Inner product computations using periodized daubechies wavelets.

Inner products of wavelets and their derivatives are presently known as connection coefficients. The numerical calculation of inner products of periodized Daubechies wavelets and their derivatives is reviewed, with the aim at providing potential users of the publicly-available numerical scheme, details of its operation. The numerical scheme for the calculation of connection coefficients is evaluated in the context of approximating differential operators, information which is useful in the solution of partial differential equations using wavelet-Galerkin techniques. Specific details of the periodization of inner products in the solution differential equations are included in the presentation. Wavelets have found a well-deserved niche in such areas of applied mathematics and engineering as approximation theory, signal analysis, and projection techniques for the solution of differential equations.

[1]  L. Ahlfors.  Complex Analysis , 1979 .

[2]  L. Trefethen,et al.  An Instability Phenomenon in Spectral Methods , 1987 .

[3]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[4]  S. Mallat,et al.  A wavelet based space-time adaptive numerical method for partial differential equations , 1990 .

[5]  William H. Press,et al.  Numerical Recipes: FORTRAN , 1988 .

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

[7]  R. Haddad,et al.  Multiresolution Signal Decomposition: Transforms, Subbands, and Wavelets , 1992 .

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

[9]  Jinchao Xu,et al.  Galerkin-wavelet methods for two-point boundary value problems , 1992 .

[10]  C. Micchelli,et al.  Using the refinement equation for evaluating integrals of wavelets , 1993 .

[11]  I. Daubechies,et al.  Wavelets on the Interval and Fast Wavelet Transforms , 1993 .

[12]  Sam Qian,et al.  Wavelets and the Numerical Solution of Partial Differential Equations , 1993 .

[13]  W. Sweldens,et al.  Quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions , 1994 .

[14]  John R. Williams,et al.  Wavelet–Galerkin solutions for one‐dimensional partial differential equations , 1994 .

[15]  Juan M. Restrepo,et al.  Wavelet-Galerkin Discretization of Hyperbolic Equations , 1995 .

[16]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[17]  G. Beylkin,et al.  On the representation of operators in bases of compactly supported wavelets , 1992 .