Mathematical Theory of Wavelets

Wavelets are functions that satisfy certain requirements and are used in representing and processing functions and signals, as well as, in compression of data and images, in many fields such as: mathematics, physics, computer science, engineering, and study of wavelet transform had been motivated by the need to the medicine. The overcome some weak points in representing functions and signals by the classical Fourier transform such as Gibbs phenomenon. In addition, wavelet transform have showed superiority over the classical Fourier transform. They converge faster than Fourier transform, leading to more efficient processing of signals and data. In this thesis, we overview the theory of wavelet transform, as well as, the theory of Fourier transform and make a comparative theoretical study between the tow major transforms proving the superiority of wavelet transform over the Fourier transforms in the speed of convergence and the accuracy or many functions