RECOVERING THE LAPLACIAN FROM CENTERED MEANS ON BALLS AND SPHERES OF FIXED RADIUS
Various issues related to restrictions on radii in meanvalue formulas are well-known in the theory of harmonic functions. In particular, using the Brown-Schreiber-Taylor theorem on spectral synthesis for motion-invariant subspaces in 𝐶(R^𝑛), one can obtain the following strengthening of the classi...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Petrozavodsk State University
2023-01-01
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| Series: | Проблемы анализа |
| Subjects: | |
| Online Access: | https://issuesofanalysis.petrsu.ru/article/genpdf.php?id=13290&lang=ru |
| Summary: | Various issues related to restrictions on radii in meanvalue formulas are well-known in the theory of harmonic functions.
In particular, using the Brown-Schreiber-Taylor theorem on spectral synthesis for motion-invariant subspaces in 𝐶(R^𝑛), one can
obtain the following strengthening of the classical mean-value theorem for harmonic functions: if a continuous function on R^𝑛
satisfies the mean-value equations for all balls and spheres of a fixed radius 𝑟, then it is harmonic on R^𝑛. In connection with this result, the
following problem arises: recover the Laplacian from the deviation
of a function from its average values on balls and spheres of a fixed
radius. The aim of this work is to solve this problem. The article
uses methods of harmonic analysis, as well as the theory of entire
and special functions. The key step in the proof of the main result
is expansion of the Dirac delta function in terms of a system of
radial distributions supported in a fixed ball, biorthogonal to some
system of spherical functions. A similar approach can be used to
invert a number of convolution operators with compactly supported
radial distributions. |
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| ISSN: | 2306-3424 2306-3432 |