Harmonic analysis of spherical functions on S U ( 1 , 1 )

Y. Benyamini; Yitzhak Weit

Annales de l'institut Fourier (1992)

  • Volume: 42, Issue: 3, page 671-694
  • ISSN: 0373-0956

Abstract

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Denote by L 1 ( K ∖ G / K ) the algebra of spherical integrable functions on S U ( 1 , 1 ) , with convolution as multiplication. This is a commutative semi-simple algebra, and we use its Gelfand transform to study the ideals in L 1 ( K ∖ G / K ) . In particular, we are interested in conditions on an ideal that ensure that it is all of L 1 ( K ∖ G / K ) , or that it is L 0 1 ( K ∖ G / K ) . Spherical functions on S U ( 1 , 1 ) are naturally represented as radial functions on the unit disk D in the complex plane. Using this representation, these results are applied to characterize harmonic and holomorphic functions on D .

How to cite

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Benyamini, Y., and Weit, Yitzhak. "Harmonic analysis of spherical functions on $SU(1,1)$." Annales de l'institut Fourier 42.3 (1992): 671-694. <http://eudml.org/doc/74969>.

@article{Benyamini1992,
abstract = {Denote by $L^ 1(K\backslash G/K)$ the algebra of spherical integrable functions on $SU(1,1)$, with convolution as multiplication. This is a commutative semi-simple algebra, and we use its Gelfand transform to study the ideals in $L^ 1(K\backslash G/K)$. In particular, we are interested in conditions on an ideal that ensure that it is all of $L^ 1(K\backslash G/K)$, or that it is $L_ 0^ 1(K\backslash G/K)$. Spherical functions on $SU(1,1)$ are naturally represented as radial functions on the unit disk $D$ in the complex plane. Using this representation, these results are applied to characterize harmonic and holomorphic functions on $D$.},
author = {Benyamini, Y., Weit, Yitzhak},
journal = {Annales de l'institut Fourier},
keywords = {spherical functions on ; algebra of spherical integrable functions; convolution; semi-simple algebra; Gelfand transform; ideals; radial functions; holomorphic functions},
language = {eng},
number = {3},
pages = {671-694},
publisher = {Association des Annales de l'Institut Fourier},
title = {Harmonic analysis of spherical functions on $SU(1,1)$},
url = {http://eudml.org/doc/74969},
volume = {42},
year = {1992},
}

TY - JOUR
AU - Benyamini, Y.
AU - Weit, Yitzhak
TI - Harmonic analysis of spherical functions on $SU(1,1)$
JO - Annales de l'institut Fourier
PY - 1992
PB - Association des Annales de l'Institut Fourier
VL - 42
IS - 3
SP - 671
EP - 694
AB - Denote by $L^ 1(K\backslash G/K)$ the algebra of spherical integrable functions on $SU(1,1)$, with convolution as multiplication. This is a commutative semi-simple algebra, and we use its Gelfand transform to study the ideals in $L^ 1(K\backslash G/K)$. In particular, we are interested in conditions on an ideal that ensure that it is all of $L^ 1(K\backslash G/K)$, or that it is $L_ 0^ 1(K\backslash G/K)$. Spherical functions on $SU(1,1)$ are naturally represented as radial functions on the unit disk $D$ in the complex plane. Using this representation, these results are applied to characterize harmonic and holomorphic functions on $D$.
LA - eng
KW - spherical functions on ; algebra of spherical integrable functions; convolution; semi-simple algebra; Gelfand transform; ideals; radial functions; holomorphic functions
UR - http://eudml.org/doc/74969
ER -

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