细分树的L_p范数或拟范数平均大小和小波包的渐近性态
|
摘要
本文研究了一般整数扩张矩阵向量细分方程所生成的细分树，亦即，当整数扩张矩阵M是一个满足lim_(n→∞)M~(-n)=0时的s×s整数扩张矩阵的可细分函数产生的细分树的L_p范数或拟范数范数平均大小估计，从而可以得到一般整数扩张矩阵向量细分方程所生成的小波包L_p平均大小公式。需要指出的是，我们得到的定理不仅对于1≤p≤∞时成立，同时也适用于0细分树的平均大小L_p估计从欧氏空间R~s推广到非交换的Heisenberg群上，建立Heisenberg群H~s上向量细分方程的细分树L_p范数估计，从而延拓了[22，23]中的相关结果。
The purpose of this dissertation is to investigate the subdivision tree of the general vector refinement equation whose integer expanding matrix M satisfying lim_(n→8) M~(-n) = 0. We establish the L_p norm or quasi-norm estimatesof the subdivision tree. On the basis of the L_p norm or quasinorm estimates of the subdivision tree, we obtain the L_p norm mean size formula of general (vector)wavelet packets. The L_p estimates is not only true for 1≤p≤∞, but also for 0 < p < 1. We remark here that there are very few discussions on L_p estimates for 0 < p < 1 in the literatures. We apply our conclusions established in above to multivariate quincunx biorthogonal wavelet packets and multiple biorthogonal wavelet packets in L_p. We obtain a asymptotic formula of the mean size for multivariate quincunx biorthogonalwavelet packets. Similarily, we also get a asymptotic mean size formula for multiple biorthogonal wavelet packets in L_p. These two concrete formulascontain important information for these two kinds of wavelet packets. Some results in [40,41,56] are special cases of our theorems. Furthermore, we generalize the L_p norm of the subdivision tree from Euclid space R~s to the noncommutative Heisenberg group H~s. We get the L_p norm estimates of the subdivision tree on Heisenberg group H~s, which extend the corresponding conclusions in [22,23].
The purpose of this dissertation is to investigate the subdivision tree of the general vector refinement equation whose integer expanding matrix M satisfying lim_(n→8) M~(-n) = 0. We establish the L_p norm or quasi-norm estimatesof the subdivision tree. On the basis of the L_p norm or quasinorm estimates of the subdivision tree, we obtain the L_p norm mean size formula of general (vector)wavelet packets. The L_p estimates is not only true for 1≤p≤∞, but also for 0 < p < 1. We remark here that there are very few discussions on L_p estimates for 0 < p < 1 in the literatures. We apply our conclusions established in above to multivariate quincunx biorthogonal wavelet packets and multiple biorthogonal wavelet packets in L_p. We obtain a asymptotic formula of the mean size for multivariate quincunx biorthogonalwavelet packets. Similarily, we also get a asymptotic mean size formula for multiple biorthogonal wavelet packets in L_p. These two concrete formulascontain important information for these two kinds of wavelet packets. Some results in [40,41,56] are special cases of our theorems. Furthermore, we generalize the L_p norm of the subdivision tree from Euclid space R~s to the noncommutative Heisenberg group H~s. We get the L_p norm estimates of the subdivision tree on Heisenberg group H~s, which extend the corresponding conclusions in [22,23].
引文
[1] I. Daubechies, Ten lectures on wavelets, SIAM, Philadelphia, 1992.
[2] S. Mallat, A wavelet tour of signal processing, Academic Press, San Diego, 1998.
[3] C. K. Chui, An introduction to wavelets, Academic Press, New York, 1992.
[4] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure. Appl. Math., 41, 909-996(1988).
[5] A. Cohen, I. Daubechies, and J. C. Feauveau, Biorthogonal bases of compactlysupported wavelets, Comm. Pure. Appl. Math., 45, 485-500(1992).
[6] A. Cohen and I. Daubechies, On the instability of arbitrary biorthogonal wavelet packets, SIAM J. Math Anal., 24, 1340-1354 (1993).
[7] A. S. Cavaretta, W. Dahmen, and C. A. Micchelli, Stationary subdivision,Memior Amer. Math. Soc, vol. 93, 1991.
[8] A. Haar, Zur Theorie der orthogonalen Funcktionen-Systeme, Math. Ann., 69, 331-371(1910).
[9] R. Coifman, Y. Meyer, and M. V. Wickerhauser, Size properties of wavelet packets,in :M.B.Ruskai.,et al.(Eds.), Wavelet and Applications, Jones and Bartlett, Boston, 453-470, 1992.
[10] R. Coifman, M. V. Wickerhauser, Entropy-based algorithms for best basis selection. IEEE Trans. Inform. Theorv.. 38: 713-718(1992).
[11] A. H. Fan, Moyenne de Localisation frequentille des paquets d'ondelettes, Rev. Mat. Iberoamericana.,14: 63-70(1998).
[12] C. K. Chui and C. Li, Nonorthogonal wavelet packets,SIAM J. Math Anal., 24, 712-738(1993).
[13] C. Heil, and D. Colella, Matrix refinement equations: existence and uniqueness, J. Fourier Anal. Appl., 2, 363-377(1996).
[14] C. Heil, and D. Colella, Characterizations of scaling functions; continuoussolutions, SIAM J. Matrix Anal. Appl., 15, 496-518(1994).
[15] C. C. Rota and G. Strang, A note on the joint spectral radius, Indag. Math., 22, 379-381(1960).
[16] Y. Wang, Two-scale dilation equations and mean spectral radius, RandomComput. Dynam., 4,49-72(1996).
[17] K. S. Lau and J. R. Wang, Characterization of IP- solutions for twoscaledilation equations, SIAM J. Math Anal., 26, 1018-1046(1995).
[18] Q. T. Jiang, P. Oswald, and S. D. Riemenschneider, (?)-subdivision schemes: maximal sum role order, Constr. Approx., 19, 437-463 (2003).
[19] Q. D. Yang, Multiresolution analysis on non-abelian locally campact groups, Ph.D. Thesis, Dept. of Math., Univ. of Saskatchewan, 1999.
[20] H. P. Liu, and L. Z. Peng, Admissible wavelets associated with the Heisenberq group. Pacific. J. Math.. 180: 101-123(1997).
[21] L. Z. Peng, Wavelets on the Heisenberg group, Geometry and Nonlinear Partial Differential Equations, AMS/IP Studies in Advanced Mathematics,29: 123-131(2002).
[22] H. P. Liu, and Y. Liu, Convergence of cascade sequence on the Heisenberggroup, Proceedings of the American Mathematical Societty, 134(5): 1413-1423(2005).
[23] H. P. Liu, Y. Liu, L. Z. Peng, X. Y. Chu, Cascade algorithm and multiresolutionanalysis on the Heisenberg group, Progress in Natural Science, 15(7): 602-608(2005).
[24] I. Pesenson, Sampling of Paley-Wiener functions on stratified groups, J. Fourier Anal. Appl., 4(3),271-281(1998).
[25] A. Mayeli, Shannon multiresolution analysis on the Heisenberg group , arXiv 0804.0245(April 2008).
[26] C. de Boor, R. DeVore, and A. Ron, Approximation from shift invariant subspaces of L_2(R~d), Trans. Amer. Math. Soc, 342(2): 787-806(1994).
[27] C. de Boor, R. DeVore, and A. Ron, The structure of finitely generatedshift invariant subspaces of L_2(R~d), J. Funct. Anal., 119(1): 37-78(1994).
[28] C. A. Micchelli and H. Prautzsch, Uniform refinement of curves, Linear Algebra and its Applications. 114/115. 841-870(1989).
[29] R. Q. Jia and C. A. Micchelli, Using the refinement equations for the construction of pre-wavelets II: Powers of two, in Curves and Surfaces (P. J. Laurent, A. Le Mehaute and L. L. Schumaker eds.), Academic Press, New York, 209-246(1991).
[30] R. Q. Jia and C. A. Micchelli, On linear independence of integer translatesof a finite number of functions, Proc. Edinburgh Math. Soc, 36, 69-85(1992).
[31] R. Q. Jia, Stability of the shifts of a finite number of functions, J. Approx.Theory., 95,149-202(1998).
[32] D. R. Chen, R. Q. Jia, and S. D. Riemenschneider, Convergence of vector subdivsion schemes in Sobolev spaces, Applied and Computational Harmonic Analysis., 12,128-149(2002).
[33] B. Han, and R. Q. Jia, Quincunx fundamental refinable functions and quincunx biorthogonal wavelets, Comp. Math., 17, 165-196(2002).
[34] R. Q. Jia, Subdivision schemes in L_p spaces, Advances in Comp. Math., 3, 309-341(1995).
[35] R. Q. Jia, S. D. Riemenschneider, and D. X. Zhou, Smoothness of multiplerefinable functions and multiple wavelets, SIAM J. Matrix Anal. Appl., 21, 1-28(1999).
[36] R. Q. Jia, K. S. Lau, and D. X. Zhou, L_p solutions of refinement equations.J. Fourier Anal. Appl.. 2. 143-169(2001).
[37] R. Q. Jia, S. D. Riemenschneider, and D. X. Zhou Vector subdivision schemes and multiple wavelets, Math. Comp., 67, 1533-1563(1998).
[38] B. Han, and R. Q. Jia, Multivariate refinement equations and convergenceof subdivision schemes, SIAM J. Math. Anal., 29, 1177-1199(1998).
[39] Q. T. Jiang, Multivariate matrix refinement functions with arbitrary matrix dilation, Tran. Amer. Math. Soc, 351, 2407-2438(1999).
[40] Z. Shen, Refinable function vectors, SIAM. Math. Anal., 29,235-250(1998).
[41] M. Nielsen, Size properties of wavelet packets, PhD Thesis, Washington University, Saint Louis, 1999.
[42] M. Nielsen and D. X. Zhou, Mean size of wavelet packets, Appl. Comput. Harmon. Anal., 13, 22-34(2002).
[43] D. X. Zhou, Norms concerning subdivision sequences and their applicationsin wavelets, Applied and Computational Harmonic Analysis, 11: 329-346(2001).
[44] D. X. Zhou, The p-norm joint spectral radius for even integers, Methods and Applications of Analysis, 5, 39-54(1998).
[45] R. Q. Jia, Shift-invariant spaces and and linear operator equations, Israel Journal of Mathematics.103.259-288(1998).
[46] S. Li, Convergence of cascade algorithms in Sobolev spaces associated with multivariate refinement equations, J. Math. Anal. Appl., 257(1): 154-169(2001).
[47] S. Li, Multivariate refinement equations and convergence of cascade algorithmsin L_p(0 < p < 1) spaces, Acta Math. Sin., (Engl. Ser.) 19(1): 97-106(2003).
[48] S. Li, Vector subdivsion schemes in (L_p(R~s))~r(1≤p≤, <∞)spaces, Sciencein China(Series A), 46(3): 364-375(2003).
[49] S. Li, Characterization of smoothness of multivariate refinable functions and convergence of cascade algorithms of nonhomogeneous refinement equations, Adv. Comput. Math.,20(4): 311-331(2004).
[50] S. Li, R. F. Hu, X. Q. Wang, L_p-solutions of vector refinement equations with general dilation matrix, Acta Math. Sin., (Engl. Ser.)22(1): 51-60(2006).
[51] S. Li, X. L. Zhou, Multivariate refinement equation with nonnegative masks, Science in China(Series A),49(4): 439-450(2006).
[52] S. Li, J. Xian, Biorthogonal multiple wavelets generated by vector refinementequation, Science in China(Series A),50(7): 1015-1025(2007).
[53] S. Li, J. J. Yang, Vector refinement equations with infinitely supported masks. J. Approx. Theory.. 148(2): 158-176(2007).
[54] A. Ron, Z. Shen, Affine systems in L_2(R~d): the analysis of the analysis operator, Journal of Functional Analysis, 148: 408-447(1997).
[55] I. Daubechies, B. Han. A. Ron, Z. Shen, Framelets:MRA-based constructionsof wavelet frames, Applied and Computationic Harmonic Analysis., 14: 1-46(2003).
[56] D. X. Zhou, The p-norm joint spectral radius and its applications in wavelet analysis , International conference on wavelet analysis and its applications, AMS/IP Studies in Advanced Mathematics, 29: 123-131(2002)
[57] A. Ron, Z. Shen, Weyl-Heisenberg frames and Riesz bases in L_2(R~d), Duke Mathematical Journal, 89: 237-282(1997).
[58] A. Ron, Z. Shen, Affine systems in L_2(R~d): dual systems, Journal of Fourier Analysis and Applications, 3: 617-637(1997).
[59] A. Ron, Z. Shen, Frames and stable bases for shift invariant subspaces of L_2(R~d), Canadian Journal of Mathematics, 473: 1051-1094(1995).
[2] S. Mallat, A wavelet tour of signal processing, Academic Press, San Diego, 1998.
[3] C. K. Chui, An introduction to wavelets, Academic Press, New York, 1992.
[4] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure. Appl. Math., 41, 909-996(1988).
[5] A. Cohen, I. Daubechies, and J. C. Feauveau, Biorthogonal bases of compactlysupported wavelets, Comm. Pure. Appl. Math., 45, 485-500(1992).
[6] A. Cohen and I. Daubechies, On the instability of arbitrary biorthogonal wavelet packets, SIAM J. Math Anal., 24, 1340-1354 (1993).
[7] A. S. Cavaretta, W. Dahmen, and C. A. Micchelli, Stationary subdivision,Memior Amer. Math. Soc, vol. 93, 1991.
[8] A. Haar, Zur Theorie der orthogonalen Funcktionen-Systeme, Math. Ann., 69, 331-371(1910).
[9] R. Coifman, Y. Meyer, and M. V. Wickerhauser, Size properties of wavelet packets,in :M.B.Ruskai.,et al.(Eds.), Wavelet and Applications, Jones and Bartlett, Boston, 453-470, 1992.
[10] R. Coifman, M. V. Wickerhauser, Entropy-based algorithms for best basis selection. IEEE Trans. Inform. Theorv.. 38: 713-718(1992).
[11] A. H. Fan, Moyenne de Localisation frequentille des paquets d'ondelettes, Rev. Mat. Iberoamericana.,14: 63-70(1998).
[12] C. K. Chui and C. Li, Nonorthogonal wavelet packets,SIAM J. Math Anal., 24, 712-738(1993).
[13] C. Heil, and D. Colella, Matrix refinement equations: existence and uniqueness, J. Fourier Anal. Appl., 2, 363-377(1996).
[14] C. Heil, and D. Colella, Characterizations of scaling functions; continuoussolutions, SIAM J. Matrix Anal. Appl., 15, 496-518(1994).
[15] C. C. Rota and G. Strang, A note on the joint spectral radius, Indag. Math., 22, 379-381(1960).
[16] Y. Wang, Two-scale dilation equations and mean spectral radius, RandomComput. Dynam., 4,49-72(1996).
[17] K. S. Lau and J. R. Wang, Characterization of IP- solutions for twoscaledilation equations, SIAM J. Math Anal., 26, 1018-1046(1995).
[18] Q. T. Jiang, P. Oswald, and S. D. Riemenschneider, (?)-subdivision schemes: maximal sum role order, Constr. Approx., 19, 437-463 (2003).
[19] Q. D. Yang, Multiresolution analysis on non-abelian locally campact groups, Ph.D. Thesis, Dept. of Math., Univ. of Saskatchewan, 1999.
[20] H. P. Liu, and L. Z. Peng, Admissible wavelets associated with the Heisenberq group. Pacific. J. Math.. 180: 101-123(1997).
[21] L. Z. Peng, Wavelets on the Heisenberg group, Geometry and Nonlinear Partial Differential Equations, AMS/IP Studies in Advanced Mathematics,29: 123-131(2002).
[22] H. P. Liu, and Y. Liu, Convergence of cascade sequence on the Heisenberggroup, Proceedings of the American Mathematical Societty, 134(5): 1413-1423(2005).
[23] H. P. Liu, Y. Liu, L. Z. Peng, X. Y. Chu, Cascade algorithm and multiresolutionanalysis on the Heisenberg group, Progress in Natural Science, 15(7): 602-608(2005).
[24] I. Pesenson, Sampling of Paley-Wiener functions on stratified groups, J. Fourier Anal. Appl., 4(3),271-281(1998).
[25] A. Mayeli, Shannon multiresolution analysis on the Heisenberg group , arXiv 0804.0245(April 2008).
[26] C. de Boor, R. DeVore, and A. Ron, Approximation from shift invariant subspaces of L_2(R~d), Trans. Amer. Math. Soc, 342(2): 787-806(1994).
[27] C. de Boor, R. DeVore, and A. Ron, The structure of finitely generatedshift invariant subspaces of L_2(R~d), J. Funct. Anal., 119(1): 37-78(1994).
[28] C. A. Micchelli and H. Prautzsch, Uniform refinement of curves, Linear Algebra and its Applications. 114/115. 841-870(1989).
[29] R. Q. Jia and C. A. Micchelli, Using the refinement equations for the construction of pre-wavelets II: Powers of two, in Curves and Surfaces (P. J. Laurent, A. Le Mehaute and L. L. Schumaker eds.), Academic Press, New York, 209-246(1991).
[30] R. Q. Jia and C. A. Micchelli, On linear independence of integer translatesof a finite number of functions, Proc. Edinburgh Math. Soc, 36, 69-85(1992).
[31] R. Q. Jia, Stability of the shifts of a finite number of functions, J. Approx.Theory., 95,149-202(1998).
[32] D. R. Chen, R. Q. Jia, and S. D. Riemenschneider, Convergence of vector subdivsion schemes in Sobolev spaces, Applied and Computational Harmonic Analysis., 12,128-149(2002).
[33] B. Han, and R. Q. Jia, Quincunx fundamental refinable functions and quincunx biorthogonal wavelets, Comp. Math., 17, 165-196(2002).
[34] R. Q. Jia, Subdivision schemes in L_p spaces, Advances in Comp. Math., 3, 309-341(1995).
[35] R. Q. Jia, S. D. Riemenschneider, and D. X. Zhou, Smoothness of multiplerefinable functions and multiple wavelets, SIAM J. Matrix Anal. Appl., 21, 1-28(1999).
[36] R. Q. Jia, K. S. Lau, and D. X. Zhou, L_p solutions of refinement equations.J. Fourier Anal. Appl.. 2. 143-169(2001).
[37] R. Q. Jia, S. D. Riemenschneider, and D. X. Zhou Vector subdivision schemes and multiple wavelets, Math. Comp., 67, 1533-1563(1998).
[38] B. Han, and R. Q. Jia, Multivariate refinement equations and convergenceof subdivision schemes, SIAM J. Math. Anal., 29, 1177-1199(1998).
[39] Q. T. Jiang, Multivariate matrix refinement functions with arbitrary matrix dilation, Tran. Amer. Math. Soc, 351, 2407-2438(1999).
[40] Z. Shen, Refinable function vectors, SIAM. Math. Anal., 29,235-250(1998).
[41] M. Nielsen, Size properties of wavelet packets, PhD Thesis, Washington University, Saint Louis, 1999.
[42] M. Nielsen and D. X. Zhou, Mean size of wavelet packets, Appl. Comput. Harmon. Anal., 13, 22-34(2002).
[43] D. X. Zhou, Norms concerning subdivision sequences and their applicationsin wavelets, Applied and Computational Harmonic Analysis, 11: 329-346(2001).
[44] D. X. Zhou, The p-norm joint spectral radius for even integers, Methods and Applications of Analysis, 5, 39-54(1998).
[45] R. Q. Jia, Shift-invariant spaces and and linear operator equations, Israel Journal of Mathematics.103.259-288(1998).
[46] S. Li, Convergence of cascade algorithms in Sobolev spaces associated with multivariate refinement equations, J. Math. Anal. Appl., 257(1): 154-169(2001).
[47] S. Li, Multivariate refinement equations and convergence of cascade algorithmsin L_p(0 < p < 1) spaces, Acta Math. Sin., (Engl. Ser.) 19(1): 97-106(2003).
[48] S. Li, Vector subdivsion schemes in (L_p(R~s))~r(1≤p≤, <∞)spaces, Sciencein China(Series A), 46(3): 364-375(2003).
[49] S. Li, Characterization of smoothness of multivariate refinable functions and convergence of cascade algorithms of nonhomogeneous refinement equations, Adv. Comput. Math.,20(4): 311-331(2004).
[50] S. Li, R. F. Hu, X. Q. Wang, L_p-solutions of vector refinement equations with general dilation matrix, Acta Math. Sin., (Engl. Ser.)22(1): 51-60(2006).
[51] S. Li, X. L. Zhou, Multivariate refinement equation with nonnegative masks, Science in China(Series A),49(4): 439-450(2006).
[52] S. Li, J. Xian, Biorthogonal multiple wavelets generated by vector refinementequation, Science in China(Series A),50(7): 1015-1025(2007).
[53] S. Li, J. J. Yang, Vector refinement equations with infinitely supported masks. J. Approx. Theory.. 148(2): 158-176(2007).
[54] A. Ron, Z. Shen, Affine systems in L_2(R~d): the analysis of the analysis operator, Journal of Functional Analysis, 148: 408-447(1997).
[55] I. Daubechies, B. Han. A. Ron, Z. Shen, Framelets:MRA-based constructionsof wavelet frames, Applied and Computationic Harmonic Analysis., 14: 1-46(2003).
[56] D. X. Zhou, The p-norm joint spectral radius and its applications in wavelet analysis , International conference on wavelet analysis and its applications, AMS/IP Studies in Advanced Mathematics, 29: 123-131(2002)
[57] A. Ron, Z. Shen, Weyl-Heisenberg frames and Riesz bases in L_2(R~d), Duke Mathematical Journal, 89: 237-282(1997).
[58] A. Ron, Z. Shen, Affine systems in L_2(R~d): dual systems, Journal of Fourier Analysis and Applications, 3: 617-637(1997).
[59] A. Ron, Z. Shen, Frames and stable bases for shift invariant subspaces of L_2(R~d), Canadian Journal of Mathematics, 473: 1051-1094(1995).