Approximation properties of

multivariate wavelets

Author:
Rong-Qing Jia

Journal:
Math. Comp. **67** (1998), 647-665

MSC (1991):
Primary 41A25, 41A63; Secondary 42C15, 65D15

DOI:
https://doi.org/10.1090/S0025-5718-98-00925-9

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 provides approximation order .

**[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*, Trans. Amer. Math. Soc.**341**(1994), 787-806. MR**94d:41028****[4]**C. de Boor and K. Höllig,*Approximation order from bivariate -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*, 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**

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

Email:
jia@xihu.math.ualberta.ca

DOI:
https://doi.org/10.1090/S0025-5718-98-00925-9

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