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Keywords:

0101 Pure Mathematics, 0102 Applied Mathematics, 4901 Applied mathematics, 4904 Pure mathematics

Abstract:

Let R be the space of rational functions with poles among {a_k,1/ã_k:k=0,...,∞} with a_0 = 0 and |a_k|< /1, k ≥ 1. We consider sequences {R_n:n=0,...,∞} of nested subspaces with U_{n=0,...,∞} R_n = R. This first part will be concerned with the construction of two distinct orthogonal bases for R. We derive an intertwined recurrence relation for these orthogonal functions which appear as denominators of certain continued fractions. By contraction of these continued fractions, these recurrences are decoupled. It is explained how, with the given problem, one can associate two sequences of interpolation data and it is shown that the approximants of the continued fractions interpolate these data in a multipoint Padé sense. In part II, interpolatory quadrature rules which are exact for all f in Rn are constructed and their convergence is discussed as n → ∞.