Biorthogonal System Associated with Uniform Asymptotic Expansions

摘要

The Soni and Sleeman (1987) family of polynomials which are related to the coefficients in a uniform asymptotic expansion of a class of integrals is introduced. In this expansion, parabolic cylinder functions (Weber functions) occur as basic approximants and the resulting series is of Bleistein type. A family of rational functions is introduced, and the two families form a biorthogonal system on a contour in the complex plane. The system can be viewed as a generalization of the families (z to the n) and (z to the -n-1) which occur in Taylor expansions and the Cauchy integrals of analytic functions. Explicit representations of the rational functions are given together with rigorous estimates. These results are used to establish convergence of expansions of certain functions in terms of the polynomials and the rational functions. The main motivation to study this system stems from the problem on the asymptotic expansion of a class of integrals. It is shown how to use the system in order to construct bounds for the remainders in the asymptotic expansion. An example is worked out.