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Lidstone series

From Wikipedia, the free encyclopedia

In mathematics, a Lidstone series, named after George James Lidstone, is a kind of polynomial expansion that can express certain types of entire functions.

Let ƒ(z) be an entire function of exponential type less than (N + 1)π, as defined below. Then ƒ(z) can be expanded in terms of polynomials An as follows:

Here An(z) is a polynomial in z of degree n, Ck a constant, and ƒ(n)(a) the nth derivative of ƒ at a.

A function is said to be of exponential type of less than t if the function

is bounded above by t. Thus, the constant N used in the summation above is given by

with

References

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  • Ralph P. Boas, Jr. and C. Creighton Buck, Polynomial Expansions of Analytic Functions, (1964) Academic Press, NY. Library of Congress Catalog 63-23263. Issued as volume 19 of Moderne Funktionentheorie ed. L.V. Ahlfors, series Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag ISBN 3-540-03123-5