Regularity of refinable function vectors. (English) Zbl 0914.42025

Compactly supported solutions of refinement equations \(\phi(x)=\sum_{n=0}^N P_n\phi(2x-n)\) are considered for vector valued functions \(\phi\) and constant square matrices \(P_n\). As in the scalar case, the smoothness of \(\phi\) and the decay of the Fourier transform \(\widehat{\phi}\) are characterized by properties of the refinement mask \(P(u)={1\over 2}\sum_n P_ne^{-inu}\). In the scalar case, the approximation order can only be \(m\) if \(P(u)\) factors as \([(1+e^{-iu})/2]^mP^{(m)}(u)\) where \(P^{(m)}\) is \(2\pi\)-periodic, and \(P^{(m)}(0)=1\). The factorization property needed in the vector case is however much more involved: a factorization of the form \[ P(u)=2^{-m}C_0(2u)\cdots C_{m-1}(2u) P^{(m)}(u) C_{m-1}(u)^{-1}\cdots C_0(u)^{-1} \] is needed. From the relation \(\widehat{\phi}(u)=P(u/2)\widehat{\phi}(u/2)\), one gets the well-known infinite product representation of \(\widehat{\phi}\). The convergence of the infinite product in the matrix case is again much more intricate than in the scalar case and needs more conditions on \(P(u)\), for example \(P(0)\) should be diagonalizable with spectral radius at most 1. Finally, some conditions are needed to make \(\widehat{\phi}(u)\) decay for \(| u| \to\infty\). These decay properties allow to prove uniqueness of the solution of the refinement equations in a large function class and also the convergence of the cascade and subdivision algorithm can be derived. Several examples illustrate these results. The scaling functions of G. C. Donovan, J. S. Geronimo, D. P. Hardin and P. R. Massopust [SIAM J. Math. Anal. 27, No. 4, 1158-1192 (1996; Zbl 0873.42021)] are a special case.


42C15 General harmonic expansions, frames
39B62 Functional inequalities, including subadditivity, convexity, etc.
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type


Zbl 0873.42021
Full Text: DOI EuDML


[1] Alpert, B. K. (1993). A class of bases inL 2 for the sparse representation of integral operators.SIAM J. Math. Anal. 24, 246–262. · Zbl 0764.42017
[2] Alpert, B. K. and Rokhlin, V. A. (1991). A fast algorithm for the evaluation of Legendre expansions.SIAM J. Sci. Stat. Comput. 12, 158–179. · Zbl 0726.65018
[3] de Boor, C. (1976). Splines as linear combinations of B-splines. A survey.Approximation Theory II (G. G. Lorentz, C. K. Chui, and L. L. Schumaker, eds.). Academic Press, New York, 1–47.
[4] Cavaretta, A., Dahmen, W., and Micchelli, C. A. (1991).Stationary subdivision.Mem. Amer. Math. Soc. 453, Amer. Math. Soc., Providence, RI, 1–186. · Zbl 0741.41009
[5] Cohen, A. and Conze, J. P. (1992). Régularité des d’ondelettes et mesures ergodiques.Rev. Math. Iberoamericana 8, 351–366. · Zbl 0781.42027
[6] Cohen, A. and Daubechies, I. (1996). A new technique to estimate the regularity of refinable functions,Rev. Math. Iberoamericana (submitted). · Zbl 0879.65102
[7] Conze, J. P. and Raugi, A. (1990). Fonction harmonique pour un opérateur de transition et application.Bull. Soc. Math. France 118, 273–310. · Zbl 0725.60026
[8] Daubechies, I. (1988). Orthonormal bases of wavelets with compact support.Comm. Pure Appl. Math. 41, 909–996. · Zbl 0644.42026
[9] ———-. (1992).Ten Lectures on Wavelets. SIAM, Philadelphia, PA. · Zbl 0776.42018
[10] Donovan, G., Geronimo, J. S., Hardin, D. P., and Massopust, P. R. (1994). Construction of orthogonal wavelets using fractal interpolation functions. Preprint. · Zbl 0873.42021
[11] Dyn, N. (1992). Subdivision schemes in computer-aided geometric design.Advances in Numerical Analysis II, Wavelets, Subdivision Algorithms and Radial Functions (W. A. Light, ed.). Oxford University Press, New York, 36–104. · Zbl 0760.65012
[12] Eirola, T. (1992). Sobolev characterization of solutions of dilation equations.SIAM J. Math. Anal. 23, 1015–1030. · Zbl 0761.42014
[13] Goodman, T. N. T. and Lee, S. L. (1994). Wavelets of multiplicityr.Trans. Amer. Math. Soc. 342, 307–324. · Zbl 0799.41013
[14] Goodman, T. N. T., Lee, S. L., and Tang, W. S. (1993). Wavelets in wandering subspaces.Trans. Amer. Math. Soc. 338, 639–654. · Zbl 0777.41011
[15] Griepenberg, G. (1993). Unconditional bases of wavelets for Sobolev spaces.SIAM J. Math. Anal. 24, 1030–1042. · Zbl 0785.42013
[16] Heil, C. and Colella, D. (1996). Matrix refinement equations: Existence and uniqueness. Preprint. · Zbl 0904.39017
[17] Heil, C., Strang, G., and Strela, V. (1996). Approximation by translates of refinable functions.Numer. Math. (to appear). · Zbl 0857.65015
[18] Heller, P., Strela, V., Strang, G., Topiwala, P., Heil, C., and Hills, L. (1995). Multiwavelet filter banks for data compression.Proc. IEEE ISCAS, Seattle, WA.
[19] Hervé, L. (1994). Multi-Resolution analysis of multiplicityd. Application to dyadic interpolation.Appl. Comput. Harmonic Anal. 1, 299–315. · Zbl 0814.42017
[20] ———-. (1995). Construction et régularité des fonctions d’échelle,SIAM J. Math. Anal. 26(5), 1361–1385. · Zbl 0848.42023
[21] Lancaster, P. and Tismenetsky, M. (1985).The Theory of Matrices. Academic Press, San Diego, CA. · Zbl 0558.15001
[22] Micchelli, C. A. and Pinkus, A. (1991). Descartes systems from corner cutting.Const. Approx. 7, 161–194. · Zbl 0784.65017
[23] Plonka, G. (1995). Approximation order provided by refinable function vectors.Constr. Approx., to appear. · Zbl 0870.41015
[24] ———-. (1995). Factorization of refinement masks for function vectors. inWavelets and Multilevel Approximation (C. K. Chui and L. L. Schumaker, eds.), Singapore, World Scientific Publ. Co., 317–324. · Zbl 0927.42028
[25] Strang, G. and Strela, V. (1994). Orthogonal multiwavelets with vanishing moments.J. Opt. Engrg. 33, 2104–2107.
[26] ———-. (1995). Short wavelets and matrix dilation equations.IEEE Trans. Signal Proc. 43, 108–115.
[27] Vetterli, M. and Strang, G. (1994). Time-varying filter banks and multiwavelets. Electrical Engineering Department, Berkeley, preprint.
[28] Villemoes, L. (1994). Wavelet analysis of refinement equations.SIAM J. Math. Anal. 25, 1433–1460. · Zbl 0809.42016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.