Wavelets in Generalized Haar Spaces
- PDF / 265,459 Bytes
- 20 Pages / 594 x 792 pts Page_size
- 55 Downloads / 154 Views
Journal of Mathematical Sciences, Vol. 251, No. 5, December, 2020
WAVELETS IN GENERALIZED HAAR SPACES Yu. K. Dem’yanovich St. Petersburg State University 28, Universitetskii pr., Petrodvorets, St. Petersburg 198504, Russia [email protected]
UDC 517.9
We consider wavelet decompositions of Haar type spaces on arbitrary nonuniform grids by methods of the nonclassical theory of wavelets. The number of nodes of the original (nonuniform) grid can be arbitrary, and the main grid can be any subset of the original one. We proposie decomposition algorithms that take into account the character of changes in the original numerical flow. The number of arithmetical operations is proportional to the length of the original flow, and successive real-time processing is possible for the original flow. We propose simple decomposition and reconstruction algorithms leading to formulas where the coefficients are independent of the grid and are equal to 1 in absolute value. Bibliography: 6 titles.
The numerical flows are often associated with nonuniform grids. Fragments of rapid changes in the numerical flow are associated with a fine grid, whereas the grid is coarsened for fragments of slow changes in the flow. In the wavelet decomposition of the original flow, the main and refined (wavelet) flows appear. The main flow serves as an approximation of the original flow and is obtained by thinning the original one. In the classical approach [1]–[3], the standard thinning is used (usually, by removing the components with odd numbers) so that it is difficult to obtain a qualitative approximation of the original flow. Within the framework of the classical approach, wavelets on a nonuniform grid were considered [2] for the Haar functions [4]. In particular, a nonuniform original grid was considered in [2] with M = 2s components. Coarsening was obtained by removing odd-numbering components. The number of nodes of the coarsened grid turned out to be equal to K = 2s−1 . Under this approach, the coarsening of the grid is not related to the properties of the original flow, so there is no reason to hope that the main flow will be a qualitative approximation of the original flow. The goal of this paper is to consider the wavelet decompositions of Haar type spaces on arbitrary nonuniform grids. Owing to the nonclassical approach [5, 6], we can consider the case where the number of nodes of the original (nonuniform) grid can be arbitrary, whereas the main grid may be any subset of the original one. To apply this approach, we need an algorithm for coarsening the original grid with taking into account the character of changes in the original flow. If no additional information concerning the original flow of length M is known, the
Translated from Problemy Matematicheskogo Analiza 106, 2020, pp. 55-71. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2515-0615
615
number of required arithmetical operations is proportional to M 2 . For convex flows there exist algorithms where the order of the number of operations is proportional to M log2 M (however, such alg
Data Loading...
