Radial functions and invariant convolution operators

Christopher Meaney*

*Corresponding author for this work

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Abstract

For 1 < p < 2 and n > 1, let Ap(Rn) denote the Figa- Talamanca-Herz algebra, consisting of functions of the form (FORMULA PRESENTED) with (FORMULA PRESENTED). We show that if 2n/(n + 1) < p < 2, then the subalgebra of radial functions in Ap(Rn) is strictly larger than the subspace of functions with expansions subject to the additional condition that /& and gk are radial for all k. This is a partial answer to a question of Eymard and is a consequence of results of Herz and Fefferman. We arrive at the statement above after examining a more abstract situation. Namely, we fix G Î [FIA]B and considerBAP(G) the subalgebra of B-invariant elements of AP(G). In particular, we show that the dual of B AP(G) is equal to the space of bounded, right-translation invariant operators on LP(G) which commute with the action of B.

Original languageEnglish
Pages (from-to)665-674
Number of pages10
JournalTransactions of the American Mathematical Society
Volume286
Issue number2
DOIs
Publication statusPublished - 1984
Externally publishedYes

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Keywords

  • B-characters
  • Central function
  • Compact semisimple Lie group
  • Convolution operator
  • Figa-Talamanca-Herz algebra
  • Fourier transform
  • Multiplier
  • Radial function
  • [FIA]g

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