72942221-d59f-41bf-bc8f-8626e8cc565d20210224245227086wseamdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON INFORMATION SCIENCE AND APPLICATIONS2224-34021790-083210.37394/23209http://wseas.org/wseas/cms.action?id=40462420202420201710.37394/23209.2020.17http://wseas.org/wseas/cms.action?id=23178On Five-Diagonal Splitting for Cubic Spline Wavelets with Six Vanishing Moments on a SegmentBorisShumilovDepartment of Applied Mathematics Tomsk State University of Architecture and Building 2 Solyanaya, Tomsk, 634003 RussiaIn this study, we use the vanishing property of the first six moments for constructing a splitting algorithm for cubic spline wavelets. First, we construct the corresponding wavelet space that satisfies the orthogonality conditions for all fifth-degree polynomials. Then, using the homogeneous Dirichlet boundary conditions, we adapt spaces to the closed interval. The originality of the study consists in obtaining implicit relations connecting the coefficients of the spline decomposition at the initial scale with the spline coefficients and wavelet coefficients at the nested scale by a tape system of linear algebraic equations with a non-degenerate matrix. After excluding the even rows of the system, in contrast to the case with two zero moments, the resulting transformation matrix has five (instead of three) diagonals. The results of numerical experiments on calculating the derivatives of a discrete function are presented.22420212242021156165https://www.wseas.org/multimedia/journals/information/2020/a385109-072.pdf10.37394/23209.2020.17.19https://www.wseas.org/multimedia/journals/information/2020/a385109-072.pdfC.K. Chui, An Introduction to Wavelets, Academic Press, New York, London,1992. E.J. Stollnitz, T.D. DeRose, D.H. Salesin, Wavelets for Computer Graphics, Morgan Kaufmann Publishers, San Francisco, 1996. M.W. Frazier, An Introduction to Wavelets through Linear Algebra, Springer-Verlag, New York, 1999. 10.1017/cbo9780511569616.007M. Lyche, K. Mǿrken, E. Quak, Theory and algorithms for non-uniform spline wavelets, Multivariate Approximation and Applications, eds. N. Dyn, D. Leviatan, D. Levin, A. Pinkus, Cambridge University Press, Cambridge, 2001, pp. 152-187. 10.1002/cpa.3160450502A. Cohen, I. Daubechies, J.C. Feauveau, Biorthogonal bases of compactly supported wavelets, Communications on Pure and Applied Mathematics, Vol.45, 1992, pp. 485-560. 10.1006/acha.1996.0013J. Wang, Cubic spline wavelet bases of sobolev spaces and multilevel interpolation, Applied and Computational Harmonic Analysis, Vol.3, No.2, 1996, pp. 154-163. 10.1016/s0955-7997(01)00036-4K. Koro, K. Abe, Non-orthogonal spline wavelets for boundary element analysis, Engineering Analysis with Boundary Elements, Vol.25, 2001, pp. 149-164. 10.1142/s0219691318500613D. Ĉerná, Cubic spline wavelets with four vanishing moments on the interval and their applications to option pricing under Kou mode, International Journal of Wavelets, Multiresolution and Information Processing, Vol.17, No.1, 2019. 1850061 10.37394/23202.2020.19.20B.M. Shumilov, Shifted cubic spline wavelets with two vanishing moments on the interval and a splitting algorithm, WSEAS Transactions on Systems, Vol.19, 2020, pp. 149-158. C. De Boor, A Practical Guide to Splines, Applied Mathematical Sciences, Vol.27, Springer- Verlag, New York, 1978. 10.1109/icmt.2011.6002586B.M. Shumilov, S.M. Matanov, Supercompact cubic multiwavelets and algorithm with splitting, 2011 International Conference on Multimedia Technology, 2011, pp. 2636-2639. 10.1134/s1995423913030087B.M. Shumilov, Cubic multiwavelets orthogonal to polynomials and a splitting algorithm, Numerical Analysis and Applications, Vol.6, No.3, 2013, pp. 247-259. S. Pissanetzky, Sparse Matrix Technology, Academic Press, London, 1984. A.A. Samarskii, E.S. Nikolaev, Numerical Methods for Grid Equations, Vol. I Direct Methods, Birkhauser, Basel, 1989. 10.1016/j.amc.2012.08.027D. Ĉerná, V. Finêk, Cubic spline wavelets with complementary boundary conditions, Applied Mathematics and Computation, Vol.219, 2012, pp. 1853-1865. 10.1016/j.amc.2014.05.065D. Ĉerná, V. Finêk, Cubic spline wavelets with short support for fourth-order problems, Applied Mathematics and Computation, Vol.243, 2014, pp. 44-56. 10.1134/s0965542517100128Z.M. Sulaimanov, B.M. Shumilov, A splitting algorithm for the wavelet transform of cubic splines on a nonuniform grid, Computational Mathematics and Mathematical Physics, Vol.57, No.10, 2017, pp. 1577-1591. B.M. Shumilov, Semi-orthogonal splinewavelets with derivatives and the algorithm with splitting, Numerical Analysis and Applications, Vol.10, No. 1, 2017, pp. 90-100.