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.