Fejér and Dirichlet Kernels: Their Associated Polynomials

We show that the Fejér kernel generates the fifth-kind Chebyshev polynomials.Keywords: Kernels in Fourier series; Chebyshev polynomials.© 2009 JSR Publications.ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v1i2.2282

The support of pseudomeasures on

PM(E) denotes the set of pseudomeasures on $\mathbb{R}$ with support in the closed set E ⊆ $\mathbb{R}$. Then y ∈ $\mathbb{R}$ is not in E if and only if there is a neighbourhood W of y with $\lim_{N\to\infty} \int_{-N}^{N}(1-{|t|}/{N})e^{2\pi itw}\cal{F} S(t)\,dt=0$ uniformly for w ∈ W and S ∈ PM(E) with ‖S‖PM ≤ 1. This improves previous results by adding “uniformly” and its scope. The proof uses the fact that squashing the central spike of the Fejer kernel leads to A-norm convergence.