Arkiv för Matematik

  • Ark. Mat.
  • Volume 34, Number 2 (1996), 199-224.

Wiener's tauberian theorem for spherical functions on the automorphism group of the unit disk

Yaakov Ben Natan, Yoav Benyamini, H»kan Hendenmalm, and Yitzhak Weit

Full-text: Open access

Abstract

Our main result gives necessary and sufficient conditions, in terms of Fourier transforms, for an ideal in the convolution algebra of spherical integrable functions on the (conformal) automorphism group of the unit disk to be dense, or to have as closure the closed ideal of functions with integral 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 Agranovskii).

Note

The second author's work was partially supported by the fund for the promotion of research at the Technion-Israel Institute of Technology. The third author's work was partially supported by the Swedish Natural Science Research Council, and by the 1992 Wallenberg Prize from the Swedish Mathematical Society.

Article information

Source
Ark. Mat., Volume 34, Number 2 (1996), 199-224.

Dates
Received: 23 August 1995
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898511

Digital Object Identifier
doi:10.1007/BF02559544

Mathematical Reviews number (MathSciNet)
MR1416664

Zentralblatt MATH identifier
0858.43003

Rights
1996 © Institut Mittag-Leffler

Citation

Natan, Yaakov Ben; Benyamini, Yoav; Hendenmalm, H»kan; Weit, Yitzhak. Wiener's tauberian theorem for spherical functions on the automorphism group of the unit disk. Ark. Mat. 34 (1996), no. 2, 199--224. doi:10.1007/BF02559544. https://projecteuclid.org/euclid.afm/1485898511


Export citation

References

  • Agranovskiî, M. L., Tests for holomorphy in symmetric domains, Sibirsk. Mat. Zh. 22:2 (1981), 7–18, 235 (Russian). English transl.: Siberian Math. J. 22 (1981), 171–179.
  • Ahlfors, L. and Heins, M., Questions of regularity connected with the Phragmén-Lindelöf principle, Ann. of Math. 50:2 (1949), 341–346.
  • Bargmann, V., Irreducible unitary representations of the Lorentz group, Ann. of Math. 48 (1947), 568–640.
  • Ben Natan, Y., Benyamini, Y., Hedenmalm, H. and Weit, Y., Wiener's tauberian theorem in L1(G//K) and harmonic functions in the unit disk, Bull. Amer. Math. Soc. 32 (1995), 43–49.
  • Benyamini, Y. and Weit, Y., Harmonic analysis of spherical functions on SU(1,1), Ann. Inst. Fourier (Grenoble) 42 (1992), 671–694.
  • Bernstein, C. A. and Zalcman, L., Pompeiu's problem on spaces of constant curvature, J. Analyse Math. 30 (1976), 113–130.
  • Beurling, A., Analytic continuation across a linear boundary, Acta Math. 128 (1972), 153–182.
  • Boas, Jr., R. P., Entire Functions, Academic Press, New York, 1954.
  • Borichev, A. and Hedenmalm, H., Approximation in a class of Banach algebras of quasi-analytically smooth analytic functions, J. Funct. Anal. 115 (1993), 359–390.
  • Borichev, A. and Hedenmalm, H., Completeness of translates in weighted spaces on the half-line, Acta Math. 174 (1995), 1–84.
  • Carleman, T., L'intégrale de Fourier et questions qui s'y rattachent, Uppsala, 1944.
  • Choquet, G. and Deny, J., Sur l'équation de convolution μ=μ*σ, C. R. Acad. Sci. Paris 250 (1960), 799–801.
  • Domar, Y., On the analytic transform of bounded linear functionals on certain Banach algebras, Studia Math. 53 (1975), 429–440.
  • Ehrenpreis, L. and Mautner, F. I., Some properties of the Fourier transform on semi simple Lie groups I, Ann. of Math. 61 (1955), 406–439.
  • Ehrenpreis, L. and Mautner, F. I., Some properties of the Fourier transform on semi simple Lie groups III, Trans. Amer. Math. Soc. 90 (1959), 431–484.
  • Furstenberg, H., A Poisson formula for semi-simple groups, Ann. of Math. 77 (1963), 335–386.
  • Furstenberg, H., Boundaries of Riemannian symmetric spaces, in Symmetric Spaces (Boothby, W. M. and Weiss, G. L., eds.), pp. 359–377, Marcel Dekker Inc., New York, 1972.
  • Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products, Academic Press, New York, 1980.
  • Gurariî, V. P., Harmonic analysis in spaces with a weight, Trudy Moskov. Mat. Obshch. 35 (1976), 21–76 (Russian). English transl.: Trans. Moscow Math. Soc. 35 (1979), 21–75.
  • Hedenmalm, H., On the primary ideal structure at infinity for analytic Beurling algebras, Ark. Mat. 23 (1985), 129–158.
  • Hedenmalm, H., Translates of functions of two variables, Duke Math. J. 58 (1989), 251–297.
  • Helgason, S., Groups and Geometric Analysis, Academic Press, Orlando, Fla., 1984.
  • Kac, M., A remark on Wiener's Tauberian theorem, Proc. Amer. Math. Soc. 16 (1965), 1155–1157.
  • Koosis, P., The Logarithmic Integral, Cambridge University Press, Cambridge-New York, 1988.
  • Lang, S., SL2(R), Addison-Wesley, Reading, Mass., 1975.
  • Lebedev, N. N., Special Functions and their Applications, Prentice-Hall, Englewood Cliffs, N. J., 1965.
  • Leptin, H., Ideal theory in group algebras of locally compact groups, Invent. Math. 31 (1976), 259–278.
  • Magnus, W., Oberhettinger, F. and Soni, R. P., Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, New York, 1966.
  • Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O. I., Integrals and Series 3, Gordon and Breach Science Publishers, New York, 1990.
  • Sugiura, M., Unitary Representations and Harmonic Analysis, an Introduction, Kodansha, Tokyo and Halsted Press (Wiley), New York-London-Sydney, 1975.
  • Vilenkin, N., Special Functions and the Theory of Group Representations, Nauka, Moscow, 1965 (Russian). English transl.: Amer. Math. Soc. Transl. 22, Amer. Math. Soc., Providence, R. I., 1968.