Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Spectral factorization of 2-block Toeplitz matrices and refinement equations


Author: V. Yu. Protasov
Translated by: the author
Original publication: Algebra i Analiz, tom 18 (2006), nomer 4.
Journal: St. Petersburg Math. J. 18 (2007), 607-646
MSC (2000): Primary 39B22, 15A23, 26C10, 26A30
DOI: https://doi.org/10.1090/S1061-0022-07-00963-6
Published electronically: May 30, 2007
MathSciNet review: 2262586
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Pairs of 2-block Toeplitz $ (N\times N)$-matrices $ (T_s)_{ij} = p_{2i - j +s -1}$, $ s = 0, 1$, $ i,j \in \lbrace 1, \ldots, N\rbrace $, are considered for arbitrary sequences of complex coefficients $ p_0,\ldots,p_N$. A complete spectral resolution of the matrices $ T_0$, $ T_1$ in the system of their common invariant subspaces is obtained. A criterion of nondegeneracy and of irreducibility of these matrices is derived, and their kernels, root subspaces, and all common invariant subspaces are found explicitly. The results are applied to the study of refinement functional equations and also subdivision and cascade approximation algorithms. In particular, the well-known formula for the exponent of regularity of a refinable function is simplified. A factorization theorem that represents solutions of refinement equations by certain convolutions is obtained, along with a characterization of the manifold of smooth refinable functions. The problem of continuity of solutions of the refinement equations with respect to their coefficients is solved. A criterion of convergence of the corresponding cascade algorithms is obtained, and the rate of convergence is computed.


References [Enhancements On Off] (What's this?)

  • 1. G. Deslauriers and S. Dubuc, Symmetric iterative interpolation processes. Fractal approximation, Constr. Approx. 5 (1989), 49-68. MR 982724 (90c:65016)
  • 2. N. Dyn, J. A. Gregory, and D. Levin, Analysis of uniform binary subdivision schemes for curve design, Constr. Approx. 7 (1991), 127-147. MR 1101059 (92d:65027)
  • 3. A. S. Cavaretta, W. Dahmen, and C. A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93 (1991), no. 453, 186 pp. MR 1079033 (92h:65017)
  • 4. I. Daubechies and 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)
  • 5. D. Colella and C. Heil, Characterization of scaling functions. I. Continuous solutions, SIAM J. Matrix Anal. Appl. 15 (1994), 496-518. MR 1266600 (95f:26004)
  • 6. G. Strang, Eigenvalues of Toeplitz matrices with $ 1\times2$ blocks, Z. Angew. Math. Mech. 76 (1996), 37-39.
  • 7. C. A. Cabrelli, C. Heil, and U. M. Molter, Polynomial reproduction by refinable functions, Advances in Wavelets (Hong Kong, 1997), Springer, Singapore, 1999, pp. 121-161. MR 1688767 (2000g:42040)
  • 8. T. N. T. Goodman, C. A. Micchelli, G. Rodriguez, and S. Seatzu, Spectral factorization of Laurent polynomials, Adv. Comput. Math. 7 (1997), no. 4, 429-454. MR 1470294 (98e:65117)
  • 9. L. Berg and G. Plonka, Spectral properties of two-slanted matrices, Results Math. 35 (1999), no. 3-4, 201-215. MR 1694902 (2000c:15009)
  • 10. -, Some notes on two-scale difference equations, Functional Equations and Inequalities, Math. Appl., vol. 518, Kluwer Acad. Publ., Dordrecht, 2000, pp. 7-29. MR 1792070 (2002f:39049)
  • 11. V. Protasov, Refinement equations with nonnegative coefficients, J. Fourier Anal. Appl. 6 (2000), no. 1, 55-78. MR 1756136 (2001i:42008)
  • 12. -, Asymptotics of the partition function, Mat. Sb. 191 (2000), no. 3, 65-98; English transl., Sb. Math. 191 (2000), no. 3-4, 381-414. MR 1773255 (2001h:11134)
  • 13. -, Fractal curves and wavelets, Izv. Ross. Akad. Nauk Ser. Mat. 70 (2006), no. 5, 123-162. (Russian) MR 2269711
  • 14. I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 61, SIAM, Philadelphia, 1992. MR 1162107 (93e:42045)
  • 15. A. I. Kostrikin and Yu. I. Manin, Linear algebra and geometry, ``Nauka'', Moscow, 1986; English transl., Gordon and Breach Sci. Publ., New York, 1989. MR 847051 (87f:00006); MR 1057342 (91h:00008)
  • 16. W. Lawton, S. L. Lee, and Z. Shen, Characterization of compactly supported refinable splines, Adv. Comput. Math. 3 (1995), no. 1-2, 137-145. MR 1314906 (95m:41020)
  • 17. V. Yu. Protasov, Piecewise-smooth refinable functions, Algebra i Analiz 16 (2004), no. 5, 101-123; English transl., St. Petersburg Math. J. 16 (2005), no. 5, 821-835. MR 2106669 (2005g:41023)
  • 18. C. A. Micchelli and H. Prautzsch, Uniform refinement of curves, Linear Algebra Appl. 114/115 (1989), 841-870. MR 986909 (90k:65088)
  • 19. K. S. Lau and J. Wang, Characterization of $ L_p$-solutions for two-scale dilation equations, SIAM J. Math. Anal. 26 (1995), 1018-1046. MR 1338372 (96f:39004)
  • 20. G. Gripenberg, Computing the joint spectral radius, Linear Algebra Appl. 234 (1996), 43-60. MR 1368770 (97c:15043)
  • 21. V. Yu. Protasov, The joint spectral radius and invariant sets of linear operators, Fundam. Prikl. Mat. 2 (1996), no. 1, 205-231. (Russian) MR 1789006 (2002d:47006)
  • 22. -, A generalized joint spectral radius. A geometric approach, Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), no. 5, 99-136; English transl., Izv. Math. 61 (1997), no. 5, 995-1030. MR 1486700 (99c:15041)
  • 23. D. X. Zhou, The p-norm joint spectral radius for even integers, Methods Appl. Anal. 5 (1998), 39-54. MR 1631335 (99e:42054)
  • 24. V. Protasov, A complete solution characterizing smooth refinable functions, SIAM J. Math. Anal. 31 (2000), no. 6, 1332-1350. MR 1766557 (2001j:42033)
  • 25. I. Ya. Novikov, V. Yu. Protasov, and M. A. Skopina, Theory of wavelets, ``Fizmatlit'', Moscow, 2006, 612 pp. (Russian)
  • 26. M. Neamtu, Convergence of subdivision versus solvability of refinement equations, East J. Approx. 5 (1999), no. 2, 183-210. MR 1705396 (2000j:42054)
  • 27. R. Q. Jia, Subdivision schemes in $ L_p$ spaces, Adv. Comput. Math. 3 (1995), no. 4, 309-341. MR 1339166 (96d:65028)
  • 28. Z. Wu, Convergence of subdivision schemes in $ L\sb p$ spaces, Appl. Math. J. Chinese Univ. Ser. B 16 (2001), no. 2, 171-177. MR 1833112 (2002c:42059)
  • 29. D. Chen and M. Han, Convergence of cascade algorithm for individual initial function and arbitrary refinement masks, Sci. China Ser. A 48 (2005), no. 3, 350-359. MR 2158275 (2006b:42050)
  • 30. A. A. Melkman, Subdivision schemes with nonnegative masks converge always--unless they obviously cannot?, Ann. Numer. Math. 4 (1997), no. 1-4, 451-460. MR 1422696 (97i:41014)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 39B22, 15A23, 26C10, 26A30

Retrieve articles in all journals with MSC (2000): 39B22, 15A23, 26C10, 26A30


Additional Information

V. Yu. Protasov
Affiliation: Department of Mechanics and Mathematics, Moscow State University, 119992, Moscow, Russia
Email: vladimir_protassov@yahoo.com

DOI: https://doi.org/10.1090/S1061-0022-07-00963-6
Keywords: Matrix factorization, spectrum, polynomial, cyclic trees, refinement equation, approximation algorithms, wavelets
Received by editor(s): April 10, 2006
Published electronically: May 30, 2007
Additional Notes: Supported by RFBR (grant no. 05-01-00066) and by Leading Scientific Schools (grant no. 5813.2006.1)
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society