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.

这篇论文有两个相对不同但相互关联的目标；第一个是在更高属的任意紧黎曼曲面上定义 Weyl-Stiltjes 变换的 Padé 近似的概念。数据由黎曼曲面中的等高线和其上的测度以及点附近局部坐标的附加数据和g的除数组成. 所得的 Padé-like 近似的分母也满足正交关系，并且是适当线丛的部分。显示了一个用于二阶方阵的黎曼-希尔伯特问题，以与普通正交多项式情况类似的方式表征这些正交部分。第二部分扩展了这个想法以探索它与可集成系统的联系。相同的数据可用于定义两个线束序列之间的配对。变形空间中配对退化为固定程度的轨迹与“tau”函数的零点重合。我们展示了这个 tau 函数如何在考虑整个 tau 函数序列时满足 Kadomtsev-Petviashvili 层次结构关于变形参数和 2-Toda 层次结构的某些修改。