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BIT Numerical Mathematics

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09-08-2021

Pointwise and uniform error estimates associated with Abel-Whittaker interpolation series and its dual

Authors: M. H. Annaby, S. R. Elsayed-Abdullah

Published in: BIT Numerical Mathematics | Issue 2/2022

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Abstract

Abel-Whittaker series (1934) is viewed as a two-point Taylor series. It solves the interpolation problem \(f^{(2n)}(1) =a_n\), \(f^{(2n+1)}(0) =b_n,\,n\ge 0,\) for an appropriate analytic function \(f(\cdot )\). The remainder of the truncated series is investigated and sharp estimates for it are established. A key result is a Cauchy-type representation of the remainder that is established in terms of Euler’s polynomials. The results are illustrated through numerical tables and sketched graphs, introducing the two-point theorem as an efficient approximation tool. Comparisons with both Taylor series and the classical sampling theorem of Whittaker (1915) are carried out. Dually, we derive and investigate the remainder associated with the dual Abel-Whittaker interpolation problem \(f^{(2n+1)}(1) =a_n\), \(f^{(2n)}(0) =b_n,\,n\ge 0.\)

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Title: Pointwise and uniform error estimates associated with Abel-Whittaker interpolation series and its dual
Authors: M. H. Annaby
S. R. Elsayed-Abdullah
Publication date: 09-08-2021
Publisher: Springer Netherlands
Published in: BIT Numerical Mathematics / Issue 2/2022
Print ISSN: 0006-3835
Electronic ISSN: 1572-9125
DOI: https://doi.org/10.1007/s10543-021-00888-7

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