Mathematic Methods and Algorithms for Real-Time Applications

摘要

A general framework of scaling functions and wavelets is developed, from whichexplicit formulas of interpolatory spline-wavelets, compactly supported spline-wavelets, their duals, wavelets on a bounded interval, and trigonometric wavelets are derived. On the other hand, by using the scale of 3 instead of 2, we are able to construct compactly supported orthonormal symmetric scaling functions and their corresponding pair of symmetric and antisymmetric wavelets. To avoid aliasing and other undesirable effects in wavelet decompositions, we introduce a continuous multiresolution analysis that generates the dyadic wavelets of Mallat and Zhong. Another approach is to consider frames. In this regard, we derive Littlewood-Paley inequalities and identities for frames. As an important application, we prove two oversampling theorems: one for generating frames from frames, and the other to insure that tight frames remain tight. For further decomposition of the higher octave bands, wavelet packets are studied and a stability result is obtained. In a different project, we use ridge functions to construct neural networks with one hidden layer, and prove that all such networks only give global approximations. Our study of systems reduction is again based on the AAK approach. We obtained rates of convergence of the rational symbol functions.... Cardinal spline-wavelets, Spline-wavelets on bounded intervals, Trigonometric wavelets, Dyadic wavelets, Compactly supported symmetric orthogonal wavelets, Bessel families, Frames, Oversampling, Cardinal interpolation, Convexity criteria, Radial basis functions, Ridge functions, Neural networks with one hidden layer, Hankel approximation.