Soit l’algèbre des fonctions sphériques intégrales sur , 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 . En particulier, nous trouvons des conditions sur un idéal qui garantissent qu’il est identique à , ou à .
Les fonctions sphériques sur se représentent naturellement comme des fonctions radiales sur le disque unité du plan complexe. À l’aide de cette représentation, nous appliquons les résultats précédents à la caractérisation des fonctions harmoniques et holomorphes sur .
Denote by the algebra of spherical integrable functions on , with convolution as multiplication. This is a commutative semi-simple algebra, and we use its Gelfand transform to study the ideals in . In particular, we are interested in conditions on an ideal that ensure that it is all of , or that it is . Spherical functions on are naturally represented as radial functions on the unit disk in the complex plane. Using this representation, these results are applied to characterize harmonic and holomorphic functions on .
@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}, volume = {42}, year = {1992}, pages = {671-694}, doi = {10.5802/aif.1305}, mrnumber = {94d:43009}, zbl = {0763.43006}, language = {en}, url = {http://dml.mathdoc.fr/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, Tome 42 (1992) pp. 671-694. doi : 10.5802/aif.1305. http://gdmltest.u-ga.fr/item/AIF_1992__42_3_671_0/
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