Isaac Scientific Publishing

Advances in Analysis

Zero Sets of Solutions of the Generalized Darboux Equation

Download PDF (612.8 KB) PP. 272 - 280 Pub. Date: October 20, 2017

DOI: 10.22606/aan.2017.24006

Author(s)

  • Valery Volchkov
    Donetsk National University, Faculty of Mathematics and Information Technologies, Universitetskaya 24, Donetsk, 83001
  • Vitaly Volchkov*

    Donetsk National University, Faculty of Mathematics and Information Technologies, Universitetskaya 24, Donetsk, 83001

Abstract

A non-Euclidean analog of the generalized Darboux equation is considered. For the case where its solutions are radial functions of the second variable we obtain an uniqueness result (Theorem 1), which deals with zero sets of these solutions. The example of the function in Theorem 2 of the paper shows that Theorem 1 cannot be essentially reinforced.

Keywords

Darboux equation, hyperbolic space, transmutation homeomorphisms.

References

[1] S. Helgason, Groups and Geometric Analysis (Academic Press, New York 1984).

[2] S. Helgason, Geometric Analysis on Symmetric Spaces (Amer. Math. Soc., Providence, Rhode Island 1994).

[3] S. Helgason, Integral Geometry and Radon Transforms (Springer, New York 2010).

[4] V. V. Volchkov, Integral Geometry and Convolution Equations (Kluwer Academic, Dordrecht 2003).

[5] V. V. Volchkov and Vit. V. Volchkov, Offbeat Integral Geometry on Symmetric Spaces (Birkh?user, Basel 2013).

[6] V. V. Volchkov and Vit. V. Volchkov, Conical injectivity sets of the Radon transform on spheres, Algebra and Analiz, 27 (5), 1–31 (2015).

[7] V. V. Volchkov and Vit. V. Volchkov, Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group (Springer Monographs in Mathematics, Springer, London 2009).

[8] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vol I (McGraw-Hill, New York 1953).

[9] B. Ya. Levin, Distribution of Zeros of Entire Functions (Gostekhizdat, Moscow 1956).

[10] T. H. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie groups, Special Functions: Group Theoretical Aspects and Applications (R. A. Askey et al. (eds.)), D. Reidel Publishing Company, Dordrecht, 1–85 (1984).

[11] L. H?rmander, The Analysis of Linear Partial Differential Operators, Vol I (Springer-Verlag, New York 1983).