Arkiv för Matematik

  • Ark. Mat.
  • Volume 20, Number 1-2 (1982), 101-109.

Absolute convergence of Fourier series on totally disconnected groups

Walter R. Bloom

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Abstract

LetG denote a totally disconnected locally compact metric abelian group with translation invariant metric d and character group ΓG. The Lipschitz spaces are defined by $Lip\left( {\alpha ;p} \right) = \left\{ {f \in L^p \left( G \right):\left\| {\tau _a f - f} \right\|_p = O\left( {d\left( {a,0} \right)^\alpha } \right),a \to 0} \right\},$ where τaf: x→f(x-a) and α∈(0,1). For a suitable choice of metric it is shown that Lip (α; p)⊂Lr(ΓG), where α>1/p+1/r−1≧0 and 1≦p≦2. In the case G is compact the corresponding result holds for α>1/r−1/2 and p>2. In addition for G non-discrete the above result is shown to be sharp, in the sense that the range of values of α cannot be extended. The results include classical theorems of S. N. Bernstein, O. Szász and E. C. Titchmarsh.

Article information

Source
Ark. Mat., Volume 20, Number 1-2 (1982), 101-109.

Dates
Received: 1 January 1980
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485896972

Digital Object Identifier
doi:10.1007/BF02390501

Mathematical Reviews number (MathSciNet)
MR660128

Zentralblatt MATH identifier
0492.43004

Subjects
Primary: 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
Secondary: 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc. 43A70: Analysis on specific locally compact and other abelian groups [See also 11R56, 22B05]

Rights
1982 © Institut Mittag Leffler

Citation

Bloom, Walter R. Absolute convergence of Fourier series on totally disconnected groups. Ark. Mat. 20 (1982), no. 1-2, 101--109. doi:10.1007/BF02390501. https://projecteuclid.org/euclid.afm/1485896972


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