Abstract
We prove an analog of the classical Titchmarsh theorem on the image under the Fourier transform of a set of functions satisfying the Lipschitz condition in L 2 for functions on noncompact rank 1 Riemannian symmetric spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Helgason S., Differential Geometry and Symmetric Spaces [Russian translation], Mir, Moscow (1964).
Helgason S., Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions [Russian translation], Mir, Moscow (1987).
Helgason S., Geometric Analysis on Symmetric Spaces, Amer. Math. Soc., Providence, RI (1994).
Helgason S., Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York (1978).
Wolf J. A., Spaces of Constant Curvature [Russian translation], Nauka, Moscow (1982).
Besse A. L., Manifolds All of Whose Geodesics Are Closed [Russian translation], Mir, Moscow (1981).
Titchmarsh E. C., Introduction to the Theory of Fourier Integrals [Russian translation], OGIZ; Gostekhizdat, Moscow (1948).
Helgason S., “A duality for symmetric spaces with applications to group representations,” Adv. Math., 5, No.1, 1–154 (1970).
Nikol'skii S. M. and Lizorkin P. I., “Function spaces on the sphere, related with approximation theory,” Mat. Zametki, 41, No.4, 509–516 (1987).
Nikol'skii S. M. and Lizorkin P. I., “Approximation of functions on the sphere,” Izv. Akad. Nauk SSSR Ser. Mat., 51, No.3, 635–651 (1987).
Lizorkin P. I. and Rustamov Kh. P., “Nikol'skii-Besov spaces on the sphere that are associated with approximation theory,” Trudy Mat. Inst. Steklov., 204, 172–200 (1993).
Rustamov Kh. P., “On approximation of functions on the sphere,” Izv. Ross. Akad. Nauk Ser. Mat., 57, No.5, 127–148 (1993).
Younis M. S., “Fourier transforms of Lipschitz functions on the hyperbolic plane,” Internat. J. Math. & Math. Sci., 21, No.2, 397–401 (1998).
Gangolli R. and Varadarajan V. S., Harmonic Analysis of Spherical Functions on Real Reductive Groups, Springer-Verlag, Berlin etc. (1988).
Platoons S. S., “Approximation of functions in the L 2-metric on noncompact rank 1 symmetric spaces,” Algebra i Analiz, 11, No.1, 244–270 (1999).
Nikiforov A. F. and Uvarov V. B., Special Functions of Mathematical Physics [in Russian], Nauka, Moscow (1978).
Author information
Authors and Affiliations
Additional information
Original Russian Text Copyright © 2005 Platoons S. S.
__________
Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 1374–1387, November–December, 2005.
Rights and permissions
About this article
Cite this article
Platonov, S.S. The Fourier Transform of Functions Satisfying the Lipschitz Condition on Rank 1 Symmetric Spaces. Sib Math J 46, 1108–1118 (2005). https://doi.org/10.1007/s11202-005-0105-z
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11202-005-0105-z