Mathematics of Computation

Author(s): Victor D. Didenko; Bernd Silbermann.
Journal: Math. Comp. 78 (2009), 1435-1466.
MSC (2000): Primary 65T60; Secondary 42C40, 39B32
Posted: March 10, 2009
Retrieve article in: PDF

Abstract: The solvability and Fredholm properties of refinement equations in spaces of square-integrable functions are studied. Necessary and jointly necessary and sufficient conditions for the solvability of homogeneous and non-homogeneous refinement equations are established. It is shown that in the space $L_2(\mathbb{R})$ the kernel space of any homogeneous equation with a non-trivial solution is infinite dimensional. Moreover, the solvability problem is reduced to the study of singular values of certain matrix sequences. These sequences arise from Galerkin approximations of auxiliary linear operators. The corresponding constructions use only the coefficients of refinement equations that generate multiresolution analysis, and the coefficients of the refinement equation studied. For the equations with polynomial symbols the most complete results are obtained if the corresponding operator is considered on an appropriate subspace of the space $L_2(\mathbb{R})$ .

1.: A.S. Cavaretta, W. Damen, C.A. Micchelli, Stationary subdivisions, Mem. Amer. Math. Soc., 93(1991), 1-186. MR 1079033 (92h:65017)
2.: C.K. Chui, Wavelets: A Mathematical Tool for Signal Processing, SIAM, 1997. MR 1443204 (99b:42012)
3.: A. Cohen, I. Daubechies, A new technique to estimate the regularity of refinable functions, Rev. Mat. Iberoamericana, 12(1996), 527-591. MR 1402677 (97g:42025)
4.: A. Cohen, K. Gröchenig, L.F. Villemoes, Regularity of multivariate refinable functions, Const. Approx., 15(1999), 241-255. MR 1668921 (2000b:42028)
5.: C.K. Chui, An Introduction to Wavelets, Academic Press, 1992. MR 1150048 (93f:42055)
6.: D. Colella, C. Heil, Characterization of scaling functions: Continuous solutions, SIAM J. Matrix Anal., 15(1994), 496-518. MR 1266600 (95f:26004)
7.: D. Colella, C. Heil, Matrix Refinement Equation: Existence and Uniqueness, J. Fourier Anal. Appl., 2(1996), 363-377. MR 1395770 (97k:39021)
8.: I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 41(1988), 909-996. MR 951745 (90m:42039)
9.: I. Daubechies, Ten Lectures on wavelets, SIAM, 1992. MR 1162107 (93e:42045)
10.: I. Daubechies, J. Lagarias, Two-scale difference equations, I. Existence and global regularity of solutions, SIAM J. Math. Anal., 22(1991), 388-1410. MR 1112515 (92d:39001)
11.: I. Daubechies, J. Lagarias, Two-scale difference equations, II. Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal., 23(1992), 1031-1079. MR 1166574 (93g:39001)
12.: W. Dahmen, C.A. Micchelli, Subdivision algorithms for the generation of box spline surfaces, Comput. Aided Geom. Des., 1(1984), 115-129.
13.: V.D. Didenko, A.A. Korenovsky, S.L. Lee, On the spectral radius of convolution dilation operators, Z. Anal. Anwendungen, 21(2002), 879-890. MR 1957302 (2004a:39049)
14.: T.B. Dinsenbacher, D.P. Hardin, Nonhomogeneous refinement equations, A. Aldroubi, E.-B. Lin (eds), Wavelets, multiwavelets, and their applications, San Diego, CA, 1997, Contemp. Math. Series, 1998, pp. 117-127. MR 1614717 (99a:39055)
15.: I. Gohberg, N. Feldman, Convolution equations and projection methods for their solutions, Akademie-Verlag, Berlin, 1974.
16.: I. Gohberg, N.Ya. Krupnik, Introduction to the theory of one-dimensional singular integral operators, Birkhäuser, 1992. MR 545507 (81d:45010)
17.: I. Gohberg, M. Krein, Introduction to the theory of linear nonselfadjoint operators, Nauka, Moscow, 1965. English translation in Translations of Mathematical Monographs, Vol. 18, Amer. Math. Soc., 1969. MR 0246142 (39:7447)
18.: R. Hagen, S. Roch, B. Silbermann, -Algebras and Numerical Analysis, Marcel Dekker. Inc., New York, 2001. MR 1792428 (2002g:46133)
19.: B. Han, Solutions in Sobolev spaces of vector refinement equations with general dilation matrix, Adv. Comp. Math., 24(2006), 375-403. MR 2222276 (2007a:42073)
20.: R.Q. Jia, K.S. Lau, D.X. Zhou, -solutions of refinement eqiuations, J. Fourier Anal. Appl., 7(2001), 143-167. MR 1817673 (2002i:42049)
21.: K.S. Lau, J.R. Wang, Characterization of solutions for the two-scale dilation equations, SIAM J. Math. Anal., 26(1995), 1018-1046. MR 1338372 (96f:39004)
22.: S. Mallat, Multiresolution approximation and wavelet orthonormal bases of , Trans. Amer. Math. Soc., 315(1989), 69-88. MR 1008470 (90e:42046)
23.: D. Malone, Solutions to Dilation Equations, Ph.D. Thesis, University of Dublin, 2000.
24.: D. Malone, $L^2(\mathbb{R})$ solutions of dilation equations and Fourier-like transforms, J. Fourier Anal. Appl., 8(2002), 309-317. MR 1906255 (2003d:42012)
25.: C.A. Micchelli, H. Prautzsch, Uniform refinement of curves, Linear Algebra Appl., 114/115(1989), 841-870. MR 986909 (90k:65088)
26.: B. Silbermann, Modified finite sections for Toeplitz operators and their singular values, SIAM J. Matrix. Anal. Appl., 24(2003), 678-692. MR 1972674 (2004f:47037)
27.: B. Silbermann, Fredholm theory and numerical linear algebra, Operator Theory Advances and Applications, 160(2005), 403-411. MR 2191100 (2006h:47016)
28.: G. Strang, D.-X. Zhou, Inhomogeneous refinement equations, J. Fourier Anal. Appl., 4(1998), 733-744. MR 1666013 (99m:42056)
29.: L.F. Villemoes, Energy moments in time and frequency for two-scale difference equation solutions and wavelets, SIAM J. Math. Anal., 23(1992), 1519-1543. MR 1185640 (94c:39002)

Retrieve articles in Mathematics of Computation with MSC (2000): 65T60, 42C40, 39B32

Victor D. Didenko
Affiliation: Department of Mathematics, Universiti Brunei Darussalam, Bandar Seri Begawan, BE1410, Brunei
Email: diviol@gmail.com

Bernd Silbermann
Affiliation: Faculty of Mathematics, University of Technology Chemnitz, 09107 Chemnitz, Germany
Email: Bernd.Silbermann@mathematik.tu-chemnitz.de

DOI: 10.1090/S0025-5718-09-02232-7
PII: S 0025-5718(09)02232-7
Keywords: Refinement equation, solvability, singular values, splitting
Received by editor(s): November 16, 2007
Received by editor(s) in revised form: August 18, 2008
Posted: March 10, 2009
Additional Notes: The first author was supported in part by Universiti Brunei Darussalam Grants PNC2/2/RG/1(66) and PNC2/2/RG/1(72)
Copyright of article: Copyright 2009, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS