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

:$$Npi leq t < (N+1)pi.,

References

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