ISSN:
1572-9265
Keywords:
AMS 65D
;
33C
;
Neumann series
;
Bessel functions
;
Padé approximation
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
,
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02141932
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