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- W2023440130 abstract "If f is analytic in the open unit disc D and A is a sequence of points in D converging to 0, then f admits the Newton series expansion f(z) f(A,)+2n= AA f(A l)(zAl)(Z-Ak )--(z-A ), where Anf(z) is the nth divided difference of f with respect to the sequence A. The Newton series reduces to the Maclaurin series in case An 0. The present paper investigates relationships between the behavior of zeros of the normalized remainders Ak f (zY=A kf + ) + ; n -k + A'Anf (A ) (z -Ak + ) ' (z-A ) of the Newton X kX n=k+1 X n-+1 k+ nkn series and zeros of the normalized remainders I? a znof the Maclaurin n=k n series for f. Let CA be the supremum of numbers c > 0 such that if f is analytic in D and each of Akf(z), 0 W, for all A, and, moreover, CX = W if A e11. In obtaining this result, we prove that functions f analy tic in D have expansions of the formf(z) = Y= An f(z )c (z), where 1? < W, for all n, and C (z) is nO =0 n n n l a polynomial of degree n determined by the conditions AI1 Ck(z ) k" @default.
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- W2023440130 date "1973-01-01" @default.
- W2023440130 modified "2023-09-23" @default.
- W2023440130 title "Successive remainders of the Newton series" @default.
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- W2023440130 doi "https://doi.org/10.1090/s0002-9947-1973-0320286-x" @default.
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