Approximation by Multiple Refinable Functions

Canadian journal of mathematics, Tome 49 (1997) no. 5, pp. 944-962

Résumé

We consider the shift-invariant space, S(Φ), generated by a set Φ = {Φ 1,..., Φ r} of compactly supported distributions on R when the vector of distributions φ:= {Φ 1,..., Φ r} T satisfies a system of refinement equations expressed in matrix form as where a is a finitely supported sequence of r x r matrices of complex numbers. Such multiple refinable functions occur naturally in the study of multiple wavelets.The purpose of the present paper is to characterize the accuracy of Φ, the order of the polynomial space contained in S(Φ), strictly in terms of the refinement mask a. The accuracy determines the Lp -approximation order of S(Φ) when the functions in (Φ) belong to Lp(R) (see Jia [10]). The characterization is achieved in terms of the eigenvalues and eigenvectors of the subdivision operator associated with the mask a. In particular, they extend and improve the results of Heil, Strang and Strela [7], and of Plonka [16]. In addition, a counterexample is given to the statement of Strang and Strela [20] that the eigenvalues of the subdivision operator determine the accuracy. The results do not require the linear independence of the shifts of Φ.

Détail

DOI : 10.4153/CJM-1997-049-8

Mots-clés : 39B12, 41A25, 65F15, Refinement equations, refinable functions, approximation order, accuracy, shift-invariant spaces, subdivision

Bibliographie

[1] 1. Barros-Neto, J., An Introduction to the Theory of Distributions. Marcel Dekker, New York, 1973. Google Scholar

[2] 2. de Boor, C. and Höllig, K., Approximation order from bivariate C1-cubics: a counterexample. Proc. Amer. Math. Soc. 87(1983), 649–655. Google Scholar

[3] 3. de Boor, C. and Jia, R.Q., A sharp upper bound on the approximation order of smooth bivariate pp functions. J. Approx. Theory 72(1993), 24–33. Google Scholar

[4] 4. Cavaretta, A.S.,Dahmen, W., and Micchelli, C.A., Stationary Subdivision. Memoirs of Amer. Math. Soc., Vol. 93, 1991. Google Scholar

[5] 5. Daubechies, I. and Lagarias, J.C., Two-scale difference equations: II. Local regularity, Infinite products of matrices and fractals. SIAMJ. Math.Anal. 23(1992), 1031–1079. Google Scholar

[6] 6. Heil, C. and Colella, D., Matrix refinement equations: existence and uniqueness. J. Fourier Anal. Appl. 2(1996), 363–377. Google Scholar

[7] 7. Heil, C., Strang, G., and Strela, V., Approximation by translates of refinable functions. Numer. Math. 73(1996), 75–94. Google Scholar

[8] 8. Jia, R.Q., A characterization of the approximation order of translation invariant spaces of functions. Proc. Amer. Math. Soc. 111(1991), 61–70. Google Scholar

[9] 9. Jia, R.Q., Subdivision schemes in Lp spaces. Advances in Comp. Math. 3(1995), 309–341. Google Scholar

[10] 10. Jia, R.Q., Shift-invariant spaces on the real line. Proc. Amer. Math. Soc. 125(1997), 785–793. Google Scholar

[11] 11. Jia, R.Q. and Micchelli, C.A., On linear independence of integer translates of a finite number of functions. Proc. Edinburgh Math. Soc. 36(1992), 69–85. Google Scholar

[12] 12. Jia, R.Q., Riemenschneider, S.D., and Zhou, D.X., Vector subdivision schemes and multiple wavelets. Math. Comp., to appear. Google Scholar

[13] 13. Jia, R.Q. and Wang, J.Z., Stability and linear independence associated with wavelet decompositions. Proc. Amer. Math. Soc. 117(1993), 1115–1124. Google Scholar

[14] 14. Lian, J.A., Characterization of the order of polynomial reproduction for multiscaling functions. In: Approximation Theory VIII, Vol. 2 (Eds. Chui, C.K. and Schumaker, L.L.),World Scientific Publishing Co., Singapore, 1995. 251–258. Google Scholar

[15] 15. Micchelli, C.A. and Prautzsch, H., Uniform refinement of curves. Linear Algebra and its Applications 114/115(1989), 841–870. Google Scholar

[16] 16. Plonka, G., Approximation order provided by refinable function vectors. Constr. Approx. 13(1997). 221–244. Google Scholar

[17] 17. Ron, A., A characterization of the approximation order of multivariate spline spaces. StudiaMath. 98(1991), 73–90. Google Scholar

[18] 18. Schoenberg, I.J., Contributions to the problem of approximation of equidistant data by analytic functions. Quart. Appl. Math. 4(1946), 45–99. 112–141. Google Scholar

[19] 19. Strang, G. and Fix, G., A Fourier analysis of the finite-element variational method. In: Constructive Aspects of Functional Analysis, (Ed. Geymonat, G.), C.I.M.E., 1973. 793–840. Google Scholar

[20] 20. Strang, G. and Strela, V., Orthogonal multiwavelets with vanishing moments. J. Optical Eng. 33(1994), 2104–2107. Google Scholar