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Padé approximants and orthogonality on Riemann surfaces

Marco Bertola
Concordia University
Wednesday, June 14, 2023 - 16:00

I will recall the classical connection between the Padéapproximation problem for the Stieltjes transform of a measure and (non-hermitean) orthogonal polynomials.Then I will show how to extend this notion to the case where instead of "polynomials" in the Padé problem we use an appropriate subspace of meromorphic functions (and differentials) on an arbitrary Riemann surface of genus $g\geq 1$. I will then show how the theory bears its fruits when applied to a concrete example in genus $1$ (i.e. on a (real) elliptic curve), where all functions can be expressed in terms of classical Weierstrass and Jacobi functions.I will also indicate asymptotic results in this case expressing the strong asymptotic of the corresponding orthogonal “polynomials" (really, elliptic functions) and the weak limit of the counting measure of their zeroes. A connection with the potential theory of measures and S-curves will be hinted at, time permitting.

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