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.)
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