AUTHORS: Yu. K. Dem’yanovich

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ABSTRACT: Adaptive algorithms of spline-wavelet decomposition in a linear space over metrized fields are proposed. The algorithms provide a priori given estimate of the deviation of the main flow from the initial one. Comparative estimates of data of the main flow under different characteristics of the irregularity of the initial flow are done. The limiting characteristics of data, when the initial flow is generated by abstract differentiable functions, are discussed. The constructions of adaptive grid and pseudo-equidistant grid and relative quantity of their knots are considered, flows of elements of linear normed spaces and formulas of decomposition and reconstruction are discussed. Wavelet decomposition of the flows is obtained with using of spline-wavelet decomposition. Sufficient condition of the construction is obtained. Applications to different spaces of matrix of fixed order and to spaces of infinite-dimension vectors with numerical elements (rational, real, complex and p-adic elements) are considered

KEYWORDS: signal processing, matrix flows, adaptive spline-wavelets, general flows, p-adic flows

REFERENCES:

[1] Richard E.Blahut, Fast Algorithms for Digital Signal Processing, Addison-Wesley Publishing Company, California. (1989).

[2] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press (1999).

[3] I.Ya.Novikov, V.Yu.Protasov, M.A.Skopina, Wavelet Theory. AMS, Translations Mathematical Monographs, V. 239 (2011).

[4] V. Wickerhauser, Adapted Wavelet Analysis from Theory to Software, AK Peters, Boston, 1994, pp. 213-214, 237, 273-274, 387.

[5] M.A. Cody, The Wavelet Packet Transform, Dr. Dobb’s Journal, Vol 19, Apr. 1994, pp. 44-46, 50-54.

[6] Marie Farge, Kai Schneider, Analysing and compressing turbulent fields with wavelets, Note IPSL, 20 (2002).

[7] S.Albeverio, S.Evdokimov, M.Skopina, p-adic multiresolution analysis and wavelet frames,J. Fourier Anal. Appl. 16, pp. 693-714 (2010).

[8] I.DaubechiesI.Guskov, W.Sweldens, Commutation for Irregular Subdivision//Const. Approx., 17(4), 2001, pp. 479-514.

[9] A.Aldrobi, C.Cabrelli, U.Molter, Wavelets on irregular grids with arbitrary dilation matrices, and frame atoms for L 2 (Rd )/ Preprint. Date: March 30, 2004.

[10] C.K.Chu, W.He. Compactly Supported tight frames associated with refinable functions//Appl. and Comp. Harm. Anal.11, 2001,pp. 313 – 327.

[11] J.A.Packer, M.A.Rieffet, Vavelet filter functions, Matrix completion problem, and projective modules over C(T n )// J.Fourier. Anal. Appl. 9, 2003,pp. 101 – 116.

[12] V. Protasov, Refinement equations with nonnegative coefficients// J. Fourier Anal. Appl., 6, 2000, pp.55 – 77.

[13] V. Protasov, A complete solution characterizing smooth refinable functons, SIAM J. of Math. Anal., 31, 2000, pp. 1332 – 1350.

[14] I.M.Johnstone. Function estimation and wavelets. Lecture Notes, Dept. of Statistics, Stanford University (1999).

[15] S.Mallat and W.L. Ywang. Singularity detection and processing with wavelets, IEEE Trans. Info. Theory, 38, 1992, pp. 617-643.

[16] W.Sweldens. The lifting scheme: accustomdesign construction of biorthogonal wavelets, J. of Appl. and Comput. Harmonic Analysis, 3, 1996, pp. 186 – 200.

[17] M. Vetterli and C. Herley, Wavelets and Filter Banks: Theory and Design, IEEE Transactions on Signal Processing, 40, 1992, pp. 2207-2232.

[18] I. Daubechies, Orthonormal Bases of Compactly Supported Wavelets, Comm. Pure Appl. Math., 41, 1988, pp. 906-966.

[19] Yu.K.Dem’yanovich, A.Yu.Ponomareva, Adaptive Spline-Wavelet Processing of a Discrete Flow, J. of Math. Sci. 210, 2015, pp. 371–390.

[20] Yu.K.Dem’yanovich, A.S.Ponomarev, On Realization of the Spline-Wavelet Decomposition of the First Order, J. of Math. Sci. 224, 2017, pp. 833–860.

[21] Yu.K.Dem’yanovich, On Embedding and Extended Smoothness of Spline Spaces, Far East Journal of Mathematical Sciences (FJMS) 102,2017, pp. 2025–2052.

[22] Yu.K.Dem’yanovich, Theory of SplineWavelets, Printed in Pulishing House of Saint-Petersburg State University, (2013) (in Russian).

[23] Albeverio, S., Evdokimov, S. and Skopina, M., “p-adic multiresolution analysis and wavelet frames”,J. Fourier Anal. Appl. 16, No. 5, 693714 (2010).

[24] Yu.K.Dem’yanovich, A.Yu.Ponomareva, “Adaptive Spline-Wavelet Processing of a Discrete Flow”, J. of Math. Sci. 210, No.4, 371–390 (2015).

[25] Yu.K.Dem’yanovich, A.S.Ponomarev, “On Realization of the Spline-Wavelet Decomposition of the First Order”, J. of Math. Sci. 224, No.6, 833–860 (2017).

[26] Yu.K.Dem’yanovich, “On Embedding and Extended Smoothness of Spline Spaces”, Far East Journal of Mathematical Sciences (FJMS) 102, No. 9, 2025-2052 (2017).

[27] Xumin Liu, Zilong Duan, Xiaojun Wang and Weixiang Xu “An Image Edge Detection Algorithm Based on Improved Wavelet Transform”, International Journal of Signal Processing, Image Processing and Pattern Recognition 9, No.4, pp.435-442 (2016).

[28] Shutao Li Qinghua Shen Jun Sun “Skew detection using wavelet decomposition and projection profile analysis”, Journal Pattern Recognition Letters archive 28, No. 5, 555-562 (2007).

[29] M. Bayazit, B. Onoz & H. Aksoy “Nonparametric streamflow simulation by wavelet or Fourier analysis”, Hydrological Sciences Journal 46, No. 4, 623-634 (2006).