Lidstone series

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In mathematics, certain types of entire functions can be expressed as a certain polynomial expansion known as the Lidstone series.

Let f(z) be an entire function of exponential type less than (N + 1)π, as defined below. Then f(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 f(n)(a) the derivative of f at a.

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

h(\theta; f) = \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= \lim \sup h(\theta; f)\,

with

N\pi \leq t < (N+1)\pi.\,

[edit] References

  • Ralph P. Boas, Jr. and C. Creighton Buck, Polynomial Expansions of Analytic Functions, (1964) Academic Press, NY. ISBN 3-540-03123-5


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