Convergence acceleration of Taylor sections by convolution (Q1963847)

The aim of this paper is to construct for certain families of functions sequences of polynomial approximants which are computable with essentially the same effort as Taylor sections and which have a better rate of convergence on some parts of the plane. For every function \(f\) of certain class \(\mathcal{F}\) the partial sums \(S_n(f)\) of the Fourier series of \(f\) are represented as a convolution product \(S_n(f)=\varphi_f\ast\psi_n\), where the sequence \(\psi_n\) is independent of \(f\) and where \(\varphi_f\) is independent of \(n\). The author uses this representation to derive results on the (rate of) convergence of \(S_n(f)\). Let \(\mathcal{F}\) be a given family of (entire) functions such that for fixed (entire) functions \(g(z)=\sum_{\nu=0}^\infty g_\nu z^\nu\) with \(g_\nu\neq 0\) we have for all \(f\in\mathcal{F}\), \(f=\varphi_f\ast g\) with a function \(\varphi_f\) which is holomorphic in \(\Omega={\mathbb C} \setminus [1,\infty)\). Let moreover \(K\) be a given compact plane set which is starlike with respect to the origin (\(K\) is the set on which the functions \(f\) are approximated). If \((P_n)\) is a sequence of polynomials of degree \(\leq n\) converging uniformly to \(g\) on \(L\), where \(L\) is a compact set with \(K\subset L^0\), then for every \(f\in\mathcal{F}\) the polynomials \(\varphi_f\ast P_n\) approximate \(f\) on \(K\) with the same error up to some factor \(M_{f,L}=M_{f,L}(g,K)\), i.e., \(\|f-\varphi_f\ast P_n\|_K\leq M_{f,L}\cdot\|g-P_n\|_L\) (\(n\in{\mathbb N}\)). So the idea is to choose \(P_n\) in such a way that the rate of convergence on \(L\) to \(g\) is fast (essentially faster than the rate for the Taylor sections \(S_n(g)\)). The resulting method is applied to Bessel functions, to confluent hypergeometric functions, or to parabolic cylinder functions.