Orthogonality Criteria for Multi-scaling Functions

Necessary and sufficient conditions for the orthonormality of a multi-scaling function φ with integer dilation factoraand multiplicityrare established. Here φ := (φ1, … , φr)Tand satisfies φ(x) =[formula]Pkφ(ax − k), for some positive integerM, withP0, … ,PMbeingr × rreal matrices andP0,PM≠ 0. These conditions also apply to the setting of the usual uni-scaling functions and uni-wavelets (i.e., multiplicityr= 1). The orthonormality of all the multi-scaling functions and multi-wavelets constructed in the wavelet literature can be verified. Extensive demonstrative examples are given.

[1]  G. Strang,et al.  Approximation by translates of refinable functions , 1996 .

[2]  C. Chui,et al.  A study of orthonormal multi-wavelets , 1996 .

[3]  W. Lawton.  Necessary and sufficient conditions for constructing orthonormal wavelet bases , 1991 .

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

[5]  Orthogonality Criteria for Compactly Supported Scaling Functions , 1994 .

[6]  G. Alexits.  Approximation theory , 1983 .

[7]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[8]  A. Cohen.  Ondelettes, analyses multirésolutions et filtres miroirs en quadrature , 1990 .

[9]  C. Chui,et al.  Construction of Compactly Supported Symmetric and Antisymmetric Orthonormal Wavelets with Scale = 3 , 1995 .

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

[11]  S. L. Lee,et al.  WAVELETS OF MULTIPLICITY r , 1994 .

[12]  S. L. Lee,et al.  Stability and orthonormality of multivariate refinable functions , 1997 .

[13]  L. Schumaker,et al.  Curves and Surfaces , 1991, Lecture Notes in Computer Science.

[14]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .

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

[16]  David K. Ruch,et al.  On the Support Properties of Scaling Vectors , 1996 .

[17]  D. Hardin,et al.  Fractal Functions and Wavelet Expansions Based on Several Scaling Functions , 1994 .

[18]  G. Plonka.  Approximation order provided by refinable function vectors , 1997 .

[19]  G. Strang,et al.  Orthogonal multiwavelets with vanishing moments , 1994 .

[20]  Jian-ao Lian.  On the Order of Polynomial Reproduction for Multi-scaling Functions , 1996 .

[21]  W. Lawton.  Tight frames of compactly supported affine wavelets , 1990 .