Convergence Rates of Cascade Algorithms with Infinitely Supported Masks

Yang, Jianbin ; Li, Song

Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 424-434

Résumé

We investigate the solutions of refinement equations of the form $$\phi (x)\,=\,\sum\limits_{\alpha \in {{\mathbb{Z}}^{S}}}{a(\alpha )\,}\phi (Mx\,-\,\alpha ),$$ where the function $\phi $ is in ${{L}_{p}}({{\mathbb{R}}^{s}})(1\,\le \,p\,\le \,\infty )$ , $a$ is an infinitely supported sequence on ${{\mathbb{Z}}^{s}}$ called a refinement mask, and $M$ is an $s\,\times \,s$ integer matrix such that ${{\lim }_{n\to \infty }}\,{{M}^{-n}}\,=\,0$ . Associated with the mask $a$ and $M$ is a linear operator ${{\text{Q}}_{a,M}}$ defined on ${{L}_{p}}({{\mathbb{R}}^{s}})$ by ${{\text{Q}}_{a,M}}{{\phi }_{0}}\,:=\,{{\sum }_{\alpha \in {{\mathbb{Z}}^{s}}}}\,a(\alpha ){{\phi }_{0}}(M\,\cdot \,-\alpha )$ . Main results of this paper are related to the convergence rates of ${{(\text{Q}_{a,M}^{n}{{\phi }_{o}})}_{n=1,2,\ldots }}$ in ${{L}_{p}}({{\mathbb{R}}^{s}})$ with mask $a$ being infinitely supported. It is proved that under some appropriate conditions on the initial function ${{\phi }_{0}}$ , $\text{Q}_{a,M}^{n}{{\phi }_{0}}$ converges in ${{L}_{p}}({{\mathbb{R}}^{s}})$ with an exponential rate.

Détail

DOI : 10.4153/CMB-2011-081-6

Mots-clés : 39B12, 41A25, 42C40, refinement equations, infinitely supported mask, cascade algorithms, rates of convergence

Bibliographie

[1] [1] Aldroubi, A. and Gröchenig, K., Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Rev. 43(2001), no. 4, 585–620. Google Scholar | DOI

[2] [2] Cavaretta, A. S., Dahmen, W., and Micchelli, C. A., Stationary subdivision. Mem. Amer. Math. Soc. 93(1991), no. 453. Google Scholar

[3] [3] Chen, D.-R., Jia, R.-Q., and Riemenschneider, S. D., Convergence of vector subdivision schemes in Sobolev spaces. Appl. Comp. Harmon. Anal. 12(2002), no. 1, 128–149. Google Scholar | DOI

[4] [4] Cohen, A. and Ryan, R. D., Wavelets and multiscale signal processing. Applied Mathematics and Mathematical Computation, 11, Chapman & Hall, London, 1995. Google Scholar

[5] [5] Daubechies, I. and Huang, Y., A decay theorem for refinable functions. Appl. Math. Lett. 7(1994), no. 4, 1–4. Google Scholar | DOI

[6] [6] Han, B., The initial functions in a cascade algorithm. In: Wavelet analysis (Hong Kong, 2001), Ser. Anal. 1,World Sci. Publ., River Edge, NJ, 2002, pp. 154–178. Google Scholar

[7] [7] Han, B., Refinable functions and cascade algorithms in weighted spaces with Hölder continuous masks. SIAM J. Math. Anal. 41(2008), no. 1, 70–102. Google Scholar | DOI

[8] [8] Han, B. and Jia, R.-Q., Multivariate refinement equations and convergence of subdivision schemes. SIAM J. Math. Anal. 29(1998), no. 5, 1177–1199. Google Scholar | DOI

[9] [9] Han, B. and Jia, R.-Q., Characterization of Riesz bases of wavelets generated from multiresolution analysis. Appl. Comput. Harmon. Anal. 23(2007), no. 3, 321–345. Google Scholar | DOI

[10] [10] Han, B. and Shen, Z., Wavelets from the Loop scheme. J. Fourier Anal. Appl. 11(2005), no. 6, 615–637. Google Scholar | DOI

[11] [11] Han, B. and Shen, Z., Wavelets with short support. SIAM J. Math. Anal. 38(2006), no. 2, 530–556. Google Scholar | DOI

[12] [12] Herley, C. and Vetterli, M.,Wavelets and recursive filter banks. IEEE Trans. Signal Process. 41(1993), no. 8, 2536–2556. Google Scholar

[13] [13] Jia, R. Q., Subdivision schemes in Lp spaces. Adv. Comput. Math. 3(1995), no. 4, 309–341. Google Scholar | DOI

[14] [14] Jia, R. Q., Approximation properties of multivariate wavelets. Math. Comp. 67(1998), no. 222, 647–665. Google Scholar | DOI

[15] [15] Jia, R. Q., Convergence rates of cascade algorithms. Proc. Amer. Math. Soc. 131(2003), no. 6, 1739–1749. Google Scholar | DOI

[16] [16] Jia, R. Q., Approximation with scaled shift-invariant spaces by quasi-projection operators. J. Approx. Theory 131(2004), no. 1, 30–46. Google Scholar | DOI

[17] [17] Jia, R. Q. and Micchelli, C. A., Using the refinement equations for the construction of pre-wavelets. II. Powers of two. In: Curves and surfaces (Chamonix-Mont-Blanc, 1990), Academic Press, Boston, MA, 1991, pp. 209–246. Google Scholar

[18] [18] Lei, J., Jia, R. Q., and Cheney, E. W., Approximation from shift-invariant spaces by integral operators. SIAM. J. Math. Anal. 28(1997), no. 2, 481–498. Google Scholar | DOI

[19] [19] Li, S., Characterization of smoothness of multivariate refinable functions and convergence of cascade algorithms of nonhomogeneous refinement equations. Adv. Comput.Math. 20(2004), no. 4, 311–331. Google Scholar | DOI

[20] [20] Li, S., Convergence rates of vector cascade algorithms in L . J. Approx. Theory 137(2005), no. 1, 123–142. Google Scholar | DOI

[21] [21] Li, S. and Pan, Y., Subdivisions with infinitely supported mask. J. Comput. Appl. Math. 214(2008), no. 1, 288–303. Google Scholar | DOI

[22] [22] Li, S. and Pan, Y., Subdivision schemes with polynomially decaying masks. Adv. Comput. Math. 32(2010), no. 4, 487–507. Google Scholar | DOI

[23] [23] Li, S. and Yang, J., Vector refinement equations with infinitely supported masks. J. Approx. Theory 148(2007), no. 2, 158–176. Google Scholar | DOI

[24] [24] Shen, Z., Refinable function vectors. SIAM J. Math. Anal. 29(1998), no. 1, 235–250. Google Scholar | DOI

[25] [25] Strang, G. and Fix, G., A Fourier analysis of the finite element variational method. In: Constructive Aspects of Functional Analysis, Edizione Cremonese, Rome, 1973, pp. 795–840. Google Scholar

[26] [26] Sun, Q., Convergence of cascade algorithms and smoothness of refinable distributions. Chinese Ann. Math. Ser. B 24(2003), no. 3, 367–386. Google Scholar | DOI

[27] [27] Unser, M. and Blu, T., Fractional splines and wavelets. SIAM Rev. 42(2000), no. 1, 43–67. Google Scholar | DOI

[28] [28] Zhang, S. R., Refinable functions and subdivision schemes. Ph.D. Thesis, University of Alberta, 1998. Google Scholar