Radial functions and invariant convolution operators

Christopher Meaney

doi:10.1090/S0002-9947-1984-0760979-0

Radial functions and invariant convolution operators

Christopher Meaney^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

For 1 < p < 2 and n > 1, let A_p(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 A_p(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 consider^BA_P(G) the subalgebra of B-invariant elements of A_P(G). In particular, we show that the dual of B A_P(G) is equal to the space of bounded, right-translation invariant operators on L_P(G) which commute with the action of B.

Original language	English
Pages (from-to)	665-674
Number of pages	10
Journal	Transactions of the American Mathematical Society
Volume	286
Issue number	2
DOIs	https://doi.org/10.1090/S0002-9947-1984-0760979-0
Publication status	Published - 1984
Externally published	Yes

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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10.1090/S0002-9947-1984-0760979-0

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title = "Radial functions and invariant convolution operators",

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 {\^I} [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.",

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

author = "Christopher Meaney",

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doi = "10.1090/S0002-9947-1984-0760979-0",

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journal = "Transactions of the American Mathematical Society",

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TY - JOUR

T1 - Radial functions and invariant convolution operators

AU - Meaney, Christopher

PY - 1984

Y1 - 1984

N2 - 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.

AB - 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.

KW - B-characters

KW - Central function

KW - Compact semisimple Lie group

KW - Convolution operator

KW - Figa-Talamanca-Herz algebra

KW - Fourier transform

KW - Multiplier

KW - Radial function

KW - [FIA]g

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U2 - 10.1090/S0002-9947-1984-0760979-0

DO - 10.1090/S0002-9947-1984-0760979-0

M3 - Article

AN - SCOPUS:84933814703

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SP - 665

EP - 674

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -

Radial functions and invariant convolution operators

Abstract

Keywords

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