Wiener’s Tauberian theorem in $L_1(G//K)$ and harmonic functions in the unit disk
HTML articles powered by AMS MathViewer
- by Y. Ben Natan, Y. Benyamini, H. Hedenmalm and Y. Weit PDF
- Bull. Amer. Math. Soc. 32 (1995), 43-49 Request permission
Abstract:
Our main result is to give necessary and sufficient conditions, in terms of Fourier transforms, on a closed ideal I in ${L^1}(G//K)$, the space of radial integrable functions on $G = SU(1,1)$, so that $I = {L^1}(G//K)$ or $I = L_0^1(G//K)$—the ideal of ${L^1}(G//K)$ functions whose integral is zero. This is then used to prove a generalization of Furstenberg’s theorem which characterizes harmonic functions on the unit disk by a mean value property and a "two circles" Morera type theorem (earlier announced by Agranovskiĭ).References
-
M. G. Agranovskiĭ, Tests for holomorphy in symmetric domains, Siberian Math. J. 22 (1981), 171-179.
- Alexander Borichev and Håkan Hedenmalm, Approximation in a class of Banach algebras of quasianalytically smooth analytic functions, J. Funct. Anal. 115 (1993), no. 2, 359–390. MR 1234396, DOI 10.1006/jfan.1993.1095
- Y. Benyamini and Y. Weit, Harmonic analysis of spherical functions on $\textrm {SU}(1,1)$, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 3, 671–694 (English, with English and French summaries). MR 1182644, DOI 10.5802/aif.1305
- T. Carleman, L’Intégrale de Fourier et Questions que s’y Rattachent, Publications Scientifiques de l’Institut Mittag-Leffler, Uppsala, 1944 (French). Vol. 1. MR 0014165
- Yngve Domar, On the analytic transform of bounded linear functionals on certain Banach algebras, Studia Math. 53 (1975), no. 3, 203–224. MR 383003, DOI 10.4064/sm-53-3-203-224
- L. Ehrenpreis and F. I. Mautner, Some properties of the Fourier transform on semi-simple Lie groups. I, Ann. of Math. (2) 61 (1955), 406–439. MR 69311, DOI 10.2307/1969808
- L. Ehrenpreis and F. I. Mautner, Some properties of the Fourier-transform on semisimple Lie Groups. III, Trans. Amer. Math. Soc. 90 (1959), 431–484. MR 102755, DOI 10.1090/S0002-9947-1959-0102755-3
- Harry Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963), 335–386. MR 146298, DOI 10.2307/1970220
- Harry Furstenberg, Boundaries of Riemannian symmetric spaces, Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970), Pure and Appl. Math., Vol. 8, Dekker, New York, 1972, pp. 359–377. MR 0438048
- Håkan Hedenmalm, On the primary ideal structure at infinity for analytic Beurling algebras, Ark. Mat. 23 (1985), no. 1, 129–158. MR 800176, DOI 10.1007/BF02384421
- Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
- M. Kac, A remark on Wiener’s Tauberian theorem, Proc. Amer. Math. Soc. 16 (1965), 1155–1157. MR 185320, DOI 10.1090/S0002-9939-1965-0185320-X
- Paul Koosis, The logarithmic integral. I, Cambridge Studies in Advanced Mathematics, vol. 12, Cambridge University Press, Cambridge, 1988. MR 961844, DOI 10.1017/CBO9780511566196
- Serge Lang, $\textrm {SL}_{2}(\textbf {R})$, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. MR 0430163
- N. N. Lebedev, Special functions and their applications, Revised English edition, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. Translated and edited by Richard A. Silverman. MR 0174795, DOI 10.1063/1.3047047
- Horst Leptin, Ideal theory in group algebras of locally compact groups, Invent. Math. 31 (1975/76), no. 3, 259–278. MR 399344, DOI 10.1007/BF01403147
- Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966. MR 0232968, DOI 10.1007/978-3-662-11761-3
- A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 3, Gordon and Breach Science Publishers, New York, 1990. More special functions; Translated from the Russian by G. G. Gould. MR 1054647
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 32 (1995), 43-49
- MSC: Primary 43A80; Secondary 46H99
- DOI: https://doi.org/10.1090/S0273-0979-1995-00554-9
- MathSciNet review: 1273399