On an infinite series of Abel occurring in the theory of interpolation |
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Authors: | W. A. J. Luxemburg |
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Affiliation: | Department of Mathematics, California Institute of Technology, Pasadena, California 91109 USA |
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Abstract: | The purpose of this paper is to show that for a certain class of functions f which are analytic in the complex plane possibly minus (−∞, −1], the Abel series f(0) + Σn = 1∞ f(n)(nβ) z(z − nβ)n − 1/n! is convergent for all β>0. Its sum is an entire function of exponential type and can be evaluated in terms of f. Furthermore, it is shown that the Abel series of f for small β>0 approximates f uniformly in half-planes of the form Re(z) − 1 + δ, δ>0. At the end of the paper some special cases are discussed. |
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