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


 * $$f(z)=\sum_{n=0}^\infty \left[ A_n(1-z) f^{(2n)}(0) + A_n(z) f^{(2n)}(1) \right] + \sum_{k=1}^N C_k \sin (k\pi z).$$

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

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


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

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


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

with


 * $$N\pi \leq t < (N+1)\pi.$$