Summation kernels for orthogonal polynomial systems (Q2702486)

This is a survey paper on convergence of weighted Fourier expansions with respect to orthogonal polynomial systems in certain Banach spaces in \(L^1(\pi)\), where the support of the orthogonality measure is compact (containing infinite many points of increase) and where the orthogonal polynomials \(\{P_n\mid n\geq 0\}\) induce a hypergroup structure on \({\mathbb N}_0\) and a convolution structure on supp\( \pi\). NEWLINENEWLINENEWLINEAfter a short introduction into the subject of discrete and polynomial hypergroups and their convolution and dual convolution structure, the authors turn to general convergence theorems NEWLINENEWLINENEWLINEThey consider the Banach spaces \(C(D_S)\) or \(L^p(D_S,\pi)\), \(1\leq p<\infty\), where \(D_S=\{x\in{\mathbb R}\mid \{P_n(x), n\geq 0\}\) is bounded\}. Then they investigate the convergence of the weighted Fourier expansion NEWLINE\[NEWLINE A_n\varphi = \sum_{k=0}^na_{n,k}\check{\varphi}(k)P_kh(k) NEWLINE\]NEWLINE where \(h\) is the Haar measure of the group, \(\{a_{n,k}\mid n \in{\mathbb N}_0, 0\leq k\leq n\}\) is a triangular scheme of complex numbers and \(\check{\varphi}\) denotes the inverse Fourier transform. NEWLINENEWLINENEWLINEMoreover, they generalize the Dirichlet, Fejér-type and de la Vallée-Poussin kernels and give some properties, specifically paying attention to analogies with ordinary Fourier analysis in the trigonometric case. NEWLINENEWLINENEWLINEThis is a good, mainly self-contained introduction into the subject that can serve as a good starting point for those who want to familiarize themselves with the subject. The list of 44 references gives quite enough leads to background material and some insight in the developments up to 1997.NEWLINENEWLINEFor the entire collection see [Zbl 0954.65001].