Wavelet and Multiresolution Analysis Schemes

Abstract

An essential aspect of spectral representations analyzed so far is the projection of the process or solution on a polynomial basis, namely on a vector space spanned by infinitely differentiable functions. It is known following the works of Wiener [Am. J. Math. 60:897–936, 1938] and Cameron and Martin [Ann. Math. 48:385–392, 1947] that such spectral representation converges in a mean square sense as N,No→∞. In the case of numerical simulation, however, on must necessarily rely on truncation, and in this case the convergence may be seriously compromised due to truncation errors and aliasing, and in extreme cases these phenomena may lead to breakdown of the computations. This chapter explores the possibility of overcoming these difficulties by using a PC expansion based on Haar wavelets or multiwavelets.