Accelerated polynomial approximation of finite order entire functions by growth reduction

Abstract

Let f f be an entire function of positive order and finite type. The subject of this note is the convergence acceleration of polynomial approximants of f f by incorporating information about the growth of f ( z ) f(z) for z → ∞ z\to \infty . We consider “near polynomial approximation” on a compact plane set K K , which should be thought of as a circle or a real interval. Our aim is to find sequences ( f n ) n (f_n)_n of functions which are the product of a polynomial of degree ≤ n \le n and an “easy computable” second factor and such that ( f n ) n (f_n)_n converges essentially faster to f f on K K than the sequence ( P n ∗ ) n (P_n^*)_n of best approximating polynomials of degree ≤ n \le n . The resulting method, which we call Reduced Growth method ( R G RG -method) is introduced in Section 2. In Section 5, numerical examples of the R G RG -method applied to the complex error function and to Bessel functions are given.