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.Riemannian symmetric spaces constitute a remarkable class of Riemannian manifolds on which various problems of geometry, function theory, and mathematical physics are actively studied (e.g., see [1][2][3][4] and the bibliography therein). For example, the Fourier series expansion (more exactly, its analog) is defined on compact symmetric spaces and the Fourier transform is defined on noncompact symmetric spaces; moreover, many problems of the classical harmonic analysis have their natural analogs for symmetric spaces. Among all Riemannian symmetric spaces we especially distinguish the class of rank 1 Riemannian symmetric spaces. These manifolds possess nice geometric properties; in particular, they are two-point homogeneous spaces (see [5, Chapter 8]), while all geodesics on compact rank 1 symmetric spaces are closed and have the same length (see [6]). The class of rank 1 Riemannian symmetric spaces includes the n-dimensional sphere S n and the n-dimensional Lobachevskiȋ space. Henceforth by a rank 1 symmetric space we mean a noncompact rank 1 Riemannian symmetric space.In this article, for rank 1 symmetric spaces, we obtain an analog of one classical Titchmarsh theorem on description of the image under the Fourier transform of a class of functions satisfying the Lipschitz condition in L 2 . We now give the exact statement of this theorem.Suppose that f (x) is a function in the L 2 (R) space (all functions below are complex-valued), · L 2 (R) is the norm of L 2 (R), and α is an arbitrary number in the interval (0, 1).
Definition 1. A function f (x) belongs to the Lipschitz classWe present some necessary facts about the Fourier transform on symmetric spaces (see [2,3]). The necessary information from the theory of semisimple Lie groups and symmetric spaces is given, for example, in [1]. We can implement an arbitrary noncompact Riemannian symmetric space X as the quotient space G/K, where G is a connected noncompact semisimple Lie group with a finite center and K is a maximal compact subgroup in G. The group G acts transitively on the set X = G/K by left Petrozavodsk.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.