RUI: Applications of Approximation Theory to Neural Networks and Wavelets

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Mhaskar will continue his investigations on the complexity problem inthe theory of neural networks and construction of multiscales based onorthogonal polynomial coefficients as well as samples of the targetfunctions. Variations of such classical methods from approximationtheory as summability methods and polynomial inequalities will bedeveloped to provide a unified theory of these apparently diverse areas.The work will be applied to the numerical construction of orthogonalpolynomials, approximation on the sphere using scattered data, and thetheory of system identification.Although approximation theory is a classical area of mathematics, activefor more than a hundred years, technological progress in the areas ofneural networks and wavelets has given rise to many new possibilitiesfor applications of approximation theory. Neural networks and waveletsare both useful in high speed parallel computing, and naturally involvethe approximation of functions. Approximation theory techniques usedwith neural networks have produced dramatically better results incertain applications to array antenna technology, pattern recognition,and financial time series prediction.