On uniform approximation of elliptic functions by Padé approximants

Diagonal Padé approximants of elliptic functions are studied. It is known that the absence of uniform convergence of such approximants is related to them having spurious poles that do not correspond to any singularities of the function being approximated. A sequence of piecewise rational functions is proposed, which is constructed from two neighbouring Padé approximants and approximates an elliptic function locally uniformly in the Stahl domain. The proof of the convergence of this sequence is based on deriving strong asymptotic formulae for the remainder function and Padé polynomials and on the analysis of the behaviour of a spurious pole.
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