On divergence-free wavelets

This paper is concerned with the construction of compactly supported divergence-free vector wavelets. Our construction is based on a large class of refinable functions which generate multivariate multiresolution analyses which includes, in particular, the non tensor product case.For this purpose, we develop a certain relationship between partial derivatives of refinable functions and wavelets with modifications of the coefficients in their refinement equation. In addition, we demonstrate that the wavelets we construct form a Riesz-basis for the space of divergence-free vector fields.

[1]  I. Daubechies,et al.  A new technique to estimate the regularity of refinable functions , 1996 .

[2]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[3]  R. Young,et al.  An introduction to nonharmonic Fourier series , 1980 .

[4]  I. Daubechies,et al.  Non-separable bidimensional wavelets bases. , 1993 .

[5]  I. Weinreich,et al.  Wavelet-Galerkin methods: An adapted biorthogonal wavelet basis , 1993 .

[6]  C. Micchelli,et al.  Stationary Subdivision , 1991 .

[7]  A. Habibi,et al.  Introduction to wavelets , 1995, Proceedings of MILCOM '95.

[8]  C. D. Boor,et al.  Box splines , 1993 .

[9]  I. Weinreich,et al.  Wavelet Bases Adapted to Pseudodifferential Operators , 1994 .

[10]  J. Stöckler.  Multivariate wavelets , 1993 .

[11]  Paul Federbush,et al.  Navier and stokes meet the wavelet , 1993 .

[12]  Charles A. Micchelli,et al.  Using the Refinement Equations for the Construction of Pre-Wavelets II: Powers of Two , 1991, Curves and Surfaces.

[13]  J. R. Higgins.  Review: Robert M. Young, An introduction to nonharmonic Fourier series , 1981 .

[14]  Charles K. Chui,et al.  An Introduction to Wavelets , 1992 .

[15]  Pierre Gilles Lemarié-Rieusset.  Analyses multi-résolutions non orthogonales, commutation entre projecteurs et derivation et ondelettes vecteurs à divergence nuIIe , 1992 .

[16]  Wolfgang Dahmen,et al.  Translates of multivarlate splines , 1983 .

[17]  Zuowei Shen,et al.  Wavelets and pre-wavelets in low dimensions , 1992 .

[18]  I. Daubechies,et al.  Biorthogonal bases of compactly supported wavelets , 1992 .

[19]  R. Jia,et al.  Approximation by multiinteger translates of functions having global support , 1993 .

[20]  Guy Battle,et al.  Space-Time Wavelet Basis for the Continuity Equation , 1994 .

[21]  Paul Federbush,et al.  Divergence-free vector wavelets. , 1993 .

[22]  T. Lyche,et al.  Box Splines and Applications , 1991 .

[23]  W. Dahmen.  Some remarks on multiscale transformations, stability, and biorthogonality , 1994 .

[24]  S. Mallat.  Multiresolution approximations and wavelet orthonormal bases of L^2(R) , 1989 .

[25]  A. Cohen,et al.  Compactly supported bidimensional wavelet bases with hexagonal symmetry , 1993 .

[26]  M. Nikolskii,et al.  Approximation of Functions of Several Variables and Embedding Theorems , 1971 .