Interpolating orthogonal wavelet bases are constructed with the use of several scaling functions. In the classical case, a basis of the space ${L}^2(\mathbb{R})$ is formed by shifts and compressions of a single function $\psi$. In contrast to the classical case, we consider several bases of the space $L^2(\mathbb{R})$, which are formed by shifts and compressions of $n$ functions $\psi^s$, $s=1,\ldots,n$. The $n$-separate wavelets constructed by the author earlier form $n$ orthonormal bases of the space $L^2(\mathbb{R})$. In 2008, Yu.N. Subbotin and N.I. Chernykh suggested a method for modifying the Meyer scaling function in such a way that the basis formed by it is simultaneously orthogonal and interpolating. In the present paper we propose a method for modifying the masks of $n$-separate scaling functions from a wide class in such a way that the resulting new scaling functions and wavelets remain orthogonal and at the same time become interpolating.
Keywords: orthogonal wavelet, interpolating wavelet, scaling function, basis, multiresolution analysis, mask of a scaling function, $n$-separate wavelet
Received September 8, 2022
Revised October 17, 2022
Accepted October 24, 2022
Funding Agency: This study is a part of the research carried out at the Ural Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-02-2022-874).
Ekaterina Aleksandrovna Pleshcheva, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: eplescheva@gmail.com
REFERENCES
1. Mallat S.G. Multiresolution approximations and wavelet orthonormal bases $L^2 (\mathbb{R})$. Trans. Amer. Math. Soc., 1989, vol. 315, no. 1, pp. 69–87. doi: 10.1090/S0002-9947-1989-1008470-5
2. Meyer Y. Ondelettes et operateurs: Ondelettes. Hermann, 1990, 215 p. ISBN: 2705661255 .
3. Pleshcheva E.A. New generalization of orthogonal wavelet bases. Proc. Steklov Inst. Math., 2011, vol. 273, suppl. 1, pp. 124–132. doi: 10.1134/S0081543811050130
4. Berkolaiko M.Z., Novikov I.Ya. On infinitely smooth compactly supported almost-wavelets. Math. Notes, 1994, vol. 56, no. 3, pp. 877–883. doi: 10.1007/BF02362405
5. Zakharov V.G. Reproducing solutions to PDEs by scaling functions. Int. J. Wavelets Multiresolut. Inf. Process., 2020, vol. 19, no. 2, art. no. 2050017. doi: 10.1142/S0219691320500174
6. Chernykh N.I., Subbotin Yu.N. Interpolating-orthogonal wavelet systems. Proc. Steklov Inst. Math., 2009, vol. 264, pp. 107–115. doi: 10.1134/S0081543809050083
Cite this article as: E.A. Pleshcheva. Interpolating orthogonal bases of n-separate MRAs and wavelets. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 4, pp. 154–163 .
