Full text (in Russian)
Complementing the authors’ earlier joint papers on the application of orthogonal wavelets to represent solutions of Dirichlet problems with the Laplace operator and its powers in a disk and a ring and of interpolating wavelets for the same problem in a disk, we develop a technique of applying interpolating periodic wavelets in a ring for the Dirichlet boundary value problem. The emphasis is not on the exact representation of the solution in the form of (double) series in a wavelet system but on the approximation of solutions with any given accuracy by finite linear combinations of dyadic rational translations of special harmonic polynomials; these combinations are constructed with the use of interpolating wavelets. The obtained approximation formulas are simply calculated, especially if the squared Fourier transform of the Meyer scaling function with the properties described in the paper is explicitly defined in terms of the corresponding elementary functions.
Keywords: interpolating wavelets, multiresolution analysis (MRA), Dirichlet problem, Laplace operator, best approximation, modulus of continuity
Received September 05, 2018
Revised November 21, 2018
Accepted November 26, 2018
Funding Agency: This work was supported by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).
Yurii Nikolaevich Subbotin, RAS Corresponding Member, Prof., Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia,
e-mail: yunsub@imm.uran.ru
Nikolai Ivanovich Chernykh, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia, Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: chernykh@imm.uran.ru
REFERENCES
1. Subbotin Yu.N., Chernykh N.I. Interpolation wavelets in boundary value problems. Proc. Steklov Inst. Math. 2018, vol. 300, suppl. 1, pp. 172–183. doi: 10.1134/S0081543818020177
2. Donoho D.L. Interpolating wavelet transforms: preprint. Stanford: Stanford University, 1992, 54 p.
3. Goluzin G.M. The solution of the basic plane problems of a mathematical physics for the case of Laplace equations and a multiply connected domains bounded by circular curves (the method of functional equations), Mat. Sat, 1934, vol. 41, no. 2, pp. 246–278. (in Russian)
4. Subbotin Yu.N., Chernykh N.I. Harmonic wavelets and asymptotics of the solution of the Dirichlet problem in a circle with close-meshed opening. Mat. modeling, 2002, vol. 14, no. 5, pp. 17–30. (in Russian)
5. Korneychuk N.P.Extreme problems of approximation theory. M., Science. 1976. 320 p. (in Russian)
6. Akhiezer N.I. Lectures on the theory of approximation. M.; L: Gostekhizdat, 1947. (in Russian)
