stt s r r sr - - PowerPoint PPT Presentation

❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s

P❡t❡r ❇♦②✈❛❧❡♥❦♦✈

■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ■♥❢♦r♠❛t✐❝s✱ ❇✉❧❣❛r✐❛♥ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s✱ ❙♦✜❛✱ ❇✉❧❣❛r✐❛

❏♦✐♥t ✇♦r❦ ✇✐t❤✿ P✳ ❉r❛❣♥❡✈ ✭❉❡♣t✳ ▼❛t❤✳ ❙❝✐❡♥❝❡s✱ ■P❋❲✱ ❋♦rt ❲❛②♥❡✱ ■◆✱ ❯❙❆✮ ❉✳ ❍❛r❞✐♥✱ ❊✳ ❙❛✛ ✭❉❡♣t✳ ▼❛t❤✳✱ ❱❛♥❞❡r❜✐❧t ❯♥✐✈❡rs✐t②✱ ◆❛s❤✈✐❧❧❡✱ ❚◆✱ ❯❙❆✮ ▼❛②❛ ❙t♦②❛♥♦✈❛ ❉❡♣t✳ ▼❛t❤✳ ■♥❢✳ ❙♦✜❛ ❯♥✐✈❡rs✐t②✱ ❙♦✜❛✱ ❇✉❧❣❛r✐❛

▼✐❞✇❡st❡r♥ ❲♦r❦s❤♦♣ ♦♥ ❆s②♠♣t♦t✐❝ ❆♥❛❧②s✐s✱ ❖❝t♦❜❡r ✼✲✾✱ ✷✵✶✻✱ ❋♦rt ❲❛②♥❡✱ ❯❙❆

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✶ ✴ ✸✶

❊♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ✭✶✮

▲❡t Sn−✶ ❞❡♥♦t❡ t❤❡ ✉♥✐t s♣❤❡r❡ ✐♥ Rn✳ ❆ ✜♥✐t❡ ♥♦♥❡♠♣t② s❡t C ⊂ Sn−✶ ✐s ❝❛❧❧❡❞ ❛ s♣❤❡r✐❝❛❧ ❝♦❞❡✳ ❉❡✜♥✐t✐♦♥ ❋♦r ❛ ❣✐✈❡♥ ✭❡①t❡♥❞❡❞ r❡❛❧✲✈❛❧✉❡❞✮ ❢✉♥❝t✐♦♥ h(t) : [−✶, ✶] → [✵, +∞]✱ ✇❡ ❞❡✜♥❡ t❤❡ h✲❡♥❡r❣② ✭♦r ♣♦t❡♥t✐❛❧ ❡♥❡r❣②✮ ♦❢ ❛ s♣❤❡r✐❝❛❧ ❝♦❞❡ C ❜② E(n, C; h) := ✶ |C|

• x,y∈C,x=y

h(x, y), ✇❤❡r❡ x, y ❞❡♥♦t❡s t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦❢ x ❛♥❞ y✳ ❚❤❡ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥ h ✐s ❝❛❧❧❡❞ k✲❛❜s♦❧✉t❡❧② ♠♦♥♦t♦♥❡ ♦♥ [−✶, ✶) ✐❢ ✐ts ❞❡r✐✈❛t✐✈❡s h(i)(t)✱ i = ✵, ✶, . . . , k✱ ❛r❡ ♥♦♥♥❡❣❛t✐✈❡ ❢♦r ❛❧❧ ✵ ≤ i ≤ k ❛♥❞ ❡✈❡r② t ∈ [−✶, ✶)✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✷ ✴ ✸✶

❊♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ✭✷✮

Pr♦❜❧❡♠ ▼✐♥✐♠✐③❡ t❤❡ ♣♦t❡♥t✐❛❧ ❡♥❡r❣② ♣r♦✈✐❞❡❞ t❤❡ ❝❛r❞✐♥❛❧✐t② |C| ♦❢ C ✐s ✜①❡❞❀ t❤❛t ✐s✱ t♦ ❞❡t❡r♠✐♥❡ E(n, M; h) := ✐♥❢{E(n, C; h) : |C| = M} t❤❡ ♠✐♥✐♠✉♠ ♣♦ss✐❜❧❡ h✲❡♥❡r❣② ♦❢ ❛ s♣❤❡r✐❝❛❧ ❝♦❞❡ ♦❢ ❝❛r❞✐♥❛❧✐t② M✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✸ ✴ ✸✶

❊♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ✭✸✮

❙♦♠❡ ✐♥t❡r❡st✐♥❣ ♣♦t❡♥t✐❛❧s✿ ❘✐❡s③ α✲♣♦t❡♥t✐❛❧✿ h(t) = (✷ − ✷t)−α/✷ = |x − y|−α✱ α > ✵❀ ◆❡✇t♦♥ ♣♦t❡♥t✐❛❧✿ h(t) = (✷ − ✷t)−(n−✷)/✷ = |x − y|−(n−✷)❀ ▲♦❣ ♣♦t❡♥t✐❛❧✿ h(t) = −(✶/✷) ❧♦❣(✷ − ✷t) = − ❧♦❣ |x − y|❀

• ❛✉ss✐❛♥ ♣♦t❡♥t✐❛❧✿ h(t) = ❡①♣(✷t − ✷) = ❡①♣(−|x − y|✷)❀

❑♦r❡✈❛❛r ♣♦t❡♥t✐❛❧✿ h(t) = (✶ + r✷ − ✷rt)−(n−✷)/✷✱ ✵ < r < ✶✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✹ ✴ ✸✶

❙♦♠❡ r❡❢❡r❡♥❝❡s

P✳ ❉❡❧s❛rt❡✱ ❏✳✲▼✳ ●♦❡t❤❛❧s✱ ❏✳ ❏✳ ❙❡✐❞❡❧✱ ❙♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s✱

• ❡♦♠✳ ❉❡❞✐❝❛t❛ ✻✱ ♣♣✳ ✸✻✸✲✸✽✽✱ ✶✾✼✼✳

❱✳ ❆✳ ❨✉❞✐♥✱ ▼✐♥✐♠❛❧ ♣♦t❡♥t✐❛❧ ❡♥❡r❣② ♦❢ ❛ ♣♦✐♥t s②st❡♠ ♦❢ ❝❤❛r❣❡s✱ ❉✐s❝r✳ ▼❛t❤✳ ❆♣♣❧✳ ✸✱ ♣♣✳ ✼✺✲✽✶✱ ✶✾✾✸✳ ❱✳ ■✳ ▲❡✈❡♥s❤t❡✐♥✱ ❯♥✐✈❡rs❛❧ ❜♦✉♥❞s ❢♦r ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s✱ ❍❛♥❞❜♦♦❦ ♦❢ ❈♦❞✐♥❣ ❚❤❡♦r②✱ ❱✳ ❙✳ P❧❡ss ❛♥❞ ❲✳ ❈✳ ❍✉✛♠❛♥✱ ❊❞s✳✱ ❊❧s❡✈✐❡r✱ ❆♠st❡r❞❛♠✱ ❈❤✳ ✻✱ ♣♣✳ ✹✾✾✕✻✹✽✱ ✶✾✾✽✳ ❍✳ ❈♦❤♥✱ ❆✳ ❑✉♠❛r✱ ❯♥✐✈❡rs❛❧❧② ♦♣t✐♠❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ ♣♦✐♥ts ♦♥ s♣❤❡r❡s✳ ❏♦✉r♥❛❧ ♦❢ ❆▼❙✱ ✷✵✱ ♥♦✳ ✶✱ ♣♣✳ ✾✾✲✶✹✽✱ ✷✵✵✻✳ P✳ ❇♦②✈❛❧❡♥❦♦✈✱ P✳ ❉r❛❣♥❡✈✱ ❉✳ ❍❛r❞✐♥✱ ❊✳ ❙❛✛✱ ▼✳ ❙t♦②❛♥♦✈❛✱ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❜♦✉♥❞s ❢♦r ♣♦t❡♥t✐❛❧ ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ✭❛r①✐✈✶✺✵✸✳✵✼✷✷✽✮✱ t♦ ❛♣♣❡❛r ✐♥ ❈♦♥str✉❝t✐✈❡ ❆♣♣r♦①✐♠❛t✐♦♥✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✺ ✴ ✸✶

❯♥✐✈❡rs❛❧ ❧♦✇❡r ❜♦✉♥❞ ✭❯▲❇✮

❚❤❡♦r❡♠ ▲❡t n✱ M ∈ (D(n, τ), D(n, τ + ✶)] ❛♥❞ h ❜❡ ✜①❡❞✳ ❚❤❡♥ E(n, M; h) ≥ M

k−✶

• i=✵

ρih(αi), E(n, M; h) ≥ M

k

• i=✵

γih(βi). ❚❤❡s❡ ❜♦✉♥❞s ❝❛♥ ♥♦t ❜❡ ✐♠♣r♦✈❡❞ ❜② ✉s✐♥❣ ✑❣♦♦❞✑ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❛t ♠♦st τ✳ ◆♦t❡ t❤❡ ✉♥✐✈❡rs❛❧✐t② ❢❡❛t✉r❡ ✕ ρi, αi ✭r❡s♣✳ γi, βi✮ ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥ h✳ ◆❡①t ✕ t♦ ❡①♣❧❛✐♥ t❤❡ ❛❜♦✈❡ ♣❛r❛♠❡t❡rs ❛♥❞ t❤❡✐r ❝♦♥♥❡❝t✐♦♥s ❛♥❞ t♦ ✐♥✈❡st✐❣❛t❡ t❤❡ ❜♦✉♥❞ ✐♥ ❝❡rt❛✐♥ ❛s②♠♣t♦t✐❝ ♣r♦❝❡ss✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✻ ✴ ✸✶

• ❡❣❡♥❜❛✉❡r ♣♦❧②♥♦♠✐❛❧s

❋♦r ✜①❡❞ ❞✐♠❡♥s✐♦♥ n✱ t❤❡ ✭♥♦r♠❛❧✐③❡❞✮ ●❡❣❡♥❜❛✉❡r ♣♦❧②♥♦♠✐❛❧s ❛r❡ ❞❡✜♥❡❞ ❜② P(n)

✵ (t) := ✶✱ P(n) ✶ (t) := t ❛♥❞ t❤❡ t❤r❡❡✲t❡r♠ r❡❝✉rr❡♥❝❡

r❡❧❛t✐♦♥ (i + n − ✷) P(n)

i+✶(t) := (✷i + n − ✷) t P(n) i

(t) − i P(n)

i−✶(t) ❢♦r i ≥ ✶.

◆♦t❡ t❤❛t {P(n)

i

(t)} ❛r❡ ♦rt❤♦❣♦♥❛❧ ✐♥ [−✶, ✶] ✇✐t❤ ❛ ✇❡✐❣❤t (✶ − t✷)(n−✸)/✷ ❛♥❞ s❛t✐s❢② P(n)

i

(✶) = ✶ ❢♦r ❛❧❧ i ❛♥❞ n✳ ❲❡ ❤❛✈❡ P(n)

i

(t) = P((n−✸)/✷,(n−✸)/✷)

i

(t)/P((n−✸)/✷,(n−✸)/✷)

i

(✶)✱ ✇❤❡r❡ P(α,β)

i

(t) ❛r❡ t❤❡ ❏❛❝♦❜✐ ♣♦❧②♥♦♠✐❛❧s ✐♥ st❛♥❞❛r❞ ♥♦t❛t✐♦♥✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✼ ✴ ✸✶

❆❞❥❛❝❡♥t ♣♦❧②♥♦♠✐❛❧s

❚❤❡ ✭♥♦r♠❛❧✐③❡❞✮ ❏❛❝♦❜✐ ♣♦❧②♥♦♠✐❛❧s P

(a+ n−✸

✷ ,b+ n−✸ ✷ )

i

(t), a, b ∈ {✵, ✶}, P

(a+ n−✸

✷ ,b+ n−✸ ✷ )

i

(✶) = ✶ ❛♥❞ ❛r❡ ❝❛❧❧❡❞ ❛❞❥❛❝❡♥t ♣♦❧②♥♦♠✐❛❧s ✭▲❡✈❡♥s❤t❡✐♥✮✳ ❙❤♦rt ♥♦t❛t✐♦♥ P(a,b)

i

(t)✳ a = b = ✵ → ●❡❣❡♥❜❛✉❡r ♣♦❧②♥♦♠✐❛❧s✳ P(a,b)

i

(t) ❛r❡ ♦rt❤♦❣♦♥❛❧ ✐♥ [−✶, ✶] ✇✐t❤ ✇❡✐❣❤t (✶ − t)a(✶ + t)b(✶ − t✷)(n−✸)/✷✳ ▼❛♥② ✐♠♣♦rt❛♥t ♣r♦♣❡rt✐❡s ❢♦❧❧♦✇✱ ✐♥ ♣❛rt✐❝✉❧❛r ✐♥t❡r❧❛❝✐♥❣ ♦❢ ③❡r♦s✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✽ ✴ ✸✶

❙♣❤❡r✐❝❛❧ ❞❡s✐❣♥s ✭P✳ ❉❡❧s❛rt❡✱ ❏✳✲▼✳ ●♦❡t❤❛❧s✱ ❏✳ ❏✳ ❙❡✐❞❡❧✱ ✶✾✼✼✮

❉❡✜♥✐t✐♦♥ ❆ s♣❤❡r✐❝❛❧ τ✲❞❡s✐❣♥ C ⊂ Sn−✶ ✐s ❛ s♣❤❡r✐❝❛❧ ❝♦❞❡ ♦❢ Sn−✶ s✉❝❤ t❤❛t ✶ µ(Sn−✶)

• Sn−✶ f (x)dµ(x) = ✶

|C|

• x∈C

f (x) ✭µ(x) ✐s t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✮ ❤♦❧❞s ❢♦r ❛❧❧ ♣♦❧②♥♦♠✐❛❧s f (x) = f (x✶, x✷, . . . , xn) ♦❢ ❞❡❣r❡❡ ❛t ♠♦st τ✳ ❚❤❡ str❡♥❣t❤ ♦❢ C ✐s t❤❡ ♠❛①✐♠❛❧ ♥✉♠❜❡r τ = τ(C) s✉❝❤ t❤❛t C ✐s ❛ s♣❤❡r✐❝❛❧ τ✲❞❡s✐❣♥✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✾ ✴ ✸✶

❉❡❧s❛rt❡✲●♦❡t❤❛❧s✲❙❡✐❞❡❧ ❜♦✉♥❞s

❋♦r ✜①❡❞ str❡♥❣t❤ τ ❛♥❞ ❞✐♠❡♥s✐♦♥ n ❞❡♥♦t❡ ❜② B(n, τ) = ♠✐♥{|C| : ∃ τ✲❞❡s✐❣♥ C ⊂ Sn−✶} t❤❡ ♠✐♥✐♠✉♠ ♣♦ss✐❜❧❡ ❝❛r❞✐♥❛❧✐t② ♦❢ s♣❤❡r✐❝❛❧ τ✲❞❡s✐❣♥s C ⊂ Sn−✶✳ ❚❤❡♥ ❉❡❧s❛rt❡✲●♦❡t❤❛❧s✲❙❡✐❞❡❧ ❜♦✉♥❞ ✐s B(n, τ) ≥ D(n, τ) =    ✷ n+k−✷

n−✶

• ,

✐❢ τ = ✷k − ✶✱ n+k−✶

n−✶

• +

n+k−✷

n−✶

• ,

✐❢ τ = ✷k.

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✶✵ ✴ ✸✶

▲❡✈❡♥s❤t❡✐♥ ❜♦✉♥❞s ❢♦r s♣❤❡r✐❝❛❧ ❝♦❞❡s ✭✶✮

❋♦r ❡✈❡r② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r m ✇❡ ❝♦♥s✐❞❡r t❤❡ ✐♥t❡r✈❛❧s Im =   

• t✶,✶

k−✶, t✶,✵ k

• ,

✐❢ m = ✷k − ✶,

• t✶,✵

k , t✶,✶ k

• ,

✐❢ m = ✷k. ❍❡r❡ t✶,✶

✵

= −✶✱ ta,b

i

✱ a, b ∈ {✵, ✶}✱ i ≥ ✶✱ ✐s t❤❡ ❣r❡❛t❡st ③❡r♦ ♦❢ t❤❡ ❛❞❥❛❝❡♥t ♣♦❧②♥♦♠✐❛❧ P(a,b)

i

(t)✳ ❚❤❡ ✐♥t❡r✈❛❧s Im ❞❡✜♥❡ ♣❛rt✐t✐♦♥ ♦❢ I = [−✶, ✶) t♦ ❝♦✉♥t❛❜❧② ♠❛♥② ♥♦♥✲♦✈❡r❧❛♣♣✐♥❣ ❝❧♦s❡❞ s✉❜✐♥t❡r✈❛❧s✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✶✶ ✴ ✸✶

▲❡✈❡♥s❤t❡✐♥ ❜♦✉♥❞s ❢♦r s♣❤❡r✐❝❛❧ ❝♦❞❡s ✭✷✮

❋♦r ❡✈❡r② s ∈ Im✱ ▲❡✈❡♥s❤t❡✐♥ ✉s❡❞ ❝❡rt❛✐♥ ♣♦❧②♥♦♠✐❛❧ f (n,s)

m

(t) ♦❢ ❞❡❣r❡❡ m ✇❤✐❝❤ s❛t✐s❢② ❛❧❧ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ❜♦✉♥❞s ❢♦r s♣❤❡r✐❝❛❧ ❝♦❞❡s✳ ❚❤✐s ②✐❡❧❞s t❤❡ ❜♦✉♥❞ A(n, s) ≤                L✷k−✶(n, s) = k+n−✸

k−✶

✷k+n−✸

n−✶

−

P(n)

k−✶(s)−P(n) k (s)

(✶−s)P(n)

k (s)

• ❢♦r s ∈ I✷k−✶,

L✷k(n, s) = k+n−✷

k

✷k+n−✶

n−✶

−

(✶+s)(P(n)

k (s)−P(n) k+✶(s))

(✶−s)(P(n)

k (s)+P(n) k+✶(s))

• ❢♦r s ∈ I✷k.

❋♦r ❡✈❡r② ✜①❡❞ ❞✐♠❡♥s✐♦♥ n ❡❛❝❤ ❜♦✉♥❞ Lm(n, s) ✐s s♠♦♦t❤ ❛♥❞ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ s✳ ❚❤❡ ❢✉♥❝t✐♦♥ L(n, s) = L✷k−✶(n, s), ✐❢ s ∈ I✷k−✶, L✷k(n, s), ✐❢ s ∈ I✷k, ✐s ❝♦♥t✐♥✉♦✉s ✐♥ s✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✶✷ ✴ ✸✶

❈♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ ❉●❙✲ ❛♥❞ ▲✲❜♦✉♥❞s ✭✶✮

❚❤❡ ❝♦♥♥❡❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❉❡❧s❛rt❡✲●♦❡t❤❛❧s✲❙❡✐❞❡❧ ❜♦✉♥❞ ❛♥❞ t❤❡ ▲❡✈❡♥s❤t❡✐♥ ❜♦✉♥❞s ❛r❡ ❣✐✈❡♥ ❜② t❤❡ ❡q✉❛❧✐t✐❡s L✷k−✷(n, t✶,✶

k−✶) = L✷k−✶(n, t✶,✶ k−✶) = D(n, ✷k − ✶),

L✷k−✶(n, t✶,✵

k ) = L✷k(n, t✶,✵ k ) = D(n, ✷k)

❛t t❤❡ ❡♥❞s ♦❢ t❤❡ ✐♥t❡r✈❛❧s Im✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✶✸ ✴ ✸✶

❈♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ ❉●❙✲ ❛♥❞ ▲✲❜♦✉♥❞s ✭✷✮

❋♦r ❡✈❡r② ✜①❡❞ ✭❝❛r❞✐♥❛❧✐t②✮ M > D(n, ✷k − ✶) t❤❡r❡ ❡①✐st ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ r❡❛❧ ♥✉♠❜❡rs −✶ < α✵ < α✶ < · · · < αk−✶ < ✶ ❛♥❞ ♣♦s✐t✐✈❡ ρ✵, ρ✶, . . . , ρk−✶✱ s✉❝❤ t❤❛t t❤❡ ❡q✉❛❧✐t② ✭q✉❛❞r❛t✉r❡ ❢♦r♠✉❧❛✮ f✵ = f (✶) M +

k−✶

• i=✵

ρif (αi) ❤♦❧❞s ❢♦r ❡✈❡r② r❡❛❧ ♣♦❧②♥♦♠✐❛❧ f (t) ♦❢ ❞❡❣r❡❡ ❛t ♠♦st ✷k − ✶✳ ❚❤❡ ♥✉♠❜❡rs αi✱ i = ✵, ✶, . . . , k − ✶✱ ❛r❡ t❤❡ r♦♦ts ♦❢ t❤❡ ❡q✉❛t✐♦♥ Pk(t)Pk−✶(s) − Pk(s)Pk−✶(t) = ✵, ✇❤❡r❡ s = αk−✶✱ Pi(t) = P(✶,✵)

i

(t) ✐s t❤❡ (✶, ✵) ❛❞❥❛❝❡♥t ♣♦❧②♥♦♠✐❛❧✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✶✹ ✴ ✸✶

❈♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ ❉●❙✲ ❛♥❞ ▲✲❜♦✉♥❞s ✭✸✮

❋♦r ❡✈❡r② ✜①❡❞ ✭❝❛r❞✐♥❛❧✐t②✮ M > D(n, ✷k) t❤❡r❡ ❡①✐st ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ r❡❛❧ ♥✉♠❜❡rs −✶ = β✵ < β✶ < · · · < βk < ✶ ❛♥❞ ♣♦s✐t✐✈❡ γ✵, γ✶, . . . , γk✱ s✉❝❤ t❤❛t t❤❡ ❡q✉❛❧✐t② f✵ = f (✶) N +

k

• i=✵

γif (βi) ✐s tr✉❡ ❢♦r ❡✈❡r② r❡❛❧ ♣♦❧②♥♦♠✐❛❧ f (t) ♦❢ ❞❡❣r❡❡ ❛t ♠♦st ✷k✳ ❚❤❡ ♥✉♠❜❡rs βi✱ i = ✶, ✷, . . . , k✱ ❛r❡ t❤❡ r♦♦ts ♦❢ t❤❡ ❡q✉❛t✐♦♥ Pk(t)Pk−✶(s) − Pk(s)Pk−✶(t) = ✵, ✇❤❡r❡ s = βk✱ Pi(t) = P(✶,✶)

i

(t) ✐s t❤❡ (✶, ✶) ❛❞❥❛❝❡♥t ♣♦❧②♥♦♠✐❛❧✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✶✺ ✴ ✸✶

❈♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ ❉●❙✲ ❛♥❞ ▲✲❜♦✉♥❞s ✭✹✮

❙♦ ✇❡ ❛❧✇❛②s t❛❦❡ ❝❛r❡ ✇❤❡r❡ t❤❡ ❝❛r❞✐♥❛❧✐t② M ✐s ❧♦❝❛t❡❞ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❉❡❧s❛rt❡✲●♦❡t❤❛❧s✲❙❡✐❞❡❧ ❜♦✉♥❞✳ ■t ❢♦❧❧♦✇s t❤❛t M ∈ [D(n, τ), D(n, τ + ✶)] ⇐ ⇒ s ∈ Iτ, ✇❤❡r❡ s ❛♥❞ M ❛r❡ ❝♦♥♥❡❝t❡❞ ❜② t❤❡ ❡q✉❛❧✐t② M = Lτ(n, s), ❛♥❞ τ := τ(n, M) ✐s ❝♦rr❡❝t❧② ❞❡✜♥❡❞✳ ❚❤❡r❡❢♦r❡ ✇❡ ❛ss♦❝✐❛t❡ M ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♥✉♠❜❡rs α✵, α✶, . . . , αk−✶, ρ✵, ρ✶, . . . , ρk−✶ ✇❤❡♥ M ∈ [D(n, ✷k − ✶), D(n, ✷k)), β✵, β✶, . . . , βk, γ✵, γ✶, . . . , γk ✇❤❡♥ M ∈ [D(n, ✷k), D(n, ✷k + ✶)).

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✶✻ ✴ ✸✶

❆s②♠♣t♦t✐❝ ♦❢ ❯▲❇ ✭✶✮

❲❡ ❝♦♥s✐❞❡r t❤❡ ❜❡❤❛✈✐♦✉r ♦❢ ♦✉r ❜♦✉♥❞s ✐♥ t❤❡ ❛s②♠♣t♦t✐❝ ♣r♦❝❡ss ✇❤❡r❡ t❤❡ str❡♥❣t❤ τ ✐s ✜①❡❞✱ ❛♥❞ t❤❡ ❞✐♠❡♥s✐♦♥ n ❛♥❞ t❤❡ ❝❛r❞✐♥❛❧✐t② M t❡♥❞ s✐♠✉❧t❛♥❡♦✉s❧② t♦ ✐♥✜♥✐t② ✐♥ ❝❡rt❛✐♥ r❡❧❛t✐♦♥✳ ❲❡ ❝♦♥s✐❞❡r s❡q✉❡♥❝❡ ♦❢ ❝♦❞❡s ♦❢ ❝❛r❞✐♥❛❧✐t✐❡s (Mn) s❛t✐s❢②✐♥❣ Mn ∈ Iτ = (R(n, τ), R(n, τ + ✶)) ❢♦r n = ✶, ✷, ✸, . . . ❛♥❞ ❧✐♠

n→∞

Mn nk−✶ =   

✷ (k−✶)! + γ,

τ = ✷k − ✶,

✶ k! + γ,

τ = ✷k, ✭❤❡r❡ γ ≥ ✵ ✐s ❛ ❝♦♥st❛♥t ❛♥❞ t❤❡ t❡r♠s

✷ (k−✶)! ❛♥❞ ✶ k! ❝♦♠❡ ❢r♦♠ t❤❡

❉❡❧s❛rt❡✲●♦❡t❤❛❧s✲❙❡✐❞❡❧ ❜♦✉♥❞✮✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✶✼ ✴ ✸✶

❆s②♠♣t♦t✐❝ ♦❢ ❯▲❇ ✭✷✮

❘❡❝❛❧❧ t❤❛t t❤❡ ♥♦❞❡s αi = αi(n, ✷k − ✶, M)✱ i = ✵, . . . , k − ✶✱ ❛r❡ ❞❡✜♥❡❞ ❢♦r ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs n✱ k✱ ❛♥❞ M s❛t✐s❢②✐♥❣ M > R(n, ✷k − ✶) ❛♥❞ t❤❛t t❤❡ ♥♦❞❡s βi = βi(n, ✷k, M)✱ i = ✵, . . . , k✱ ❛r❡ ❞❡✜♥❡❞ ✐❢ M > R(n, ✷k)✳ ▲❡♠♠❛ ■❢ τ = ✷k − ✶ ❢♦r s♦♠❡ ✐♥t❡❣❡r k✱ t❤❡♥ ❧✐♠

n→∞ α✵(n, ✷k − ✶, Mn) = −✶/(✶ + γ(k − ✶)!), ❛♥❞

❧✐♠

n→∞ αi(n, ✷k − ✶, Mn) = ✵,

i = ✶, . . . , k − ✶. ■❢ τ = ✷k ❢♦r s♦♠❡ ✐♥t❡❣❡r k✱ t❤❡♥ ❧✐♠

n→∞ βi(n, ✷k, Mn) = ✵,

i = ✶, . . . , k.

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✶✽ ✴ ✸✶

❆s②♠♣t♦t✐❝ ♦❢ ❯▲❇ ✭✸✮

❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢ ❧✐♠n→∞ αi = ✵✱ i = ✶, . . . , k − ✶✱ ❢♦❧❧♦✇ ❢r♦♠ t❤❡ ✐♥❡q✉❛❧✐t✐❡s t✶,✶

k

> |αk−✶| > |α✶| > |αk−✷| > |α✷| > · · · . ❋♦r α✵ ✕ ✉s❡ t❤❡ ❱✐❡t❛ ❢♦r♠✉❧❛

k−✶

• i=✵

αi = (n + ✷k − ✶)(n + k − ✷) (n + ✷k − ✷)(n + ✷k − ✸) · P✶,✵

k (s)

P✶,✵

k−✶(s)

− k n + ✷k − ✷ t♦ ❝♦♥❝❧✉❞❡ t❤❛t ❧✐♠

n→∞ α✵ = ❧✐♠ n→∞

P(✶,✵)

k

(s) P(✶,✵)

k−✶ (s)

.

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✶✾ ✴ ✸✶

❆s②♠♣t♦t✐❝ ♦❢ ❯▲❇ ✭✹✮

❚❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ r❛t✐♦ P(✶,✵)

k

(s)/P(✶,✵)

k−✶ (s) ❝❛♥ ❜❡ ❢♦✉♥❞ ❜② ✉s✐♥❣ ❝❡rt❛✐♥

✐❞❡♥t✐t✐❡s ❜② ▲❡✈❡♥s❤t❡✐♥✿ Mn =

• ✶ − P(✶,✵)

k−✶ (s)

P(n)

k (s)

• D(n, ✷k − ✷) =
• ✶ − P(✶,✵)

k

(s) P(n)

k (s)

• D(n, ✷k).

❚❤❡s❡ ✐♠♣❧② ❧✐♠

n→∞

P(n)

k (s)

P(✶,✵)

k−✶ (s)

= − ✶ ✶ + γ(k − ✶)!, ❧✐♠

n→∞

P(✶,✵)

k

(s) P(n)

k (s)

= ✶, ❝♦rr❡s♣♦♥❞✐♥❣❧②✳ ❚❤❡r❡❢♦r❡ ❧✐♠

n→∞ α✵ = P(✶,✵) k

(s) P(✶,✵)

k−✶ (s)

= − ✶ ✶ + γ(k − ✶)!.

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✷✵ ✴ ✸✶

❆s②♠♣t♦t✐❝ ♦❢ ❯▲❇ ✭✺✮

❙✐♠✐❧❛r❧②✱ ❧✐♠n→∞ βi = ✵ ❢♦❧❧♦✇s ❡❛s② ❢♦r i ≥ ✷✱ t❤❡♥ ❧✐♠n→∞ β✶ = ✵ ✐s ♦❜t❛✐♥❡❞ ❜② ✉s✐♥❣ t❤❡ ❢♦r♠✉❧❛

k−✶

• i=✶

βi = (n − k − ✶)P(✶,✶)

k

(s) nP(✶,✶)

k−✶ (s)

❛♥❞ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ t❤❡ r❛t✐♦ P(✶,✶)

k

(s)/P(✶,✶)

k−✶ (s) ✐♥ t❤❡ ✐♥t❡r✈❛❧ I✷k ✕ ✐t ✐s

♥♦♥✲♣♦s✐t✐✈❡✱ ✐♥❝r❡❛s✐♥❣✱ ❡q✉❛❧ t♦ ③❡r♦ ✐♥ t❤❡ r✐❣❤t ❡♥❞ s = t✶,✶

k ✱ ❛♥❞

t❡♥❞✐♥❣ t♦ ✵ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t② ✐♥ t❤❡ ❧❡❢t ❡♥❞ s = t✶,✵

k ✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✷✶ ✴ ✸✶

❆s②♠♣t♦t✐❝ ♦❢ ❯▲❇ ✭✻✮

❘❡❝❛❧❧ t❤❛t ✐♥ t❤❡ ❝❛s❡ τ = ✷k − ✶ t❤❡r❡ ❛r❡ ❛ss♦❝✐❛t❡❞ ✇❡✐❣❤ts ρi = ρi(n, ✷k − ✶, Mn)✱ i = ✵, . . . , k − ✶✱ ❛♥❞✱ s✐♠✐❧❛r❧②✱ ✐♥ t❤❡ ❝❛s❡ τ = ✷k t❤❡r❡ ❛r❡ ✇❡✐❣❤ts γi = γi(n, ✷k − ✶, Mn)✱ i = ✵, . . . , k✳ ■♥ ✈✐❡✇ ♦❢ t❤❡ ▲❡♠♠❛ ✇❡ ♥❡❡❞ t❤❡ ❛s②♠♣t♦t✐❝ ♦❢ ρ✵(n, ✷k − ✶, Mn)Mn ♦♥❧②✳ ▲❡♠♠❛ ■❢ τ = ✷k − ✶✱ t❤❡♥ ❧✐♠

n→∞ ρ✵(n, ✷k − ✶, Mn)Mn = (✶ + γ(k − ✶)!)✷k−✶.

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✷✷ ✴ ✸✶

❆s②♠♣t♦t✐❝ ♦❢ ❯▲❇ ✭✼✮

❚❤✐s ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❛s②♠♣t♦t✐❝ ♦❢ α✵ ❛♥❞ t❤❡ ❢♦r♠✉❧❛ ρ✵(n, ✷k − ✶, Mn)Mn = − (✶ − α✷

✶)(✶ − α✷ ✷) · · · (✶ − α✷ k−✶)

α✵(α✷

✵ − α✷ ✶)(α✷ ✵ − α✷ ✷) · · · (α✷ ✵ − α✷ k−✶)

✭❝❛♥ ❜❡ ❞❡r✐✈❡❞ ❜② s❡tt✐♥❣ f (t) = t, t✸, . . . , t✷k−✶ ✐♥ t❤❡ q✉❛❞r❛t✉r❡ r✉❧❡ ❛♥❞ r❡s♦❧✈✐♥❣ t❤❡ ♦❜t❛✐♥❡❞ ❧✐♥❡❛r s②st❡♠ ✇✐t❤ r❡s♣❡❝t t♦ ρ✵, . . . , ρk−✶✮✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✷✸ ✴ ✸✶

❆s②♠♣t♦t✐❝ ♦❢ ❯▲❇ ✭✽✮

❚❤❡♦r❡♠ ❧✐♠ ✐♥❢

n→∞

E(n, Mn; h) Mn ≥ h(✵).

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✷✹ ✴ ✸✶

❆s②♠♣t♦t✐❝ ♦❢ ❯▲❇ ✭✾✮

▲❡t τ = ✷k − ✶✳ ❲❡ ❞❡❛❧ ✇✐t❤ t❤❡ ♦❞❞ ❜r❛♥❝❤ ♦❢ ♦✉r ❯▲❇ E(n, Mn; h) ≥ Mn

k−✶

• i=✵

ρih(αi) = Mn

• ρ✵h(α✵) + h(✵)

k−✶

• i=✶

ρi + o(✶)

• =

Mn

• ρ✵(h(α✵) − h(✵)) + h(✵)
• ✶ − ✶

Mn + o(✶)

• =

h(✵)Mn + c✸ + Mno(✶), ✇❤❡r❡ o(✶) ✐s ❛ t❡r♠ t❤❛t ❣♦❡s t♦ ✵ ❛s n → ∞ ❛♥❞ c✸ =

• (✶ + γ(k − ✶)!)✷k−✶

h

• −

✶ ✶ + γ(k − ✶)!

• − h(✵)
• − h(✵).

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✷✺ ✴ ✸✶

❆s②♠♣t♦t✐❝ ♦❢ ❯▲❇ ✭✶✵✮

❙✐♠✐❧❛r❧②✱ ✐♥ t❤❡ ❡✈❡♥ ❝❛s❡ ✇❡ ♦❜t❛✐♥ E(n, Mn; h) ≥ Mn

• γ✵(h(−✶) − h(✵)) + h(✵)
• ✶ − ✶

Mn

• + o(✶)
• =

h(✵)Mn + c✹ + Mno(✶), ✇❤❡r❡ c✹ = γ✵Mn(h(−✶) − h(✵)) − h(✵) ✭❤❡r❡ γ✵Mn ∈ (✵, ✶)✮✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✷✻ ✴ ✸✶

▼♦r❡ ♣r❡❝✐s❡ ❛s②♠♣t♦t✐❝ ✭✶✮

❚❤❡♦r❡♠ ■❢ τ = ✷k − ✶ t❤❡♥ ❧✐♠

n→∞ Mn

 

k−✶

• i=✵

ρ(n)

i

h(α(n)

i

) −

k−✶

• j=✵

h(✷j)(✵) (✷j)! · b✷j   = γ✷k−✶

k

• h
• − ✶

γk

• − P✷k−✶
• − ✶

γk

• ,

✇❤❡r❡ b✷j = ✶

−✶ t✷j(✶ − t✷)(n−✸)/✷dt = (✷j−✶)!! n(n+✷)...(n+✷j−✷)✱

γk = ✶ + γ(k − ✶)! ❛♥❞ P✷k−✶(t) = k−✶

j=✵ h(✷j)(✵) (✷j)! tj✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✷✼ ✴ ✸✶

▼♦r❡ ♣r❡❝✐s❡ ❛s②♠♣t♦t✐❝ ✭✷✮

❖❜s❡r✈❡ t❤❛t h(t) ≥ P✷k−✶(t) ❢♦r ❡✈❡r② t ∈ [−✶, ✶)✱ ❛♥❞✱ ❢✉rt❤❡r♠♦r❡✱ ✵ ≤ h(α(n)

i

) − P✷k−✶(α(n)

i

) ≤ h(✷k)(ξ) (✷k)! · |α(n)

i

|✷k, ✭✶✮ ✇❤❡r❡ |ξ| ∈ (✵, |α(n)

i

|)✱ i = ✶, ✷ . . . , k − ✶✱ ❜② t❤❡ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ❢♦r♠✉❧❛✳ ❙✐♥❝❡

c✶ √n ≤ t✶,✶ k

≤ c✷

√n ❢♦r s♦♠❡ ❝♦♥st❛♥ts c✶ ❛♥❞ c✷✱ ❛♥❞ ❢♦r ❡✈❡r② n✱ ✐t

❢♦❧❧♦✇s t❤❛t Mn

k−✶

• i=✶

ρ(n)

i

• h(α(n)

i

) − P✷k−✶(α(n)

i

)

• = O(✶/n).

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✷✽ ✴ ✸✶

▼♦r❡ ♣r❡❝✐s❡ ❛s②♠♣t♦t✐❝ ✭✸✮

❈♦r♦❧❧❛r② ■❢ τ = ✷k − ✶ t❤❡♥ ❧✐♠ ✐♥❢

n→∞

E(n, Mn; h) Mn = h(✵) ❛♥❞ ❧✐♠ ✐♥❢

n→∞

E(n, Mn; h) − h(✵)Mn Mn · n = h′′(✵) ✷ .

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✷✾ ✴ ✸✶

❲❤❛t ♥❡①t❄

❲❡ ❞♦ ♥♦t ❦♥♦✇ t❤❡ ❛s②♠♣t♦t✐❝ ♦❢ ♦✉r ❜♦✉♥❞s ✐♥ t❤❡ ❝❛s❡ ✇❤❡♥ t❤❡ ❞✐♠❡♥s✐♦♥ n ✐s ✜①❡❞✱ ❛♥❞ t❤❡ ❝❛r❞✐♥❛❧✐t② M t❡♥❞s t♦ ✐♥✜♥✐t② ✭✇✐t❤ τ✮✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✸✵ ✴ ✸✶

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ❋♦rt ❲❛②♥❡ ✷✵✶✻ ✸✶ ✴ ✸✶