Lidstone series

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 "A"_n as follows:

:$f(z)=sum\_\{n=0\}^infty\; left\; [\; A\_n(1-z)\; f^\{(2n)\}(0)\; +\; A\_n(z)\; f^\{(2n)\}(1)\; ight]\; +\; sum\_\{k=1\}^N\; C\_k\; sin\; (kpi\; z)$.

Here "A"_"n"("z") is a polynomial in "z" of degree "n", "C"_"k" 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

:$h(\; heta;\; f)\; =\; lim\; sup\; frac\{1\}\{r\}\; log\; |f(r\; e^\{i\; heta\})|,$

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

:$t=\; lim\; sup\; h(\; heta;\; f),$

with

:

References

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