Rational approximation to Neumann series of Bessel functions

Abstract

LetJ _n (z) be the Bessel function of the first kind and ordern, and letf(z) be an analytic function in|z|≤r (r>0); then it is known that the Bessel expansion

$$f(z) = \sum\limits_{n = 0}^\infty {a_n J_n (z) (Neumann series)} $$

converges for|z|≤r. In this paper, we shall be concerned with the construction of “approximating” functions to (1) which are easily computable (rational functions). Namely, making use of the generating function for the family {J _n (z)}, a rational functionf _k (z) with prescribed poles can be obtained such thatf _k (z) “approximates” tof(z) in the following sense:

$$f_k (z) = \sum\limits_{n = 0}^\infty {\bar a_n J_n (z)} with \bar a_n = a_n , n = 0,1,...,k - 1;$$

and it will be said thatf _k is an “approximant” of orderk. When orthogonality conditions with respect to a linear functional defined from the sequence {a _n} are used, then the order of approximation may be increased up to2k. An algebraic approach of these approximants is carried out.