The purpose of this paper is to construct new types of wavelets for minimal splines on an irregular grid. The approach applied to construct spline-wavelet decompositions uses approximation relations as an initial structure for constructing the spaces of minimal splines. The advantages of this approach are the possibilities of using irregular grids and sufficiently arbitrary nonpolynomial spline-wavelets.
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References
A. A. Makarov, “On wavelet decomposition of spaces of first order splines,” Probl. Mat. Anal., 38, 47–60 (2008).
A. A. Makarov, “Algorithms of wavelet compression of linear spline spaces,” Vestn. St. Peterburg Univ., 1, No. 2, 41–51 (2012).
A. A. Makarov, “Algorithms for wavelet refinement of spaces of the first-order splines,” Tr. St.Peterburg Inst. Inf. Automat. RAN, 19, 203–220 (2011).
Yu. K. Dem’yanovich and I. D. Miroschnichenko, “Nested spline-wavelet decompositions,” Probl. Mat. Anal., 64, 51–61 (2012).
Yu. K. Dem’yanovich, “Spline-wavelets in the case of a single local coarsening of a grid,” Zap. Nauchn. Semin. POMI, 405, 97–118 (2012).
Yu. K. Dem’yanovich and A. S. Ponomarev, “Realization of the spline-wavelet decomposition of the first order,” Zap. Nauchn. Semin. POMI, 453, 33–73 (2016).
W. Sweldens, “The lifting scheme: A custom-design construction of biorthogonal wavelets,” Appl. Comput. Harm. Anal. 3, No. 2, 186–200 (1996).
B. M. Shumilov, “Algorithms with splitting of the wavelet transform of the first-order splines on nonuniform grids,” Vychisl. Mat. Mat. Fiz., 56, No. 7, 1236–1247 (2016).
A. A. Makarov, “Construction of splines of maximal smoothness,” Probl. Mat. Anal., 60, 25–38 (2011).
A. A. Makarov, “On one algebraic identity in the theory of B φ-splines of degree two,” Vestn. St.Peterburg Univ., 1, No. 1, 96–98 (2007).
E. Stolnitz, T. D. DeRose, and D. H. Salezin, Wavelets for Computer Grapfics [Russian translation], Izhevsk (2002).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 463, 2017, pp. 277–293.
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Makarov, A.A. On Two Algorithms of Wavelet Decomposition for Spaces of Linear Splines. J Math Sci 232, 926–937 (2018). https://doi.org/10.1007/s10958-018-3920-z
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DOI: https://doi.org/10.1007/s10958-018-3920-z