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

Soit L 1 (K∖G/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 (K∖G/K). En particulier, nous trouvons des conditions sur un idéal qui garantissent qu’il est identique à L 1 (K∖G/K), ou à L 0 1 (K∖G/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.

Denote by L 1 (K∖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∖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 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.

@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},
     zbl = {0763.43006},
     mrnumber = {94d:43009},
     language = {en},
     url = {https://aif.centre-mersenne.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  - https://aif.centre-mersenne.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 https://aif.centre-mersenne.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, Tome 42 (1992) no. 3, pp. 671-694. doi : 10.5802/aif.1305. https://aif.centre-mersenne.org/articles/10.5802/aif.1305/

[A] M.G. Agranovskii, Tests for holomorphy in symmetric domains, Siberian Math. J., 22 (1981), 171-179. | Zbl

[BY] C.A. Berenstein and P. Yang, An overdetermined Neuman problem in the unit disk, Advances in Math., 44 (1982), 1-17. | MR | Zbl

[BZ] C.A. Berenstein and L. Zalcman, Pomepin's problem on spaces of constant curvature, J. Analyse Math., 30 (1976), 113-130. | MR | Zbl

[CD] G. Choquet and J. Deny, Sur l'équation de convolution µ = µ * σ, C.R. Acad. Sc. Paris, 250 (1960), 779-801. | MR | Zbl

[EM1] L. Ehrenpreis and F.I. Mautner, Some properties of the Fourier transform on semi-simple Lie groups I, Ann. of Math., 61 (1955), 406-439. | MR | Zbl

[EM2] L. Ehrenpreis and F.I. Mautner, Some properties of the Fourier transform on semi-simple Lie groups III, Trans. Amer. Math. Soc., 90 (1959), 431-484. | MR | Zbl

[Fo] S.R. Foguel, On iterates of convolutions, Proc. Amer. Math. Soc., 47 (1975), 368-370. | MR | Zbl

[FW] S.R. Foguel and B. Weiss, On convex power series of conservative Markov operators, Proc. Amer. Math. Soc., 38 (1973), 325-330. | MR | Zbl

[Fu1] H. Furstenberg, A Poisson formula for semi-simple groups, Ann. of Math., 77 (1963), 335-386. | MR | Zbl

[Fu2] H. Furstenberg, 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] V.P. Gurarii, Harmonic analysis in spaces with weights, Trans. Moscow Math. Soc., 35 (1979), 21-75. | MR | Zbl

[H] H. Hedenmalm, On the primary ideal structure at inifinity for analytic Beurling algebras, Ark. Mat., 23 (1985), 129-158. | MR | Zbl

[Ho] K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, New Jersy, 1962. | MR | Zbl

[K] B.I. Korenblum, A generalization of Wiener's Tauberian Theorem and harmonic analysis of rapidly increasing functions, (Russian), Trudy Moskov. Mat. Obš, 7 (1958), 121-148.

[KT] Y. Katznelson and L. Tzafriri, On power-bounded operators, J. Func. Anal., 68 (1986), 313-328. | MR | Zbl

[L] S. Lang, SL2(R), Addison-Wesley, Reading, Mass, 1975.

[RW] T. Ramsey and Y. Weit, Ergodic and mixing properties of measures on locally compact Abelian groups, Proc. Amer. Math. Soc., 92 (1984), 519-520. | MR | Zbl

[S] M. Sugiura, Unitary Representations and Harmonic Analysis, an Introduction, North-Holland, 1975. | MR | Zbl

[T] E.C. Titchmarsh, Theory of Functions, 2nd ed., Oxford University Press, 1939. | JFM

[Z] L. Zalcman, Analyticity and the Pompeiu Problem, Arch. Rat. Mech. Anal., 47 (1972), 237-254. | MR | Zbl

Cité par Sources :