Accelerated polynomial approximation of finite order entire functions by growth reduction

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