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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Successive remainders of the Newton series


Authors: G. W. Crofts and J. K. Shaw
Journal: Trans. Amer. Math. Soc. 181 (1973), 369-383
MSC: Primary 30A08
MathSciNet review: 0320286
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Abstract: If $ f$ is analytic in the open unit disc $ D$ and $ \lambda $ is a sequence of points in $ D$ converging to 0, then $ f$ admits the Newton series expansion $ f(z) = f({\lambda _1}) + \sum\nolimits_{n = 1}^\infty {\Delta _\lambda ^nf({\lambda _{n + 1}})(z - {\lambda _1})(z - {\lambda _2}) \cdots (z - {\lambda _n})} $, where $ \Delta _\lambda ^nf(z)$ is the $ n$th divided difference of $ f$ with respect to the sequence $ \lambda $. The Newton series reduces to the Maclaurin series in case $ {\lambda _n} \equiv 0$. The present paper investigates relationships between the behavior of zeros of the normalized remainders $ \Delta _\lambda ^kf(z) = \Delta _\lambda ^kf({\lambda _{k + 1}}) + \sum\nolimi... ...bda ^nf({\lambda _{n + 1}})(z - {\lambda _{k + 1}}) \cdots (z - {\lambda _n})} $ of the Newton series and zeros of the normalized remainders $ \sum\nolimits_{n = k}^\infty {{a_n}{z^{n - k}}} $ of the Maclaurin series for $ f$. Let $ {C_\lambda }$ be the supremum of numbers $ c > 0$ such that if $ f$ is analytic in $ D$ and each of $ \Delta _\lambda ^kf(z),\;0 \leqslant k < \infty $, has a zero in $ \vert z\vert \leqslant c$, then $ f \equiv 0$. The corresponding constant for the Maclaurin series ( $ {C_\lambda }$, where $ {\lambda _n} \equiv 0$) is called the Whittaker constant for remainders and is denoted by $ W$. We prove that $ {C_\lambda } \geqslant W$, for all $ \lambda $, and, moreover, $ {C_\lambda } = W$ if $ \lambda \in {l_1}$. In obtaining this result, we prove that functions $ f$ analytic in $ D$ have expansions of the form $ f(z) = \sum\nolimits_{n = 0}^\infty {\Delta _\lambda ^nf({z_n}){C_n}(z)} $, where $ \vert{z_n}\vert \leqslant W$, for all $ n$, and $ {C_n}(z)$ is a polynomial of degree $ n$ determined by the conditions $ \Delta _\lambda ^j{C_k}({z_j}) = {\delta _{jk}}$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0320286-X
PII: S 0002-9947(1973)0320286-X
Keywords: Newton series, zeros of remainders, extremal functions, matrix transformations
Article copyright: © Copyright 1973 American Mathematical Society