For citation:
Komissarova N. E. Generalized non-stationary multiresolution analysis on locally compact zero-dimensional groups with an arbitrary generating sequence. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2026, vol. 26, iss. 2, pp. 165-174. DOI: 10.18500/1816-9791-2026-26-2-165-174, EDN: EJPJVI
Generalized non-stationary multiresolution analysis on locally compact zero-dimensional groups with an arbitrary generating sequence
We consider a locally compact zero-dimensional group $(G, \dot+),$ whose coset orders are arbitrary prime numbers. If the orders of the cosets are constant and equal to some prime number, the multiresolution analysis can be constructed by using a dilation operator. If the orders of the cosets are different, it is not possible to define the dilation operator. In this paper, we describe a method for constructing a multiresolution analysis without using the dilation operator and construct the corresponding wavelet basis.
- Farkov Yu. A. Orthogonal wavelets on direct products of cyclic groups. Mathematical Notes, 2007, vol. 82, iss. 6, pp. 843–859. DOI: https://doi.org/10.1134/S0001434607110296, EDN: LKELPN
- Khrennikov A. Yu., Shelkovich V. M., Skopina M. p-Adic refinable functions and MRA-based wavelets. arXiv: 0711.2820v1 [math.GM].
- Lukomskii S. F. Haar series on compact zero-dimensional groups. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2009, vol. 9, iss. 1, pp. 14–19 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2009-9-1-14-19, EDN: KAMPFR
- Lukomskii S. F. Multiresolution analysis on zero-dimensional Abelian groups and wavelets bases. Sbornik: Mathematics, 2010, vol. 201, iss. 5, pp. 669–661. DOI: http://dx.doi.org/10.1070/SM2010v201n05ABEH004088, EDN: OHNDUH
- Novikov I. Ya., Protasov V. Yu., Skopina M. A. Teoriya vspleskov [Wavelet theory]. Moscow, Fizmatlit, 2005. 616 p. (in Russian).
- Komissarova N. Ye. Non-stationary multiresolution analysis on locally compact zero-dimensional groups with arbitrary generating sequence. Research in Algebra, Number Theory, Functional Analysis and Related Problems, 2019, iss. 9, pp. 7–16 (in Russian). EDN: HVPJRH
- Lukomskii S. F. Lebesgue constants for characters of the compact zero-dimensional Abelian group. East Journal on Approximation, 2009, vol. 15, iss. 2, pp. 219–231.
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