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Abstract
The paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.
Highlights
The theory of Hermite–Padé approximation is intimately connected with the theory of orthogonal polynomials
The Hankel determinant of the corresponding moments n(t) = det n xa+b−2dμ(x; t) a,b=1 provide tau functions for the Kadomtsev–Petviashvili hierarchy. This type of interplay between Padé approximation and integrable systems has been exploited in numerous papers, to name a few [1,5,6,14,16,17]
(2.47) are compatible under the assumption Dn = 0, we obtain a contradiction with the uniqueness of the solution of the Riemann–Hilbert problem since neither α nor β are uniquely defined
Summary
The theory of Hermite–Padé approximation is intimately connected with the theory of (mutliple) orthogonal polynomials. The Hankel determinant of the corresponding moments n(t) = det n xa+b−2dμ(x; t) a,b=1 provide tau functions for the Kadomtsev–Petviashvili hierarchy This type of interplay between (multiple) Padé approximation (and related multiple orthogonality) and integrable systems has been exploited in numerous papers, to name a few [1,5,6,14,16,17]. 2. We define the Padé approximation problem in Definition 2.3: instead of a ratio of polynomials the relevant generalization requires the ratio of a meromorphic differential Qn−1 and a meromorphic function Pn such that it approximates the Weyl–Stiltjes function at the point ∞ ∈ C to appropriate order. We define two sequences of biorthogonal sections of certain line bundles and a pairing between them in terms of integration along the curve γ with the given measure. We identify the sequence as an instance of a (suitably modified) solution of the 2-Toda hierarchy, as presented in [1,20]
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