Fejér kernels for Fourier series on \(\mathbb{T}^ n\) and on compact Lie groups
type: Item
instance of: scholarly article
title: Fejér kernels for Fourier series on \(\mathbb{T}^ n\) and on compact Lie groups
zbMATH Open document ID: 0807.42011
DOI: 10.1007/BF02572322
author: Giancarlo Travaglini
published in: Mathematische Zeitschrift
publication date: 1994-12-21T00:00:00Z
full work available at URL: https://eudml.org/doc/174651
review text: We discuss two different generalizations of the one-dimensional Fejér kernel to the polyhedral summability of multiple Fourier series and we obtain fairly explicit extensions of the well-known identity \[ \sum^ N_{k=-N}\left(1- {| k|\over N+1}\right) e^{ikt}= {1\over N+1} \left({\sin ((N+ 1)t/2) \over \sin (t/2)}\right)^ 2. \] We then apply these results to the study of Fourier series on a compact simple simply connected Lie group \(G\). For each definition of Fejér kernel on \(G\) we compute the Lebesgue constants (i.e., the \(L^ 1\)-norms), which in one case turn out to be unbounded even if we substitute the Fejér kernel with a Cesàro kernel of arbitrary order.
Mathematics Subject Classification ID: 42B08
Mathematics Subject Classification ID: 43A50
zbMATH DE Number: 704918
zbMATH Keywords: polyhedral Dirichlet kernels
zbMATH Keywords: polyhedral summability of multiple Fourier series
zbMATH Keywords: compact simple simply connected Lie group
zbMATH Keywords: Fejér kernel
zbMATH Keywords: Cesàro kernel
Wikidata QID: Q115391976
MaRDI profile type: Publication
cites work: Exponential sums and integrals over convex polytopes
cites work: Points entiers dans les polyèdres convexes
cites work: Jackson's theorem for compact connected Lie groups
cites work: Best conditions for the norm convergence of Fourier series
cites work: Sommes de Riesz et multiplicateurs sur un groupe de Lie compact
cites work: Sharp results for the mean summability of Fourier series on compact Lie groups
cites work: L p Norms of Certain Kernels on the N-Dimensional Torus
cites work: Estimates for Riesz kernels of eigenfunction expansions of elliptic differential operators on compact manifolds
cites work: On the asymptotics of the Bochner-Riesz kernel on compact Lie groups
cites work: On the convergence of multiple Fourier series
cites work: Q3697574
cites work: Sharp estimates for Lebesgue constants on compact Lie groups
cites work: ON THE MEAN INVERSION OF FOURIER AND HANKEL TRANSFORMS
cites work: Q3934931
cites work: Polyhedral Summability of Fourier Series on Compact Lie Groups
cites work: Q5640160
cites work: Polyhedral summability of multiple Fourier series (and explicit formulas for Dirichlet kernels on $
cites work: Q4146925
cites work: The approximation of functions of many variables by their Fejer sums
cites work: Q3254471
description: scientific article; zbMATH DE number 704918