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Absolute convergence of Fourier series on totally disconnected groups

Bloom, W.R. (1982) Absolute convergence of Fourier series on totally disconnected groups. Arkiv för matematik, 20 (1-2). pp. 101-109.

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Let G denote a totally disconnected locally compact metric abelian group with translation invariant metric d and character group ΓG. The Lipschitz spaces are defined by {Mathematical expression} 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.

Item Type: Journal Article
Murdoch Affiliation(s): School of Mathematical and Physical Sciences
Publisher: Kluwer Academic Publishers
Copyright: © 1982 Institut Mittag Leffler.
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