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:

${\displaystyle f(z)=\sum _{n=0}^{\infty }\left[A_{n}(1-z)f^{(2n)}(0)+A_{n}(z)f^{(2n)}(1)\right]+\sum _{k=1}^{N}C_{k}\sin(k\pi z).}$

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

${\displaystyle h(\theta ;f)={\underset {r\to \infty }{\limsup }}\,{\frac {1}{r}}\log |f(re^{i\theta })|}$

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

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

with

${\displaystyle N\pi \leq t<(N+1)\pi .}$