The aim of the course is to introduce the main constructions of wavelets and frames, show their applications to the modern signal processing and to numerical algorithms, and consider optimization problems solved in that theory. The course provides students with all necessary tools from functional and harmonic analysis to construct and analyze systems of wavelets and with the methods of optimal control for the corresponding optimization problems.
A short overview of the Fourier analysis. Direct and inverse theorems of the approximation theory. Discrete and fast Fourier transform. Haar, Shannon-Kotelnikov, and Meyer systems. The cascade algorithms for the wavelet transform. The noise analysis. Compactly supported wavelets and frames. Daubechies wavelets. Optimization problems in wavelets: the best approximation rates, the smallest support, the minimal uncertainty constants. Methods of optimal control. Calculus of variations and the Pontryagin maximum principle.