Chapter 3 Fundamentals of wavelet transforms

Abstract

This chapter discusses various aspects of the wavelet transform when applied to continuous functions or signals. Wavelets form a set of basis functions, which linearly combine to represent a function f(t), from the space of square integrable functions L 2 (R). Functions in this space have finite energy. Because wavelet basis functions linearly combine to represent functions from L 2 (R) they are from this space as well. The reason for choosing this space largely relates to the properties of the L 2 norm. The wavelet basis functions are derived by translating and dilating one basic wavelet, called a “mother wavelet.” The dilated and translated wavelet basis functions are called “children wavelets.” The wavelet coefficients are the coefficients in the expansion of the wavelet basis functions. The wavelet transform is the procedure for computing the wavelet coefficients. The wavelet coefficients convey information about the weight that a wavelet basis function contributes to the function. The chapter introduces the continuous wavelet transform and discusses the conditions required for invertibility and the inverse or reconstruction formula.