Convolution equations and mean value theorems for solutions of linear elliptic equations with constant coefficients in the complex plane.
Анотація
Convolution equations generated by distributions with compact supports and the corresponding mean value theorems was investigated by many authors. In particular, Volchkov described a wide class of radial distributions with compact supports such that solutions of the corresponding convolution equations in open Euclidean balls can be efficiently characterized in terms of the Bessel functions. This characterization implies different corollaries such as uniqueness theorems for solutions of the corresponding convolution equations and two-radius theorems that go back to John (1934) and Delsarte (1961), respectively.Посилання
John F. Plane Waves and Spherical Means, Applied to Partial Differential Equations, Dover, New York, 1971.
Delsarte J. Lectures on Topics in Mean Periodic Functions and the Two-Radius Theorem, Tata Institute of Fundamental Research, Bombay, 1961.
Berenstein C. A. and Struppa D. C., Complex Analysis and Convolution Equations, Enciclopedia of Mathematical Sciences, V. 54, pp. 1–108, Springer-Verlag, New York, 1993.
Volchkov V. V. Integral Geometry and Convolution Equations, Kluwer Academic Publishers, Dordrecht, 2003.
Trofymenko O. D., Two-radii theorem for solutions of some mean value equations, Matematychni Studii, 40:2, 137-143 (2013).
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