Spurious poles in Pade´ approximation

In the theory of Pade approximation locally uniform convergence has been proved only for special classes of functions: for much larger classes convergence in capacity has been shown to hold true. The reason for one type of convergence to hold true, but the other one not, can be found in poles of the approximants that may occur apparently anywhere in the complex plane. Because of their unwanted nature and since they do not correspond to singularities of the function f to be approximated, these poles are called spurious. The denominators of Pade approximants satisfy orthogonality relations, and consequently the location and distribution of spurious poles depend on properties of the orthogonality relations. In the present paper the possibility of spurious poles is studied from the perspective of these orthogonal relations.

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