Offbeat Integral Geometry on Symmetric Spaces

Abstract

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Zero Sets of Solutions of the Generalized Darboux Equation

“…Such equations are of considerable interest in their own right, but they are also important for many applications in geometric analysis (see [1], [2]) and integral geometry (see [3]- [5]). In particular, equations of type (1) are closely connected with the mean value operators on symmetric spaces (see [1]- [3], [5]). …”

An analog of the Schwartz theorem on spectral analysis on a hyperbolic plane

“…Without any exaggeration, we can say that the new modern stage of studies of this old problem began after the publication of work [7]. Details can be found in reviews [3,8,9] and books [4][5][6] containing the proofs of the most strong results that are related to the Pompeiu transformation and its generalizations. Some analogs of the theorem by L. Brown, B. M. Schreiber, and B.…”

“…In the last decade, the authors of the present work have developed a new approach to the above problems that is based on the application of operators with transmutation property (see [5,6,15]). This allows us to get the final results for some problems, in particular, local versions of the above-cited results (see [5,6]).…”

“…, 6, is nonzero and does not contain a function of the form (1.1) for any λ ∈ C n . The question about the additional conditions of validity of analogs of the Schwartz theorem for C(R n ) and other spaces of continuous functions was considered by many researchers (see, e.g., review [3] with large number of references and [4][5][6]). One of the most essential results in this direction was obtained by L. Brown, B. M. Schreiber, and B.…”

“…. , µ 6 with compact supports on R n , n ≥ 2, such that the space U ⊂ C ∞ (R n ) of solutions of the system of convolution equations f * µ i = 0, i = 1, . .…”