Multivariate padé approximants, fourier series, and applications

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Padé approximants are a rational generalization of Taylor series and have been studied for over a century. Hermite's proof of transcendence of e involved constructing such approximants. His student Padé explored the approximants systematically and got his name attached to them. The study of multivariate Padé approximants is relatively new, partly because of the computation difficulties involved, and lends itself to experimentation with computer algebra packages. Part of our intention is to make as much of the study as systematic as possible, and to construct as many of the explicit approximants as we can. In the past few years, we successfully constructed explicit multivariate Padé approximants to some multivariate functions. Part of this proposal is to keep the momentum and explore more explicit constructions.There are many exciting mathematical and physical applications of these constructions that cover the spectrum from Number Theory to Signal Processing. We already have had several applications in Number Theory, and are going to explore more in Number Theory, Analysis, and Signal Processing.Another part of this proposal is to study various convergence problems of Fourier series. Fourier series are a type of trigonometric series and their convergence problems are always fundamental in Fourier Analysis and applications. Finding the weakest condition on the coefficients of the series so that the series is unformly convergent, i.e., finding the largest group of uniformly convergent series of some type, has been always important and we have made some important contributions. Our goals include more adventures in theory and applications.