Lidstone series explained

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 ^{inf ty} _{n =0} \left[ A _{n(1-z)} f ^{(2n)} ( 0 ) + A _{n(z)} f ^{(2n)} ( 1 ) \right] + \sum ^{N} _{k =1} 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 inf ty}{\limsup} (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 = \sup _{\theta\in} h(\theta; f)

with

N\pi \leq 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