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❖♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r② ✐♥ ❙♦❜♦❧❡✈ s♣❛❝❡s✳ ❈♦♥♥❡❝t✐♦♥s t♦ s♦♠❡ s♣❡❝tr❛❧ ♣r♦❜❧❡♠s✳ ■❣♦r ❙❤❡✐♣❛❦ ✭❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❚❛t✐❛♥❛ ●❛r♠❛♥♦✈❛✮ ❖♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r② ✐♥ ❙♦❜♦❧❡✈ s♣❛❝❡s✳ ■♥tr♦❞✉❝t✐♦♥ ❈♦♥♥❡❝t✐♦♥s t♦ s♦♠❡ s♣❡❝tr❛❧ ♣r♦❜❧❡♠s✳ ◆❡✇ r❡s✉❧ts✱ ✸✱ ✺ ❆r❜✐tr❛r②✱ ✶ ✶ ❀ ✷ ✶ ■❣♦r ❙❤❡✐♣❛❦ ✭❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❚❛t✐❛♥❛ ●❛r♠❛♥♦✈❛✮ ❆r❜✐tr❛r② ❡✈❡♥ ✱ ✶ ✶ ❀ ✷ ✶ ▼♦s❝♦✇ ▲♦♠♦♥♦s♦✈ ❙t❛t❡ ❯♥✐✈❡rs✐t② ❱✐❡♥♥❛✱ ❉❡❝❡♠❜❡r✱ ✷✵✶✾ ✶ ✴ ✷✹

❖♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r② ✐♥ ❙♦❜♦❧❡✈ s♣❛❝❡s✳ ❈♦♥♥❡❝t✐♦♥s t♦ ▲❡t n ∈ N ✱ ✵ � k < n − ✶ s♦♠❡ s♣❡❝tr❛❧ ♣r♦❜❧❡♠s✳ ■❣♦r ❙❤❡✐♣❛❦ ✭❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❚❛t✐❛♥❛ ●❛r♠❛♥♦✈❛✮ ▲❡t ✉s ❝♦♥s✐❞❡r s♣❡❝tr❛❧ ♣r♦❜❧❡♠ ■♥tr♦❞✉❝t✐♦♥ ◆❡✇ r❡s✉❧ts✱ ✸✱ ✺ ( − ✶ ) n y ( ✷ n ) = λ � δ ( k ) ( x − a ) , y � δ ( k ) ( x − a ) , ❆r❜✐tr❛r②✱ ✶ ✶ ❀ x , a ∈ ( ✵ , ✶ ) ✷ ✶ y ( j ) ( ✵ ) = y ( j ) ( ✶ ) = ✵ , ❆r❜✐tr❛r② ❡✈❡♥ ✱ j = ✵ , ✶ , . . . , n − ✶ ✶ ✶ ❀ ✷ ✶ ◗✉❡st✐♦♥✿ ❢♦r ✇❤❛t a t❤❡ ❡✐❣❡♥✈❛❧✉❡ ✐s ♠✐♥✐♠❛❧❄ ✷ ✴ ✷✹

❖♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r② ✐♥ ❙♦❜♦❧❡✈ s♣❛❝❡s✳ ❈♦♥♥❡❝t✐♦♥s t♦ ▲❡t n ∈ N ✱ ✵ � k < n − ✶ s♦♠❡ s♣❡❝tr❛❧ ♣r♦❜❧❡♠s✳ ■❣♦r ❙❤❡✐♣❛❦ ✭❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❚❛t✐❛♥❛ ●❛r♠❛♥♦✈❛✮ ▲❡t ✉s ❝♦♥s✐❞❡r s♣❡❝tr❛❧ ♣r♦❜❧❡♠ ■♥tr♦❞✉❝t✐♦♥ ◆❡✇ r❡s✉❧ts✱ ✸✱ ✺ ( − ✶ ) n y ( ✷ n ) = λ � δ ( k ) ( x − a ) , y � δ ( k ) ( x − a ) , ❆r❜✐tr❛r②✱ ✶ ✶ ❀ x , a ∈ ( ✵ , ✶ ) ✷ ✶ y ( j ) ( ✵ ) = y ( j ) ( ✶ ) = ✵ , ❆r❜✐tr❛r② ❡✈❡♥ ✱ j = ✵ , ✶ , . . . , n − ✶ ✶ ✶ ❀ ✷ ✶ ◗✉❡st✐♦♥✿ ❢♦r ✇❤❛t a t❤❡ ❡✐❣❡♥✈❛❧✉❡ ✐s ♠✐♥✐♠❛❧❄ ✷ ✴ ✷✹

❖♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r② ✐♥ ❙♦❜♦❧❡✈ s♣❛❝❡s✳ ❈♦♥♥❡❝t✐♦♥s t♦ ❆♥♦t❤❡r q✉❡st✐♦♥ s♦♠❡ s♣❡❝tr❛❧ ♣r♦❜❧❡♠s✳ ■❣♦r ❙❤❡✐♣❛❦ ✭❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❚❛t✐❛♥❛ ●❛r♠❛♥♦✈❛✮ ■♥tr♦❞✉❝t✐♦♥ ◆❡✇ r❡s✉❧ts✱ ❋♦r ✇❤❛t a t❤❡ ❡✐❣❡♥❢✉♥❝t✐♦♥ ✐s s②♠♠❡tr✐❝ ♦✈❡r ✐♥t❡r✈❛❧ ✸✱ ✺ [ ✵ ; ✶ ] ❄ ❆r❜✐tr❛r②✱ ✶ ✶ ❀ ✷ ✶ ❚❤❡ ❛♥s✇❡rs ❝❧♦s❡❧② ❝♦♥♥❡❝t❡❞ ✇✐t❤ t❤❡♦r② ♦❢ ❡♠❜❡❞❞✐♥❣ ❝♦♥st❛♥ts ❆r❜✐tr❛r② ❡✈❡♥ ✱ ✶ ✶ ❀ ✐♥ ❙♦❜♦❧❡✈ s♣❛❝❡s✳ ✷ ✶ ✸ ✴ ✷✹

❖♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r② ✐♥ ❙♦❜♦❧❡✈ s♣❛❝❡s✳ ❈♦♥♥❡❝t✐♦♥s t♦ s♦♠❡ s♣❡❝tr❛❧ ♣r♦❜❧❡♠s✳ ■❣♦r ❙❤❡✐♣❛❦ ◦ ✭❜❛s❡❞ ♦♥ ❥♦✐♥t ▲❡t W n ✷ [ − ✶ ; ✶ ] ✖ ❙♦❜♦❧❡✈ s♣❛❝❡ ✇✐t❤ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✇♦r❦ ✇✐t❤ ❚❛t✐❛♥❛ y ( j ) ( − ✶ ) = y ( j ) ( ✶ ) = ✵✱ j = ✵ , ✶ , . . . , n − ✶✳ ●❛r♠❛♥♦✈❛✮ ■♥tr♦❞✉❝t✐♦♥ ◆❡✇ r❡s✉❧ts✱ ❚❤❡ ❣❡♥❡r❛❧ ♣r♦❜❧❡♠ ♦♥ ❡♠❜❡❞❞✐♥❣ ❝♦♥st❛♥ts ❢♦r ❉✐r✐❝❤❧❡t ✸✱ ✺ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❆r❜✐tr❛r②✱ ✶ ✶ ❀ ✷ ✶ ❆r❜✐tr❛r② ❡✈❡♥ ✱ � � f � ˚ � ✶ ✶ ❀ ◦ W k q [ − ✶ ; ✶ ] W n Λ n , k , p , q := sup , f ∈ p [ − ✶ ; ✶ ] ✷ ✶ � f � ˚ W n p [ − ✶ ; ✶ ] ◦ ◦ W n W k Λ n , k , p , q ✐s t❤❡ ♥♦r♠ ♦❢ ❡♠❜❡❞❞✐♥❣ ♦♣❡r❛t♦r J : p [ − ✶ ; ✶ ] ֒ → q [ − ✶ ; ✶ ] ✳ ✹ ✴ ✷✹

❖♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r② ✐♥ ❙♦❜♦❧❡✈ s♣❛❝❡s✳ ❈♦♥♥❡❝t✐♦♥s t♦ s♦♠❡ s♣❡❝tr❛❧ ♣r♦❜❧❡♠s✳ ■❣♦r ❙❤❡✐♣❛❦ ◦ ✭❜❛s❡❞ ♦♥ ❥♦✐♥t ▲❡t W n ✷ [ − ✶ ; ✶ ] ✖ ❙♦❜♦❧❡✈ s♣❛❝❡ ✇✐t❤ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✇♦r❦ ✇✐t❤ ❚❛t✐❛♥❛ y ( j ) ( − ✶ ) = y ( j ) ( ✶ ) = ✵✱ j = ✵ , ✶ , . . . , n − ✶✳ ●❛r♠❛♥♦✈❛✮ ■♥tr♦❞✉❝t✐♦♥ ◆❡✇ r❡s✉❧ts✱ ❚❤❡ ❣❡♥❡r❛❧ ♣r♦❜❧❡♠ ♦♥ ❡♠❜❡❞❞✐♥❣ ❝♦♥st❛♥ts ❢♦r ❉✐r✐❝❤❧❡t ✸✱ ✺ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❆r❜✐tr❛r②✱ ✶ ✶ ❀ ✷ ✶ ❆r❜✐tr❛r② ❡✈❡♥ ✱ � � f � ˚ � ✶ ✶ ❀ ◦ W k q [ − ✶ ; ✶ ] W n Λ n , k , p , q := sup , f ∈ p [ − ✶ ; ✶ ] ✷ ✶ � f � ˚ W n p [ − ✶ ; ✶ ] ◦ ◦ W n W k Λ n , k , p , q ✐s t❤❡ ♥♦r♠ ♦❢ ❡♠❜❡❞❞✐♥❣ ♦♣❡r❛t♦r J : p [ − ✶ ; ✶ ] ֒ → q [ − ✶ ; ✶ ] ✳ ✹ ✴ ✷✹

❖♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r② ✐♥ ❙♦❜♦❧❡✈ s♣❛❝❡s✳ ❈♦♥♥❡❝t✐♦♥s t♦ s♦♠❡ s♣❡❝tr❛❧ ♣r♦❜❧❡♠s✳ ■❣♦r ❙❤❡✐♣❛❦ ◦ ✭❜❛s❡❞ ♦♥ ❥♦✐♥t ▲❡t W n ✷ [ − ✶ ; ✶ ] ✖ ❙♦❜♦❧❡✈ s♣❛❝❡ ✇✐t❤ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✇♦r❦ ✇✐t❤ ❚❛t✐❛♥❛ y ( j ) ( − ✶ ) = y ( j ) ( ✶ ) = ✵✱ j = ✵ , ✶ , . . . , n − ✶✳ ●❛r♠❛♥♦✈❛✮ ■♥tr♦❞✉❝t✐♦♥ ◆❡✇ r❡s✉❧ts✱ ❚❤❡ ❣❡♥❡r❛❧ ♣r♦❜❧❡♠ ♦♥ ❡♠❜❡❞❞✐♥❣ ❝♦♥st❛♥ts ❢♦r ❉✐r✐❝❤❧❡t ✸✱ ✺ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❆r❜✐tr❛r②✱ ✶ ✶ ❀ ✷ ✶ ❆r❜✐tr❛r② ❡✈❡♥ ✱ � � f � ˚ � ✶ ✶ ❀ ◦ W k q [ − ✶ ; ✶ ] W n Λ n , k , p , q := sup , f ∈ p [ − ✶ ; ✶ ] ✷ ✶ � f � ˚ W n p [ − ✶ ; ✶ ] ◦ ◦ W n W k Λ n , k , p , q ✐s t❤❡ ♥♦r♠ ♦❢ ❡♠❜❡❞❞✐♥❣ ♦♣❡r❛t♦r J : p [ − ✶ ; ✶ ] ֒ → q [ − ✶ ; ✶ ] ✳ ✹ ✴ ✷✹

❖♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r② ✐♥ ❙♦❜♦❧❡✈ s♣❛❝❡s✳ ❈♦♥♥❡❝t✐♦♥s t♦ ❘❡♠❛r❦s ♦♥ ❍✐st♦r②✿ n = ✶✱ k = ✵ s♦♠❡ s♣❡❝tr❛❧ ♣r♦❜❧❡♠s✳ ■❣♦r ❙❤❡✐♣❛❦ p = q = ✷ ❱✳❆✳❙t❡❦❧♦✛✱ ✶✾✵✶ ✭❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❚❛t✐❛♥❛ ●❛r♠❛♥♦✈❛✮ ❆r❜✐tr❛r② p = q ✱ ❱✳■✳▲❡✈✐♥✱ ✶✾✸✽ ■♥tr♦❞✉❝t✐♦♥ ◆❡✇ r❡s✉❧ts✱ ✸✱ ✺ ❆r❜✐tr❛r②✱ ✶ ✶ ❀ ❆r❜✐tr❛r② p ✱ q ✱ ❊✳❙❝♠✐❞t✱ ✶✾✹✵ ✷ ✶ ❆r❜✐tr❛r② ❡✈❡♥ ✱ ✶ ✶ ❀ G ( ✶ q + ✶ p ′ ) ✷ Λ ✶ , ✵ , p , q = ✶ p ′ ) , G ( s ) = Γ( s + ✶ ) p ✶ p ′ = , G ( ✶ q ) G ( ✶ ✷ s s p − ✶ Γ( ✶ / q + ✶ / p ′ + ✶ ) ( ✶ / q ) ✶ / q ( ✶ / p ′ ) ✶ / p ′ Λ ✶ , ✵ , p , q = ✶ Γ( ✶ / q + ✶ )Γ( ✶ / p ′ + ✶ ) ( ✶ / q + ✶ / p ′ ) ✶ / q + ✶ / p ′ ✷ ✺ ✴ ✷✹