ s♠♦♦t❤ ❡♥♦✉❣❤ ❢✉♥❝t✐♦♥s✳ ❉❡✜♥❡ t❤❡ ❝♦r❞✐❛❧ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦rs (Vϕu)(t) = t

✵

✶ t ϕ(s t )u(s)ds, (Vϕ,au)(t) = t

✵

✶ t a(t, s)ϕ(s t )u(s)ds. ▼❛❦❡ ❛ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s s = tx✱ ❞❡♥♦t❡ b(t, x) = a(t, tx) ❛♥❞ ❞❡✜♥❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ( Vϕ,bu)(t) = ✶

✵

ϕ(x)b(t, x)u(tx)dx.

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✶✸ ✴ ✷✺

❈♦r❞✐❛❧ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦rs ✐♥ s♣❛❝❡s C m

❉❡♥♦t❡ ∆T = {(s, t) : t ∈ [✵, T], s ∈ [✵, t]}✳

❚❤❡♦r❡♠ ✾

▲❡t ϕ ∈ L✶(✵, ✶)✱ a ∈ C m(∆T)✳ ❚❤❡♥ Vϕ,a ∈ L(C m[✵, T]) ❛♥❞ Vϕ,aC m[✵,T] ≤ CϕL✶(✵,✶)aC m(∆T )✳

❚❤❡♦r❡♠ ✶✵

▲❡t

✶ ✵ ✶ ❛♥❞ ❧❡t

❈ ✇✐t❤ ❘❡ ✵✳ ❚❤❡♥ ✐s ❛♥ ❡✐❣❡♥❢✉♥❝t✐♦♥ ♦❢ ✐♥ ✵ ✱ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡✐❣❡♥✈❛❧✉❡ ✐s

✶ ✵

✳ ■❢ ❘❡ ✱ t❤❡♥ t❤❡ ❡✐❣❡♥❢✉♥❝t✐♦♥ ❜❡❧♦♥❣s t♦ ✵ ✳

❚❤❡♦r❡♠ ✶✶

▲❡t

✶ ✵ ✶ ✱

✳ ❚❤❡♥ t❤❡ s♣❡❝tr✉♠ ♦❢ ✐♥ ✵ ✐s ❣✐✈❡♥ ❜② ✵ ✵ ✵ ✵ ✵ ✵ ❘❡ ✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✶✹ ✴ ✷✺

❈♦r❞✐❛❧ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦rs ✐♥ s♣❛❝❡s C m

❉❡♥♦t❡ ∆T = {(s, t) : t ∈ [✵, T], s ∈ [✵, t]}✳

❚❤❡♦r❡♠ ✾

▲❡t ϕ ∈ L✶(✵, ✶)✱ a ∈ C m(∆T)✳ ❚❤❡♥ Vϕ,a ∈ L(C m[✵, T]) ❛♥❞ Vϕ,aC m[✵,T] ≤ CϕL✶(✵,✶)aC m(∆T )✳

❚❤❡♦r❡♠ ✶✵

▲❡t ϕ ∈ L✶(✵, ✶) ❛♥❞ ❧❡t λ ∈ ❈ ✇✐t❤ ❘❡ λ > ✵✳ ❚❤❡♥ tλ ✐s ❛♥ ❡✐❣❡♥❢✉♥❝t✐♦♥ ♦❢ Vϕ ✐♥ C[✵, T]✱ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡✐❣❡♥✈❛❧✉❡ ✐s ˆ ϕ(λ) = ✶

✵ ϕ(x)xλdx✳

■❢ ❘❡ λ > m✱ t❤❡♥ t❤❡ ❡✐❣❡♥❢✉♥❝t✐♦♥ ❜❡❧♦♥❣s t♦ C m[✵, T]✳

❚❤❡♦r❡♠ ✶✶

▲❡t

✶ ✵ ✶ ✱

✳ ❚❤❡♥ t❤❡ s♣❡❝tr✉♠ ♦❢ ✐♥ ✵ ✐s ❣✐✈❡♥ ❜② ✵ ✵ ✵ ✵ ✵ ✵ ❘❡ ✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✶✹ ✴ ✷✺

❈♦r❞✐❛❧ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦rs ✐♥ s♣❛❝❡s C m

❉❡♥♦t❡ ∆T = {(s, t) : t ∈ [✵, T], s ∈ [✵, t]}✳

❚❤❡♦r❡♠ ✾

▲❡t ϕ ∈ L✶(✵, ✶)✱ a ∈ C m(∆T)✳ ❚❤❡♥ Vϕ,a ∈ L(C m[✵, T]) ❛♥❞ Vϕ,aC m[✵,T] ≤ CϕL✶(✵,✶)aC m(∆T )✳

❚❤❡♦r❡♠ ✶✵

▲❡t ϕ ∈ L✶(✵, ✶) ❛♥❞ ❧❡t λ ∈ ❈ ✇✐t❤ ❘❡ λ > ✵✳ ❚❤❡♥ tλ ✐s ❛♥ ❡✐❣❡♥❢✉♥❝t✐♦♥ ♦❢ Vϕ ✐♥ C[✵, T]✱ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡✐❣❡♥✈❛❧✉❡ ✐s ˆ ϕ(λ) = ✶

✵ ϕ(x)xλdx✳

■❢ ❘❡ λ > m✱ t❤❡♥ t❤❡ ❡✐❣❡♥❢✉♥❝t✐♦♥ ❜❡❧♦♥❣s t♦ C m[✵, T]✳

❚❤❡♦r❡♠ ✶✶

▲❡t ϕ ∈ L✶(✵, ✶)✱ a ∈ C m(∆T)✳ ❚❤❡♥ t❤❡ s♣❡❝tr✉♠ ♦❢ Vϕ,a ✐♥ C m[✵, T] ✐s ❣✐✈❡♥ ❜② σm(Vϕ,a) = {✵}∪{a(✵, ✵) ˆ ϕ(k), k = ✵, . . . , m}∪{a(✵, ✵) ˆ ϕ(λ), ❘❡ λ > m}✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✶✹ ✴ ✷✺

❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥

t

✵

✶ t a(t, s)ϕ(s t )u(s)ds = f (t), ✵ ≤ t ≤ T ✭✸✮

❚❤❡♦r❡♠ ✶✷

▲❡t ϕ ∈ L✶(✵, ✶)✱ x(✶ − x)ϕ′(x) ∈ L✶(✵, ✶)✱ ✶

✵

ϕ(x)dx > ✵ ❛♥❞ t❤❡r❡ ❡①✐sts β < ✶ s✉❝❤ t❤❛t (xβϕ(x))′ ≥ ✵ ❢♦r x ∈ (✵, ✶)✳ ❆ss✉♠❡ ❛❧s♦ t❤❛t a ∈ C m+✶(∆T) ❛♥❞ a(t, t) = ✵✳ ❚❤❡♥ Vϕ,a ✐s ✐♥❥❡❝t✐✈❡ ✐♥ C[✵, T]✱ C m+✶[✵, T] ⊂ Vϕ,a(C m[✵, T]) ⊂ C m[✵, T]✱ ❛♥❞ V −✶

ϕ,a ∈ L(C m+✶[✵, T], C m[✵, T])✳

❈♦r♦❧❧❛r② ✶✸

▲❡t t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ✶✷ ❜❡ s❛t✐s✜❡❞ ❛♥❞ ❧❡t

✶ ✵

❜❡ ❣✐✈❡♥✳ ❚❤❡♥ t❤❡ ❡q✉❛t✐♦♥ ✐s ✉♥✐q✉❡❧② s♦❧✈❛❜❧❡ ✐♥ ✵ ❛♥❞ ✐ts s♦❧✉t✐♦♥ ✐s ✐♥ ✵ ✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✶✺ ✴ ✷✺

❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥

t

✵

✶ t a(t, s)ϕ(s t )u(s)ds = f (t), ✵ ≤ t ≤ T ✭✸✮

❚❤❡♦r❡♠ ✶✷

▲❡t ϕ ∈ L✶(✵, ✶)✱ x(✶ − x)ϕ′(x) ∈ L✶(✵, ✶)✱ ✶

✵

ϕ(x)dx > ✵ ❛♥❞ t❤❡r❡ ❡①✐sts β < ✶ s✉❝❤ t❤❛t (xβϕ(x))′ ≥ ✵ ❢♦r x ∈ (✵, ✶)✳ ❆ss✉♠❡ ❛❧s♦ t❤❛t a ∈ C m+✶(∆T) ❛♥❞ a(t, t) = ✵✳ ❚❤❡♥ Vϕ,a ✐s ✐♥❥❡❝t✐✈❡ ✐♥ C[✵, T]✱ C m+✶[✵, T] ⊂ Vϕ,a(C m[✵, T]) ⊂ C m[✵, T]✱ ❛♥❞ V −✶

ϕ,a ∈ L(C m+✶[✵, T], C m[✵, T])✳

❈♦r♦❧❧❛r② ✶✸

▲❡t t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ✶✷ ❜❡ s❛t✐s✜❡❞ ❛♥❞ ❧❡t f ∈ C m+✶[✵, T] ❜❡ ❣✐✈❡♥✳ ❚❤❡♥ t❤❡ ❡q✉❛t✐♦♥ ✐s ✉♥✐q✉❡❧② s♦❧✈❛❜❧❡ ✐♥ C[✵, T] ❛♥❞ ✐ts s♦❧✉t✐♦♥ ✐s ✐♥ C m[✵, T]✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✶✺ ✴ ✷✺

P♦❧②♥♦♠✐❛❧ ❝♦❧❧♦❝❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ❝♦r❞✐❛❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s

▲♦♦❦ ❢♦r s♦❧✉t✐♦♥s ✐♥ t❤❡ ❢♦r♠ un(t) =

N

• k=✵

cktk✳ ❈♦❧❧♦❝❛t✐♦♥ ♠❡t❤♦❞✿ ❝❤♦♦s❡ t❤❡ ❝♦❧❧♦❝❛t✐♦♥ ♣♦✐♥ts tk✱ k = ✵, . . . , n✱ s♦❧✈❡ t❤❡ ❝♦❧❧♦❝❛t✐♦♥ ❡q✉❛t✐♦♥s

N

• j=✵

cj T

✵

✶ tk a(tk, s)ϕ( s tk )sjds = f (tk), k = ✵, . . . n. ✭✹✮ ◆❡❡❞ t♦ ❝❛❧❝✉❧❛t❡ ❡①❛❝t❧② ♦r ✏✇❡❧❧ ❡♥♦✉❣❤✑ t❤❡ ✐♥t❡❣r❛❧s

✵

✶

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✶✻ ✴ ✷✺

P♦❧②♥♦♠✐❛❧ ❝♦❧❧♦❝❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ❝♦r❞✐❛❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s

▲♦♦❦ ❢♦r s♦❧✉t✐♦♥s ✐♥ t❤❡ ❢♦r♠ un(t) =

N

• k=✵

cktk✳ ❈♦❧❧♦❝❛t✐♦♥ ♠❡t❤♦❞✿ ❝❤♦♦s❡ t❤❡ ❝♦❧❧♦❝❛t✐♦♥ ♣♦✐♥ts tk✱ k = ✵, . . . , n✱ s♦❧✈❡ t❤❡ ❝♦❧❧♦❝❛t✐♦♥ ❡q✉❛t✐♦♥s

N

• j=✵

cj T

✵

✶ tk a(tk, s)ϕ( s tk )sjds = f (tk), k = ✵, . . . n. ✭✹✮ ◆❡❡❞ t♦ ❝❛❧❝✉❧❛t❡ ❡①❛❝t❧② ♦r ✏✇❡❧❧ ❡♥♦✉❣❤✑ t❤❡ ✐♥t❡❣r❛❧s T

✵

✶ tk a(tk, s)ϕ( s tk )sjds

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✶✻ ✴ ✷✺

■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ E = F = C[✵, T]✱ En ✐s t❤❡ s♣❛❝❡ ♦❢ ♣♦❧②♥♦♠✐❛❧s ♦❢ ♦r❞❡r ✉♣ t♦ n ❛♥❞ Zn ✐s t❤❡ ❧✐♥❡❛r s♣❛♥ ♦❢ δ✲❢✉♥❝t✐♦♥s ✇✐t❤ s✉♣♣♦rts tk✱ k = ✵, . . . , n✳ ▲❡t a(t, s) ≡ ✶✳ ❚❤❡♥ Vϕ : En → En ❛♥❞ τn ✐s s✐♠♣❧② t❤❡ ♥♦r♠ ♦❢ t❤❡ ✐♥t❡r♣♦❧❛t✐♦♥ ♣r♦❥❡❝t♦r ❢r♦♠ C t♦ C ✇✐t❤ t❤❡ ✐♥t❡r♣♦❧❛t✐♦♥ ♥♦❞❡s tk✱ k = ✵, . . . , n✳ ■❢ tk ❛r❡ t❤❡ ❈❤❡❜②s❤❡✈ ♥♦❞❡s✱ t❤❡♥ τn = ✷

π ❧♥(n + ✶) + ✶✳

❲❡ ❝❤♦s❡ ❝❡rt❛✐♥ ♥♦✐s❡ ❧❡✈❡❧s ❛♥❞ t❤❡ ♥♦✐s❡ ✇❛s ❣❡♥❡r❛t❡❞ ❜② r❛♥❞♦♠ ♥✉♠❜❡rs ✇✐t❤ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ❛t t❤❡ ❝♦❧❧♦❝❛t✐♦♥ ♥♦❞❡s✳ ❲❡ ❛❧s♦ ❢♦✉♥❞ t❤❡ ♦♣t✐♠❛❧ ♥✉♠❜❡r ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡rr♦r ♠✐♥ ❲❡ ✉s❡❞ t❤❡ ♠♦❞✐✜❡❞ ❞✐s❝r❡♣❛♥❝② ♣r✐♥❝✐♣❧❡ t♦ ✜♥❞ t❤❡ ✜rst s❛t✐s❢②✐♥❣ t❤❡ ✐♥❡q✉❛❧✐t② ✇✐t❤ ✶ ✵✵✶ ✶ ✳ ❲❡ ❞❡♥♦t❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡rr♦r ❜② ✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✶✼ ✴ ✷✺

■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ E = F = C[✵, T]✱ En ✐s t❤❡ s♣❛❝❡ ♦❢ ♣♦❧②♥♦♠✐❛❧s ♦❢ ♦r❞❡r ✉♣ t♦ n ❛♥❞ Zn ✐s t❤❡ ❧✐♥❡❛r s♣❛♥ ♦❢ δ✲❢✉♥❝t✐♦♥s ✇✐t❤ s✉♣♣♦rts tk✱ k = ✵, . . . , n✳ ▲❡t a(t, s) ≡ ✶✳ ❚❤❡♥ Vϕ : En → En ❛♥❞ τn ✐s s✐♠♣❧② t❤❡ ♥♦r♠ ♦❢ t❤❡ ✐♥t❡r♣♦❧❛t✐♦♥ ♣r♦❥❡❝t♦r ❢r♦♠ C t♦ C ✇✐t❤ t❤❡ ✐♥t❡r♣♦❧❛t✐♦♥ ♥♦❞❡s tk✱ k = ✵, . . . , n✳ ■❢ tk ❛r❡ t❤❡ ❈❤❡❜②s❤❡✈ ♥♦❞❡s✱ t❤❡♥ τn = ✷

π ❧♥(n + ✶) + ✶✳

❲❡ ❝❤♦s❡ ❝❡rt❛✐♥ ♥♦✐s❡ ❧❡✈❡❧s ❛♥❞ t❤❡ ♥♦✐s❡ ✇❛s ❣❡♥❡r❛t❡❞ ❜② r❛♥❞♦♠ ♥✉♠❜❡rs ✇✐t❤ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ❛t t❤❡ ❝♦❧❧♦❝❛t✐♦♥ ♥♦❞❡s✳ ❲❡ ❛❧s♦ ❢♦✉♥❞ t❤❡ ♦♣t✐♠❛❧ ♥✉♠❜❡r nopt ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡rr♦r eopt = ♠✐♥

n∈N un − u∗E = unopt − u∗E.

❲❡ ✉s❡❞ t❤❡ ♠♦❞✐✜❡❞ ❞✐s❝r❡♣❛♥❝② ♣r✐♥❝✐♣❧❡ t♦ ✜♥❞ t❤❡ ✜rst n = nDP s❛t✐s❢②✐♥❣ t❤❡ ✐♥❡q✉❛❧✐t② Aun − f δF ≤ bnδ ✇✐t❤ bn = ✶.✵✵✶(✶ + τn)✳ ❲❡ ❞❡♥♦t❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡rr♦r ❜② eDP = unDP − u∗✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✶✼ ✴ ✷✺

◆✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡s✿ ❝♦r❞✐❛❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥

❊①❛♠♣❧❡ ✶✳ ❈♦♥s✐❞❡r t❤❡ ❝♦r❞✐❛❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ✭❤❡r❡ φ(x) =

✶ √x ✮

t

✵

u(s)ds √st = ✶ t✷ + ✶, t ∈ [✵, T] ✇✐t❤ ❡①❛❝t s♦❧✉t✐♦♥ u(s) =

✶−✸s✷ ✷(s✷+✶)✷ ✳

❋♦r t❤✐s ❡q✉❛t✐♦♥ κn ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ✉s✐♥❣ ▼❛r❦♦✛✬s ✐♥❡q✉❛❧✐t②✱ ❜② Cn✷✳ ❙✐♥❝❡ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✐s ❛♥❛❧②t✐❝✱ ❞✐st(f , AEn) ❝♦♥✈❡r❣❡s t♦ ③❡r♦ ❡①♣♦♥❡♥t✐❛❧❧②✱ ❤❡♥❝❡ t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ✽ ✭♠♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❞✐s❝r❡♣❛♥❝② ♣r✐♥❝✐♣❧❡✮ ❛r❡ s❛t✐s✜❡❞✳ ❲❡ t♦♦❦ T = ✶✵ ❛♥❞ ✉s❡❞ ♥♦✐s② ❞❛t❛ ✇✐t❤ ♥♦✐s❡ ❧❡✈❡❧s δ = ✶✵−✹, ✶✵−✻, . . . , ✶✵−✶✹✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ❝♦❧❧♦❝❛t✐♦♥ ♥♦❞❡s ✇❛s ✶✵, ✶✺, ✷✵, . . . , ✶✶✵✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✶✽ ✴ ✷✺

❚❤❡ ♦♣t✐♠❛❧ ❡rr♦rs ❛♥❞ t❤❡ ❡rr♦rs ♦❜t❛✐♥❡❞ ❜② ✉s✐♥❣ t❤❡ ❞✐s❝r❡♣❛♥❝② ♣r✐♥❝✐♣❧❡ ✇✐t❤ bnDP ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ t❛❜❧❡✳ δ eopt nopt eDP nDP bnDP ✶✵−✹ ✻ · ✶✵−✷ ✷✺ ✽ · ✶✵−✷ ✷✵ ✸.✾✹ ✶✵−✻ ✶.✵✶ · ✶✵−✸ ✹✵ ✷.✹ · ✶✵−✸ ✸✵ ✹.✶✾ ✶✵−✽ ✶.✺✶ · ✶✵−✺ ✹✵ ✶.✺✶ · ✶✵−✺ ✹✵ ✹.✸✻ ✶✵−✶✵ ✶.✽ · ✶✵−✼ ✺✵ ✶.✽ · ✶✵−✼ ✺✵ ✹.✺✻ ✶✵−✶✷ ✹.✻✾ · ✶✵−✾ ✼✺ ✾.✺✽ · ✶✵−✾ ✻✵ ✹.✻✷ ✶✵−✶✹ ✼.✵✹ · ✶✵−✶✶ ✶✵✺ ✼.✺✼ · ✶✵−✶✶ ✼✵ ✹.✼✶

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✶✾ ✴ ✷✺

❊①❛♠♣❧❡ ✷✳ ❈♦♥s✐❞❡r t❤❡ ❡q✉❛t✐♦♥ t

✵

u(s)ds √st = t✸/✷(✷ − t)✺/✷, t ∈ [✵, ✷]. ❚❤❡ ❡①❛❝t s♦❧✉t✐♦♥ ✐s ✷t✸/✷(✷ − t)✺/✷ − ✺ ✷t✺/✷(✷ − t)✸/✷✳ ❚❤❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✐s t❤❡ s❛♠❡ ❛s ✐♥ ❊①❛♠♣❧❡ ✶✱ ❤❡♥❝❡ κn ≤ Cn✷✳ ❚❤❡ ❞✐st❛♥❝❡ ❞✐st(f , AEn) ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ❜② Cn−✸✱ ❤❡♥❝❡ t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ✽ ❛r❡ s❛t✐s✜❡❞✳ ❲❡ ✉s❡❞ ♥♦✐s② ❞❛t❛ ✇✐t❤ ♥♦✐s❡ ❧❡✈❡❧s δ = ✶✵−✸, ✶✵−✹, . . . , ✶✵−✼✳ ❚❤❡ ♥♦✐s❡ ✇❛s ❣❡♥❡r❛t❡❞ ❜② r❛♥❞♦♠ ♥✉♠❜❡rs ✇✐t❤ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ❛t t❤❡ ❝♦❧❧♦❝❛t✐♦♥ ♥♦❞❡s✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ❝♦❧❧♦❝❛t✐♦♥ ♥♦❞❡s ✇❛s ✶✵, ✷✵, ✸✵, . . . , ✸✵✵✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✷✵ ✴ ✷✺

❚❤❡ ♦♣t✐♠❛❧ ❡rr♦rs ❛♥❞ t❤❡ ❡rr♦rs ♦❜t❛✐♥❡❞ ❜② ✉s✐♥❣ t❤❡ ❞✐s❝r❡♣❛♥❝② ♣r✐♥❝✐♣❧❡ ✇✐t❤ bnDP ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ t❛❜❧❡✳ δ eopt nopt eDP nDP bnDP ✶✵−✸ ✶.✺ · ✶✵−✶ ✶✵ ✶.✺ · ✶✵−✶ ✶✵ ✸.✺✸ ✶✵−✹ ✺ · ✶✵−✷ ✹✵ ✶.✶ · ✶✵−✶ ✸✵ ✹.✶✾ ✶✵−✺ ✺.✷✹ · ✶✵−✸ ✷✵ ✷ · ✶✵−✷ ✺✵ ✹.✺ ✶✵−✻ ✻.✶✸ · ✶✵−✹ ✹✵ ✺.✶✻ · ✶✵−✸ ✶✵✵ ✹.✾✹ ✶✵−✼ ✾.✶✼ · ✶✵−✺ ✾✵ ✺.✼✼ · ✶✵−✸ ✷✸✵ ✺.✹✻

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✷✶ ✴ ✷✺

◆✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡s✿ s♣❧✐♥❡✲❝♦❧❧♦❝❛t✐♦♥ ❢♦r ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥

❈♦♥s✐❞❡r ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ✜rst ❦✐♥❞ (Au)(t) := t

✵

K(t, s)u(s) ds = f (t), t ∈ [✵, ✶] ✇✐t❤ t❤❡ ♦♣❡r❛t♦r A ∈ L(Lp(✵, ✶), C[✵, ✶]), ✶ ≤ p ≤ ∞✳ ❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ s♣❛❝❡ ✐s En = S(−✶)

k−✶ (I∆)✱ t❤❡ s♣❛❝❡ ♦❢ ❞✐s❝♦♥t✐♥✉♦✉s ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧s

♦❢ ♦r❞❡r k − ✶ ✇✐t❤ ♠❡s❤ ∆✳ ❲❡ ✜♥❞ un ∈ En s✉❝❤ t❤❛t Aun(ti,j) = f δ(ti,j), i = ✶, . . . , n, j = ✶, . . . , k ✇❤❡r❡ ti,j = (i − ✶ + cj)h ∈ [✵, ✶]✱ i = ✶, . . . , n✱ j = ✶, . . . , k ❛r❡ ❝♦❧❧♦❝❛t✐♦♥ ♥♦❞❡s ❛♥❞ ✵ < c✶ < . . . < ck ≤ ✶ ❛r❡ ❝♦❧❧♦❝❛t✐♦♥ ♣❛r❛♠❡t❡rs ✇❤♦s❡ ❝❤♦✐❝❡ ✐s ❡ss❡♥t✐❛❧✳ ◆♦✇ ˘ κn ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ✉s✐♥❣ ❚❤❡♦r❡♠ ✻ ✭❊st✐♠❛t❡ ♦❢ ˘ κn✮❀ ✐t ❞❡♣❡♥❞s ♦♥ ❤♦✇ ♠✉❝❤ A ✐s s♠♦♦t❤✐♥❣✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✷✷ ✴ ✷✺

❊①❛♠♣❧❡ ✸✳ ❈♦♥s✐❞❡r t❤❡ ❡q✉❛t✐♦♥ Au(t) = t

✵

u(s)ds = tq q , t ∈ [✵, ✶], q ∈ {✸/✷, ✺/✷} ✇✐t❤ ♦♣❡r❛t♦r A : L✶(✵, ✶) → C[✵, ✶]✳ ❚❤❡ ❡①❛❝t s♦❧✉t✐♦♥ ✐s u(s) = sq−✶✳ ❲❡ ✉s❡❞ ❢♦r En t❤❡ s♣❛❝❡ ♦❢ ❞✐s❝♦♥t✐♥✉♦✉s ❧✐♥❡❛r s♣❧✐♥❡s ✇✐t❤ ✉♥✐❢♦r♠ ♠❡s❤ ih✱ i = ✵, . . . , n✱ ✇❤❡r❡ h = ✶/n✳ ❚❤❡ ❝♦❧❧♦❝❛t✐♦♥ ♣♦✐♥ts ❛r❡ ti✶ = (i − ✶ + c))h✱ ti✷ = ih✱ c ∈ (✵, ✶)✳ ❲❡ t♦♦❦ ❢♦r Fn t❤❡ s♣❛❝❡ ♦❢ ❝♦♥t✐♥✉♦✉s ❧✐♥❡❛r s♣❧✐♥❡s ❛♥❞ t❤❡ ✐♥✈❡rs❡ ♣r♦♣❡rt② ♦❢ s♣❧✐♥❡s ❣✐✈❡s w′

n ≤ Cnwn ∀wn ∈ Fn✱ ❤❡♥❝❡ ˘

κn ≤ Cn✳ ❚❤❡ ❞✐st❛♥❝❡ ❞✐st(f , AEn) ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ❜② Cn−q✳ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t ❤❡r❡ τ =

• ✶ +

c✷ ✷(✶−c),

✐❢ c ≥ ✶

✷,

✶ + (✶−c)✷

✷c

✐❢ c ≤ ✶

✷.

τ ✐s ♠✐♥✐♠❛❧ ❢♦r c = ✶

✷✱ t❤❡♥ τ = ✶.✷✺✳ ■♥ t❤✐s ❡①❛♠♣❧❡ τn = τ✳

❲❡ ✉s❡❞ c = ✶

✷ ❛♥❞ t♦♦❦ b = ✶.✵✶ + τ = ✷.✷✻ ❢♦r t❤❡ ❞✐s❝r❡♣❛♥❝② ♣r✐♥❝✐♣❧❡✳

❚❤❡ ♥♦✐s② ❞❛t❛ ✇❡r❡ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❢♦r♠✉❧❛ f δ(ti,j) = f (ti,j) + δθi,j✱ ✇❤❡r❡ δ = ✶✵−m, m ∈ {✷, . . . , ✼} ❛♥❞ θi,j ❛r❡ r❛♥❞♦♠ ♥✉♠❜❡rs ✇✐t❤ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✱ ♥♦r♠❡❞ ❛❢t❡r ❜❡✐♥❣ ❣❡♥❡r❛t❡❞✿ ♠❛①i,j |θi,j| = ✶.

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✷✸ ✴ ✷✺

❚❤❡ ♦♣t✐♠❛❧ ❡rr♦rs ❛♥❞ t❤❡ ❡rr♦rs ♦❜t❛✐♥❡❞ ❜② ✉s✐♥❣ t❤❡ ❞✐s❝r❡♣❛♥❝② ♣r✐♥❝✐♣❧❡ ✇✐t❤ q = ✸/✷ ❛♥❞ q = ✺/✷✳

δ eopt nopt eDP nDP eopt nopt eDP nDP ✶✵−✶ ✷.✺ · ✶✵−✶ ✶ ✷.✺ · ✶✵−✶ ✶ ✷.✾ · ✶✵−✶ ✶ ✷.✾ · ✶✵−✶ ✶ ✶✵−✷ ✻.✽ · ✶✵−✷ ✷ ✻.✽ · ✶✵−✷ ✷ ✺.✹ · ✶✵−✷ ✷ ✺.✹ · ✶✵−✷ ✷ ✶✵−✸ ✶.✸ · ✶✵−✷ ✽ ✶.✽ · ✶✵−✷ ✺ ✾ · ✶✵−✸ ✻ ✶.✶ · ✶✵−✷ ✺ ✶✵−✹ ✸.✷ · ✶✵−✸ ✷✹ ✸.✸ · ✶✵−✸ ✷✵ ✶.✼ · ✶✵−✸ ✶✺ ✸ · ✶✵−✸ ✽ ✶✵−✺ ✼.✻ · ✶✵−✹ ✼✷ ✽.✹ · ✶✵−✹ ✽✻ ✸.✺ · ✶✵−✹ ✸✷ ✻.✷ · ✶✵−✹ ✶✽ ✶✵−✻ ✶.✾ · ✶✵−✹ ✶✷✽ ✸.✸ · ✶✵−✹ ✺✶✷ ✻.✽ · ✶✵−✺ ✼✷ ✾.✾ · ✶✵−✺ ✹✻ ✶✵−✼ ✹.✺ · ✶✵−✺ ✺✶✷ ✶.✷ · ✶✵−✹ ✷✵✹✽ ✶.✺ · ✶✵−✺ ✶✷✽ ✶.✺ · ✶✵−✺ ✶✷✽

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✷✹ ✴ ✷✺

❯✳ ❍ä♠❛r✐❦✱ ❯✳ ❑❛♥❣r♦✱ ❖♥ s❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ♦❢ ✐❧❧✲♣♦s❡❞ ♣r♦❜❧❡♠s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❜② ♣r♦❥❡❝t✐♦♥ ♠❡t❤♦❞s✳ ■♥✿ ❚r❡♥❞s ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ❇✐r❦❤ä✉s❡r✱ ✷✵✶✽ ✭t♦ ❛♣♣❡❛r✮✳ ❯✳ ❍ä♠❛r✐❦✱ ❇✳ ❑❛❧t❡♥❜❛❝❤❡r✱ ❯✳ ❑❛♥❣r♦✱ ❊✳ ❘❡s♠❡r✐t❛✱ ❘❡❣✉❧❛r✐③❛t✐♦♥ ❜② ❉✐s❝r❡t✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ ❙♣❛❝❡s✱ ■♥✈❡rs❡ Pr♦❜❧❡♠s✱ ✸✷✱ ✸✱ ✵✸✺✵✵✹✱ ✷✵✶✻✳ ❯✳ ❑❛♥❣r♦✱ ❈♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ❛♥❞ s✐♥❣✉❧❛r ❢r❛❝t✐♦♥❛❧ ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✐♥ s♣❛❝❡s ♦❢ ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s✱ ▼❛t❤✳ ▼♦❞❡❧❧✐♥❣ ❆♥❛❧✳✱ ✷✷✭✹✮✱ ✷✵✶✼✱ ♣♣✳ ✺✹✽✕✺✻✼✳ ❘✳ ❑r❡ss✱ ▲✐♥❡❛r ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥s✳ ❙♣r✐♥❣❡r✱ ✷✵✶✹✳

• ✳ ❱❛✐♥✐❦❦♦✱ ❋✐rst ❑✐♥❞ ❈♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ✶✱ ◆✉♠❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❖♣t✐♠✳

✸✸✿✻✱ ✷✵✶✷✱ ♣♣✳ ✻✽✵✕✼✵✹✳

• ✳ ❱❛✐♥✐❦❦♦✱ ❋✐rst ❑✐♥❞ ❈♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ✷✱ ◆✉♠❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❖♣t✐♠✳

✸✺✿✶✷✱ ✷✵✶✹✱ ♣♣✳ ✶✻✵✼✕✶✻✸✼✳

• ✳ ❱❛✐♥✐❦❦♦✱ ❈♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ✶✱ ◆✉♠❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❖♣t✐♠✳ ✸✵✱ ✷✵✵✾✱

♣♣✳ ✶✶✹✺✕✶✶✼✷✳

• ✳ ❱❛✐♥✐❦❦♦✱ ❈♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ✷✱ ◆✉♠❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❖♣t✐♠✳ ✸✶✱ ✷✵✶✵✱

♣♣✳ ✶✾✶✕✷✶✾✳

❯✳ ❑❛♥❣r♦✱ ❯✳ ❍ä♠❛r✐❦ ✭❯♥✐✈✳ ♦❢ ❚❛rt✉✮ ❙❡❧❢✲r❡❣✉❧❛r✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❘✐♦ ✷✵✶✼ ✷✺ ✴ ✷✺