ACTA issues

Powers of the Dirichlet kernel with respect to orthogonal polynomials and related operators

Josef Obermaier

Acta Sci. Math. (Szeged) 83:3-4(2017), 539-549

Abstract. Let $\{P_n\}_{n=0}^\infty $ be an orthogonal polynomial sequence on the real line with respect to a probability measure $\mu $ with compact and infinite support and $D_N=\sum_{n=0}^N P_nh_n$ the $N$th element of the Dirichlet kernel, where $h_n=(\int P_n^2d\mu )^{-1}$. We are investigating the $r$th integer power $D_N^r$ and prove for special orthogonal polynomials that in the case $r\in\mathbb {N}\setminus\{1\}$ the sequence $\{D_N^r\}_{N=0}^\infty $ gives rise to an approximate identity. This applies for example for Jacobi polynomials.

DOI: 10.14232/actasm-016-072-z

AMS Subject Classification (1991): 42C05, 42C10

Keyword(s): orthogonal polynomials, approximate identities, Dirichlet kernel

Received December 21, 2016, and in revised form May 24, 2017. (Registered under 72/2016.)