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How well can the concept of Padé approximant be generalized to the multivariate case? (English) Zbl 0945.41012
Author’s abstract: “What we know about multivariate Padé approximation has been developed in the last 25 years. In the next sections we compare and discuss many of these results. It will become clear that simple properties and requirements, such as the uniqueness of Padé approximant and consequently its consistency property, can play a crucial role in the development of the multivariate theory. A separate section is devoted to a discussion of the convergenc properties. At the end we include an extensive reference list on the topic.” The principal reason emphasizing the origin of this paper is due to the non existence of a single unique definition for the notion of multivariate Padé approximant. Thus, starting from a given Taylor series expansion (*) \(f(x,y)=\sum_{(i,j)\in\mathbb{N}^2}c_{ij}x^iy^j\) (for reasons of notational simplicity, only the bivariate case is described) and rewriting suitable the data \(c_{ij}\) in (*), the author classifies the different definitions that until now exist into four main categories: a variety of approximation schemes covers the first group of definitions through a certain so-called equation lattice; the next way is the homogeneous approach by setting \(f(x,y)= \sum^\infty_{k=0} (\sum_{i+j=k} c_{ij}x^iy^j)\); a third group looks at the series development \(f(x,y)= \sum^\infty_{i=0} (\sum^\infty_{j=0} c_{ij} y^j) x^i =\sum^\infty_{i=0} c_i(y)x^i\) which treats the problem partly in a symbolic way, and finally, a fourth approach is introduced by considering convergence of the corresponding branched continued fractions. For each of the discussed multivariate definitions, some computational algorithms are briefly commented. Summing up, the work stands for a clear account and an excellent overview on the development of multivariate Padé approximation theory.

MSC:
41A21 Padé approximation
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
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