Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Approximation properties of
multivariate wavelets

Author: Rong-Qing Jia
Journal: Math. Comp. 67 (1998), 647-665
MSC (1991): Primary 41A25, 41A63; Secondary 42C15, 65D15
MathSciNet review: 1451324
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

  • [1] J. Barros-Neto, An Introduction to the Theory of Distributions, Marcel Dekker, New York, 1973. MR 57:1113
  • [2] C. de Boor, The polynomials in the linear span of integer translates of a compactly supported function, Constr. Approx. 3 (1987), 199-208. MR 88e:41054
  • [3] C. de Boor, R. DeVore, and A. Ron, Approximation from shift-invariant subspaces of $L_{2}({\Bbb R}^{d})$, Trans. Amer. Math. Soc. 341 (1994), 787-806. MR 94d:41028
  • [4] C. de Boor and K. Höllig, Approximation order from bivariate $C^{1}$-cubics: a counterexample, Proc. Amer. Math. Soc. 87 (1983), 649-655. MR 84j:41014
  • [5] C. de Boor, K. Höllig, and S. Riemenschneider, Box Splines, Springer-Verlag, New York, 1993. MR 94k:65004
  • [6] 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), 24-33. MR 94e:41012
  • [7] A. S. Cavaretta, W. Dahmen, and C. A. Micchelli, Stationary Subdivision, Memoirs of Amer. Math. Soc., vol. 93, no. 453, 1991. MR 92h:65017
  • [8] I. Daubechies and J. C. Lagarias, Two-scale difference equations: II. Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal. 23 (1992), 1031-1079. MR 93g:39001
  • [9] R. DeVore, B. Jawerth, and V. Popov, Compression of wavelet decompositions, Amer. J. Math. 114 (1992), 737-785. MR 94a:42045
  • [10] T. N. T. Goodman, C. A. Micchelli, and J. D. Ward, Spectral radius formulas for subdivision operators, Recent Advances in Wavelet Analysis (L. L. Schumaker and G. Webb, eds.), Academic Press, 1994, pp. 335-360. MR 94m:47076
  • [11] K. Gröchenig and W. R. Madych, Multiresolution analysis, Haar bases, and self-similar tilings of ${\Bbb R}^{n}$, IEEE Transactions on Information Theory 38 (1992), 556-568. MR 93i:42001
  • [12] B. Han and R. Q. Jia, Multivariate refinement equations and subdivision schemes, manuscript.
  • [13] C. Heil, G. Strang, and V. Strela, Approximation by translates of refinable functions, Numer. Math. 73 (1996), 75-94. MR 97c:65033
  • [14] R. Q. Jia, A dual basis for the integer translates of an exponential box spline, Rocky Mountain J. Math. 23 (1993), 223-242. MR 94a:41022
  • [15] R. Q. Jia, A Bernstein type inequality associated with wavelet decomposition, Constr. Approx. 9 (1993), 299-318. MR 94h:41026
  • [16] R. Q. Jia, The Toeplitz theorem and its applications to Approximation Theory and linear PDE's, Trans. Amer. Math. Soc. 347 (1995), 2585-2594. MR 95i:41014
  • [17] 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.
  • [18] 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.
  • [19] R. Q. Jia and C. A. Micchelli, On linear independence of integer translates for a finite number of functions, Proc. Edinburgh Math. Soc. 36 (1993), 69-85. MR 94e:41044
  • [20] 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), 61-72. MR 94a:42049
  • [21] A. Ron, A characterization of the approximation order of multivariate spline spaces, Studia Math. 98 (1991), 73-90. MR 92g:41017

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 41A25, 41A63, 42C15, 65D15

Retrieve articles in all journals with MSC (1991): 41A25, 41A63, 42C15, 65D15

Additional Information

Rong-Qing Jia
Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Keywords: Refinement equations, refinable functions, wavelets, accuracy, approximation order, smoothness, subdivision operators, transition operators
Received by editor(s): April 17, 1996
Additional Notes: Supported in part by NSERC Canada under Grant OGP 121336.
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society