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 .
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 .
@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}, pages = {671--694}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {42}, number = {3}, year = {1992}, doi = {10.5802/aif.1305}, mrnumber = {94d:43009}, zbl = {0763.43006}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.1305/} }
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 SP - 671 EP - 694 VL - 42 IS - 3 PB - Institut Fourier PP - Grenoble UR - http://archive.numdam.org/articles/10.5802/aif.1305/ DO - 10.5802/aif.1305 LA - en ID - AIF_1992__42_3_671_0 ER -
%0 Journal Article %A Benyamini, Y. %A Weit, Yitzhak %T Harmonic analysis of spherical functions on $SU(1,1)$ %J Annales de l'Institut Fourier %D 1992 %P 671-694 %V 42 %N 3 %I Institut Fourier %C Grenoble %U http://archive.numdam.org/articles/10.5802/aif.1305/ %R 10.5802/aif.1305 %G en %F 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://archive.numdam.org/articles/10.5802/aif.1305/
[A] Tests for holomorphy in symmetric domains, Siberian Math. J., 22 (1981), 171-179. | Zbl
,[BY] An overdetermined Neuman problem in the unit disk, Advances in Math., 44 (1982), 1-17. | MR | Zbl
and ,[BZ] Pomepin's problem on spaces of constant curvature, J. Analyse Math., 30 (1976), 113-130. | MR | Zbl
and ,[CD] Sur l'équation de convolution µ = µ * σ, C.R. Acad. Sc. Paris, 250 (1960), 779-801. | MR | Zbl
and ,[EM1] Some properties of the Fourier transform on semi-simple Lie groups I, Ann. of Math., 61 (1955), 406-439. | MR | Zbl
and ,[EM2] Some properties of the Fourier transform on semi-simple Lie groups III, Trans. Amer. Math. Soc., 90 (1959), 431-484. | MR | Zbl
and ,[Fo] On iterates of convolutions, Proc. Amer. Math. Soc., 47 (1975), 368-370. | MR | Zbl
,[FW] On convex power series of conservative Markov operators, Proc. Amer. Math. Soc., 38 (1973), 325-330. | MR | Zbl
and ,[Fu1] A Poisson formula for semi-simple groups, Ann. of Math., 77 (1963), 335-386. | MR | Zbl
,[Fu2] Boundaries of Riemannian symmetric spaces, in Symmetric Spaces, Marcel Dekker Inc., New-York, 1972 (W.M. Boothby and G.L. Weiss, Editors), 359-377. | MR | Zbl
,[G] Harmonic analysis in spaces with weights, Trans. Moscow Math. Soc., 35 (1979), 21-75. | MR | Zbl
,[H] On the primary ideal structure at inifinity for analytic Beurling algebras, Ark. Mat., 23 (1985), 129-158. | MR | Zbl
,[Ho] Banach Spaces of Analytic Functions, Prentice-Hall, New Jersy, 1962. | MR | Zbl
,[K] A generalization of Wiener's Tauberian Theorem and harmonic analysis of rapidly increasing functions, (Russian), Trudy Moskov. Mat. Obš, 7 (1958), 121-148.
,[KT] On power-bounded operators, J. Func. Anal., 68 (1986), 313-328. | MR | Zbl
and ,[L] SL2(R), Addison-Wesley, Reading, Mass, 1975.
,[RW] Ergodic and mixing properties of measures on locally compact Abelian groups, Proc. Amer. Math. Soc., 92 (1984), 519-520. | MR | Zbl
and ,[S] Unitary Representations and Harmonic Analysis, an Introduction, North-Holland, 1975. | MR | Zbl
,[T] Theory of Functions, 2nd ed., Oxford University Press, 1939. | JFM
,[Z] Analyticity and the Pompeiu Problem, Arch. Rat. Mech. Anal., 47 (1972), 237-254. | MR | Zbl
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