ABSTRACT
Daubechies compact and ortho-normal wavelets were discovered by none other than Ingrid Daubechies in the year 1988. This is an important milestone in the development of wavelet transform theory. This chapter describes before the construction of Daubechies wavelets and provides a quantitative definition of smoothness or regularity. Regularity of a function is related to its moments. It is possible to build wavelets with different levels of smoothness. Smoothness of a function is related to its rate of decay. As wavelets have a compact support, smoothness is certainly one of its desired features. Regularity of the wavelet function implies its localization in the frequency domain. Thus smoothness and the moments of a function are closely related. Using Bezout’s theorem, Daubechies developed expressions for scaling coefficients. Using these coefficients, wavelet coefficients are determined. Finally, a scheme for computing scaling and mother wavelet functions is indicated.
