On the estimation of wavelet coefficients

In wavelet representations, the magnitude of the wavelet coefficients depends on both the smoothness of the represented function f and on the wavelet. We investigate the extreme values of wavelet coefficients for the standard function spaces Ak=f| ∥fk)∥2 ≤ 1}, k∈N. In particular, we compare two important families of wavelets in this respect, the orthonormal Daubechies wavelets and the semiorthogonal spline wavelets. Deriving the precise asymptotic values in both cases, we show that the spline constants are considerably smaller.

[1]  Andreas Rathsfeld,et al.  A Wavelet Algorithm for the Solution of the Double Layer Potential Equation over Polygonal Boundaries , 1995 .

[2]  F. Olver.  Asymptotics and Special Functions , 1974 .

[3]  Christoph Schwab,et al.  Wavelet approximations for first kind boundary integral equations on polygons , 1996 .

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

[5]  Thierry Blu,et al.  Quantitative L/sup 2/ error analysis for interpolation methods and wavelet expansions , 1997, Proceedings of International Conference on Image Processing.

[6]  C. Chui,et al.  On compactly supported spline wavelets and a duality principle , 1992 .

[7]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[8]  S. Ehrich.  Explicit inequalities for wavelet coefficients , 2001 .

[9]  Michael Unser,et al.  Ten good reasons for using spline wavelets , 1997, Optics & Photonics.

[10]  Guido Walz,et al.  On the Chebyshev Norm of Polynomial B-Splines , 1995 .

[11]  Truong Q. Nguyen,et al.  Wavelets and filter banks , 1996 .

[12]  Andreas Rathsfeld,et al.  A Wavelet Algorithm for the Solution of a Singular Integral Equation over a Smooth Two-Dimensional Manifold , 1998 .

[13]  Michael Unser,et al.  A family of polynomial spline wavelet transforms , 1993, Signal Process..

[14]  Fritz Keinert,et al.  Biorthogonal Wavelets for Fast Matrix Computations , 1994 .

[15]  Wolfgang Dahmen,et al.  Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solution , 1993, Adv. Comput. Math..