Approximation properties of multivariate wavelets

Jia, Rong-Qing

doi:10.1090/S0025-5718-98-00925-9

Approximation properties of multivariate wavelets
HTML articles powered by AMS MathViewer

by Rong-Qing Jia PDF

Math. Comp. 67 (1998), 647-665 Request permission

Abstract:

Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the approximation properties of multivariate refinable functions. We give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask. We connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators. Finally, we demonstrate that a refinable function in $W_{1}^{k-1}(\mathbb {R}^{s})$ provides approximation order $k$.

References

José Barros-Neto, An introduction to the theory of distributions, Pure and Applied Mathematics, vol. 14, Marcel Dekker, Inc., New York, 1973. MR 0461128
Carl de Boor, The polynomials in the linear span of integer translates of a compactly supported function, Constr. Approx. 3 (1987), no. 2, 199–208. MR 889555, DOI 10.1007/BF01890564
Carl de Boor, Ronald A. DeVore, and Amos Ron, Approximation from shift-invariant subspaces of $L_2(\mathbf R^d)$, Trans. Amer. Math. Soc. 341 (1994), no. 2, 787–806. MR 1195508, DOI 10.1090/S0002-9947-1994-1195508-X
C. de Boor and K. Höllig, Approximation order from bivariate $C^{1}$-cubics: a counterexample, Proc. Amer. Math. Soc. 87 (1983), no. 4, 649–655. MR 687634, DOI 10.1090/S0002-9939-1983-0687634-4
C. de Boor, K. Höllig, and S. Riemenschneider, Box splines, Applied Mathematical Sciences, vol. 98, Springer-Verlag, New York, 1993. MR 1243635, DOI 10.1007/978-1-4757-2244-4
C. de Boor and R. Q. Jia, A sharp upper bound on the approximation order of smooth bivariate PP functions, J. Approx. Theory 72 (1993), no. 1, 24–33. MR 1198370, DOI 10.1006/jath.1993.1003
Alfred S. Cavaretta, Wolfgang Dahmen, and Charles A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93 (1991), no. 453, vi+186. MR 1079033, DOI 10.1090/memo/0453
Ingrid Daubechies and Jeffrey C. Lagarias, Two-scale difference equations. II. Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal. 23 (1992), no. 4, 1031–1079. MR 1166574, DOI 10.1137/0523059
Ronald A. DeVore, Björn Jawerth, and Vasil Popov, Compression of wavelet decompositions, Amer. J. Math. 114 (1992), no. 4, 737–785. MR 1175690, DOI 10.2307/2374796
T. N. T. Goodman, Charles A. Micchelli, and J. D. Ward, Spectral radius formulas for subdivision operators, Recent advances in wavelet analysis, Wavelet Anal. Appl., vol. 3, Academic Press, Boston, MA, 1994, pp. 335–360. MR 1244611
K. Gröchenig and W. R. Madych, Multiresolution analysis, Haar bases, and self-similar tilings of $\textbf {R}^n$, IEEE Trans. Inform. Theory 38 (1992), no. 2, 556–568. MR 1162214, DOI 10.1109/18.119723
B. Han and R. Q. Jia, Multivariate refinement equations and subdivision schemes, manuscript.
Christopher Heil, Gilbert Strang, and Vasily Strela, Approximation by translates of refinable functions, Numer. Math. 73 (1996), no. 1, 75–94. MR 1379281, DOI 10.1007/s002110050185
Rong Qing Jia, A dual basis for the integer translates of an exponential box spline, Rocky Mountain J. Math. 23 (1993), no. 1, 223–242. MR 1212738, DOI 10.1216/rmjm/1181072618
Rong Qing Jia, A Bernstein-type inequality associated with wavelet decomposition, Constr. Approx. 9 (1993), no. 2-3, 299–318. MR 1215774, DOI 10.1007/BF01198008
Rong Qing Jia, The Toeplitz theorem and its applications to approximation theory and linear PDEs, Trans. Amer. Math. Soc. 347 (1995), no. 7, 2585–2594. MR 1277117, DOI 10.1090/S0002-9947-1995-1277117-8
R. Q. Jia, Refinable shift-invariant spaces: from splines to wavelets, Approximation Theory VIII (C. K. Chui and L. L. Schumaker, eds.), vol. 2, World Scientific Publishing Co., Inc., 1995, pp. 179–208.
R. Q. Jia, The subdivision and transition operators associated with a refinement equation, Advanced Topics in Multivariate Approximation (F. Fontanella, K. Jetter and P.-J. Laurent, eds.), World Scientific Publishing Co., Inc., 1996, pp. 139–154.
Rong Qing Jia and Charles A. Micchelli, On linear independence for integer translates of a finite number of functions, Proc. Edinburgh Math. Soc. (2) 36 (1993), no. 1, 69–85. MR 1200188, DOI 10.1017/S0013091500005903
R.-Q. Jia and C. A. Micchelli, Using the refinement equation for the construction of pre-wavelets. V. Extensibility of trigonometric polynomials, Computing 48 (1992), no. 1, 61–72 (English, with German summary). MR 1162384, DOI 10.1007/BF02241706
Amos Ron, A characterization of the approximation order of multivariate spline spaces, Studia Math. 98 (1991), no. 1, 73–90. MR 1110099, DOI 10.4064/sm-98-1-73-90