Lidstone series

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 A_n as follows:

$f(z)=\sum _{n=0}^{\mathrm{\infty }}[A_n(1-z)f^{(2n)}(0)+A_n(z)f^{(2n)}(1)]+\sum _{k=1}^N{C}_k\sin (k\pi z).$

Here A_n(z) is a polynomial in z of degree n, C_k 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

$h(\theta ;f)=\underset{r\to \mathrm{\infty }}{\mathrm{lim\; sup}}\frac{1}{r}\log |f(r{e}^{i\theta })|$

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

$t=\underset{\theta \in [0,2\pi )}{\sup }h(\theta ;f)$

with

$N\pi \le t<(N+1)\pi .$

References

• 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