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 $z\to \infty$. 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 $(f_n)_n$ of functions which are the product of a polynomial of degree $\le n$ and an ``easy computable'' second factor and such that $(f_n)_n$ converges essentially faster to $f$ on $K$ than the sequence $(P_n^*)_n$ of best approximating polynomials of degree $\le n$. 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.