Bounds for convergence and uniqueness in Abel-Goncharov interpolation problems

Popov, A. Yu.

doi:10.1070/sm2002v193n02abeh000629

Cited by 2 publications

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“…We start with any sequence w = (w n ) n≥0 of complex numbers. Following (Gontcharoff, 1930) (see also (Evgrafov, 1954), (Popov, 2002)), we define a sequence of polynomials (Ω w 0 ,w 1 ,...,w n−1 ) n≥0 in C[z] as follows: we set Ω ∅ = 1, Ω w 0 (z) = z − w 0 , and, for n ≥ 1, we define Ω w 0 ,w 1 ,w 2 ,...,wn (z) as the polynomial of degree n + 1 which is the primitive of Ω w 1 ,w 2 ,...,wn vanishing at w 0 . For n ≥ 0, we write Ω n;w for Ω w 0 ,w 1 ,...,w n−1 , a polynomial of degree n which depends only on the first n terms of the sequence w. The leading term of…”

Section: Abel-gontcharoff Interpolationmentioning

confidence: 97%

On transcendental entire functions with infinitely many derivatives taking integer values at several points

Waldschmidt

2019

Preprint

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Let s 0 , s 1 , . . . , s m−1 be complex numbers and r 0 , . . . , r m−1 rational integers in the range 0 ≤ r j ≤ m − 1. Our first goal is to prove that if an entire function f of sufficiently small exponential type satisfies f (mn+r j ) (s j ) ∈ Z for 0 ≤ j ≤ m − 1 and all sufficiently large n, then f is a polynomial. Under suitable assumptions on s 0 , s 1 , . . . , s m−1 and r 0 , . . . , r m−1 , we introduce interpolation polynomials Λ nj , (n ≥ 0, 0 ≤ j ≤ m − 1) satisfying Λ (mk+r ) nj (s ) = δ j δ nk , for n, k ≥ 0 and 0 ≤ j, ≤ m − 1 and we show that any entire function f of sufficiently small exponential type has a convergent expansionThe case r j = j for 0 ≤ j ≤ m − 1 amounts to take a periodic sequence w = (w n ) n≥0 of elements in the set {s 0 , s 1 , . . . , s m−1 }. More generally, given a bounded sequence (not necessarily periodic) w = (w n ) n≥0 of complex numbers, we consider similar interpolation formulaeinvolving polynomials Ω w,n (z) which were introduced by W. Gontcharoff in 1930. Under suitable assumptions, we show that the hypothesis f (n) (w n ) ∈ Z for all sufficiently large n implies that f is a polynomial.

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Section: Abel-gontcharoff Interpolationmentioning

confidence: 97%