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Convergence of vector subdivision schemes in Sobolev spaces. (English) Zbl 1006.65153

Let \(M\) be a dilation matrix, i.e. an invertible integer-valued matrix such that \(\lim_{n\to \infty }M^{-n}=0, \) and let \(a\) be a matrix-valued finitely supported function on the lattice \( {{\mathbb Z}}^{s}\), the so-called refinement mask. Further let \(A\) be the mask symbol defined by \[ A( \omega) =\frac{1}{|\det M|}\sum_{\alpha \in Z^{s}}a( \alpha) e^{-i\alpha \cdot \omega }. \] The authors start with the following interesting observation: Assume that \(M \) is isotropic, \(\varphi _{i}\), \(i=1,\dots,r\), are compactly supported functions in the Sobolev space \(W_{p}^{k}( {{\mathbb R}}^{s}) \) such that \(\widehat{\Phi } ( 0) \neq 0\) and span\(\{ \widehat{\Phi }( 2\pi \beta) :\beta \in {\mathbb Z}^{s}\} ={\mathbb C}^{r}\) . If \(\Phi =( \varphi _{1},\dots ,\varphi _{r}) \) is a solution of the refinement equation \[ \Phi =\sum_{\alpha \in {{\mathbb Z}}^{s}}a( \alpha) \Phi ( M\cdot -\alpha) \] then \(A( 0) \) satisfies the so-called eigenvalue condition of order \(k\): \(A( 0) \) has \(1\) as a simple eigenvalue and the other eigenvalues are of modulus less than \(\rho ^{-k}\) where \(\rho \) denotes the spectral radius of the dilation matrix \(M.\)
The cascade operator \(Q_{a}\) is defined by \( Q_{a}\Phi =\sum_{\alpha \in {{\mathbb Z}}^{s}}a( \alpha) \Phi ( M\cdot -\alpha). \) The aim of the paper is to discuss the subdivision scheme \(\Phi _{n}=Q_{a}\Phi _{n-1}\) (or the cascade algorithm) with respect to the Sobolev norm under the assumption that the eigenvalue condition is satisfied.
The first main result is: if \(\Phi _{0}\) is an initial vector of compactly supported functions in \(W_{p}^{k}( {{\mathbb R}}^{s}) \) and \( Q_{a}^{n}\Phi _{0}\) converges to a compactly supported function \(\Phi \) in \( W_{p}^{k}( {{\mathbb R}}^{s}) \) then the Fourier transform \(\widehat{\Phi _{0} }\) of the initial vector \(\Phi _{0}\) satisfies Strang-Fix type conditions up to the order \(k\). This class of initial vectors \(\Phi _{0}\) is denoted by \(Y_{k}.\) A similar result was obtained by Q. Chen, J. Liu and W. Zhang [J. Comput. Math. 20, No. 4, 363-372 (2002; Zbl 1006.65153), reviewed above].
The subdivision scheme is said to be convergent in the Sobolev space \(W_{p}^{k}( {{\mathbb R}}^{s}) \) if there exists a compactly supported function \(\Phi \) in \(W_{p}^{k}( {{\mathbb R}}^{s}) \) such that for any initial vector \(\Phi _{0}\) in the class \(Y_{k}\) the scheme \(Q_{a}^{n}\Phi _{0}\) converges to \(\Phi\). The second main result shows that the subdivision operator \(S_{a}\) associated with a mask \(a\) possesses a natural invariant subspace defined by means of the eigenvalue condition provided that the subdivision scheme \(Q_{a}^{n}\) converges in \( W_{p}^{k}( {{\mathbb R}}^{s}) .\)
The third main result characterizes the convergence of a subdivison scheme in \(W_{p}^{k}( {{\mathbb R}}^{s}) \) in terms of the \(p\)-norm joint spectral radius of a finite collection of transition operators determined by the sequence \(a\) restricted to a certain invariant subspace.
An application of these results is given by D. Chen and X. Zheng [J. Math. Anal. Appl. 268, No. 1, 41-52 (2002; Zbl 1006.65155), reviewed below].

MSC:

65T60 Numerical methods for wavelets
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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References:

[1] de Boor, C.; DeVore, R.; Ron, A., Approximation orders of FSI spaces in \(L_2(R^d)\), Constr. Approx., 14, 631-652 (1998) · Zbl 0919.41009
[2] Cabrelli, C.; Heil, C.; Molter, U., Accuracy of lattice translates of several multidimensional refinable functions, J. Approx. Theory, 95, 5-52 (1998) · Zbl 0911.41008
[3] Cavaretta, A. S.; Dahmen, W.; Micchelli, C. A., Stationary subdivision, Mem. Amer. Math. Soc., 93, i-186 (1991) · Zbl 0741.41009
[4] Chen, D. R., Algebraic properties of subdivision operators with matrix mask and their applications, J. Approx. Theory, 97, 294-310 (1999) · Zbl 0942.42021
[5] Dahmen, W.; Micchelli, C. A., Biorthogonal wavelet expansions, Constr. Approx., 13, 293-328 (1997) · Zbl 0882.65148
[6] Daubechies, I.; Lagarias, J., Two-scale difference equations: II. Local regularity, infinite products of matrices, and fractals, SIAM J. Math. Anal., 23, 1031-1079 (1992) · Zbl 0788.42013
[7] Evans, L. C., Partial Differential Equations. Partial Differential Equations, Graduate Studies in Mathematics, 19 (1998), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0902.35002
[8] Goodman, T. N.T.; Lee, S. L., Convergence of nonstationary cascade algorithms, Numer. Math., 84, 1-33 (1999) · Zbl 0945.65157
[9] Han, B.; Jia, R. Q., Multivariate refinement equations and convergence of subdivision schemes, SIAM J. Math. Anal., 29, 1177-1199 (1998) · Zbl 0915.65143
[10] Heil, C.; Colella, D., Matrix refinement equations: existence and uniqueness, J. Fourier Anal. Appl., 2, 363-377 (1996) · Zbl 0904.39017
[11] Jia, R. Q., Subdivision schemes in \(L_p\) spaces, Adv. Comput. Math., 3, 309-341 (1995) · Zbl 0833.65148
[12] Jia, R. Q., Approximation properties of multivariate wavelets, Math. Comp., 67, 647-665 (1998) · Zbl 0889.41013
[13] Jia, R. Q., Shift-invariant spaces and linear operator equations, Israel J. Math., 103, 259-288 (1998) · Zbl 0927.41011
[14] Jia, R. Q., Stability of the shifts of a finite number of functions, J. Approx. Theory, 95, 194-202 (1998) · Zbl 0957.42019
[15] R. Q. Jia, Q. T. Jiang, and, S. L. Lee, Convergence of cascade algorithms in Sobolev spaces and integrals of wavelets, Numer. Math, to appear.; R. Q. Jia, Q. T. Jiang, and, S. L. Lee, Convergence of cascade algorithms in Sobolev spaces and integrals of wavelets, Numer. Math, to appear. · Zbl 1019.65108
[16] Jia, R. Q.; Micchelli, C. A., On linear independence of integer translates of a finite number of functions, Proc. Edinburgh Math. Soc., 36, 69-85 (1992)
[17] Jia, R. Q.; Riemenschneider, S. D.; Zhou, D. X., Approximation by multiple refinable functions, Canad. J. Math., 49, 944-962 (1997) · Zbl 0904.41010
[18] Jia, R. Q.; Riemenschneider, S. D.; Zhou, D. X., Vector subdivision schemes and multiple wavelets, Math. Comp., 67, 1533-1563 (1998) · Zbl 0905.65139
[19] Jiang, Q. T., Multivariate matrix refinable functions with arbitrary matrix dilation, Trans. Amer. Math. Soc., 351, 2407-2438 (1999) · Zbl 0931.42021
[20] Jiang, Q. T.; Shen, Z. W., On existence and weak stability of matrix refinable functions, Constr. Approx., 15, 337-353 (1999) · Zbl 0932.42028
[21] Kelley, J. L.; Namioka, I., Linear Topological Spaces (1963), Springer-Verlag: Springer-Verlag New York
[22] Micchelli, C. A.; Sauer, T., Regularity of multiwavelets, Adv. Comput. Math., 7, 455-545 (1997) · Zbl 0902.65095
[23] Micchelli, C. A.; Sauer, T., Sobolev norm convergence of stationary subdivision schemes, (Le Méhauté, A.; Rabut, C.; Schumaker, L. L., Surface Fitting and Multiresolution Methods (1997), Vanderbilt University Press: Vanderbilt University Press Nashville), 245-260 · Zbl 0937.65148
[24] Rota, G. C.; Strang, G., A note on the joint spectral radius, Indag. Math., 22, 379-381 (1960) · Zbl 0095.09701
[25] Strang, G.; Fix, G., A Fourier analysis of the finite-element variational method, (Geymonat, G., Constructive Aspects of Functional Analysis (1973), Centro Internationale Matematico Estivo: Centro Internationale Matematico Estivo Rome), 793-840 · Zbl 0278.65116
[26] Wang, Y., Two-scale dilation equations and the mean spectral radius, Random Comput. Dynam., 4, 49-72 (1996) · Zbl 0872.39007
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