Harmonic analysis of spherical functions on SU(1,1)
Annales de l'Institut Fourier, Volume 42 (1992) no. 3, p. 671-694

Denote by L 1 (KG/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 (KG/K). In particular, we are interested in conditions on an ideal that ensure that it is all of L 1 (KG/K), or that it is L 0 1 (KG/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.

Soit L 1 (KG/K) l’algèbre des fonctions sphériques intégrales sur SU(1,1), munie de l’opération de convolution comme multiplication. C’est une algèbre commutative semi-simple. Nous utilisons la transformation de Gelfand pour étudier les idéaux de L 1 (KG/K). En particulier, nous trouvons des conditions sur un idéal qui garantissent qu’il est identique à L 1 (KG/K), ou à L 0 1 (KG/K).

Les fonctions sphériques sur SU(1,1) se représentent naturellement comme des fonctions radiales sur le disque unité D du plan complexe. À l’aide de cette représentation, nous appliquons les résultats précédents à la caractérisation des fonctions harmoniques et holomorphes sur D.

@article{AIF_1992__42_3_671_0,
     author = {Benyamini, Y. and Weit, Yitzhak},
     title = {Harmonic analysis of spherical functions on $SU(1,1)$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {42},
     number = {3},
     year = {1992},
     pages = {671-694},
     doi = {10.5802/aif.1305},
     zbl = {0763.43006},
     mrnumber = {94d:43009},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1992__42_3_671_0}
}
Benyamini, Y.; Weit, Yitzhak. Harmonic analysis of spherical functions on $SU(1,1)$. Annales de l'Institut Fourier, Volume 42 (1992) no. 3, pp. 671-694. doi : 10.5802/aif.1305. http://www.numdam.org/item/AIF_1992__42_3_671_0/

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