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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, San Diego, CA (1999).
F. Dubeau, S. Elmejdani, and R. Ksantini, “Non-uniform Haar wavelets,” Appl. Math. Comput. 159, 675–691 (2004).
I. Ya. Novikov, V. Yu. Protasov, and M. A. Skopina, Wavelet Theory, Am. Math. Soc., Providence, RI (2011).
A. Haar, “Zur Theorie der orthogonalen Funktionensysteme,” Math. Ann. 69, 331–371 (1910).
Yu. K. Dem’yanovich, Theory of Spline-Wavelets [in Russian], St. Petersbg. Univ. Press, St. Petersburg (2013).
Yu. K. Dem’yanovich and A. Yu. Ponomareva, “Adaptive spline-wavelet processing of a discrete flow,” J. Math. Sci., New York 210, No. 4, 371-390 (2015).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Problemy Matematicheskogo Analiza 106, 2020, pp. 55-71.
Rights and permissions
About this article
Cite this article
Dem’yanovich, Y.K. Wavelets in Generalized Haar Spaces. J Math Sci 251, 615–634 (2020). https://doi.org/10.1007/s10958-020-05120-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-020-05120-5