Polynomial approximations in the complex plane

摘要

Good polynomial approximations for analytic functions are potentially useful but are in short supply. A new approach introduced here involves the Lanczos τ-method, with perturbations proportional to Faber or Chebyshev polynomials for specific regions of the complex plane. The results show that suitable forms of the τ-method, which are easy to use, can produce near-minimax polynomial approximations for functions which satisfy linear differential equations with polynomial coefficients. In particular, some accurate approximations of low degree for Bessel functions are presented. An appendix describes a simple algorithm which generates polynomial approximations for the Bessel function Jν(z) of any given order ν.