Padé approximants to certain elliptic-type functions

Abstract

Given non-collinear points a_1, a_2, a_3, there is a unique compact, say Δ, that has minimal logarithmic capacity among all continua joining a_1, a_2, and a_3. For h be a complex-valued non-vanishing Dini-continuous function on Δ, we consider f_h(z) := (1/πi)\int_Δh(t)/(t-z) dt/w^+(t), where w(z) := \sqrt{\prod_{k=0}^3(z-a_k)} and w^+ the one-sided value according to some orientation of Δ. In this work we present strong asymptotics of diagonal Padé approximants to f_h and describe the behavior of the spurious pole and the regions of locally uniform convergence from a generic perspective.

41 pages. arXiv admin note: substantial text overlap with arXiv:1103.3556