Abstract
Diagonal Padé approximants to hyperelliptic functions
Highlights
The central topic of the paper is the convergence of diagonal Padé approximants
Instead of the continued fractions (1.4) we study diagonal Padé approximants, which are the same in substance, and instead of the square root of a fourth order polynomial, we study the approximation of hyperelliptic functions (for a definition see (3.1) at the beginning of Section 3)
A central place is taken by the investigation of spurious poles
Summary
The central topic of the paper is the convergence of diagonal Padé approximants. DEFINITION 1.1. - Let the function f be analytic al inftnity. Even in the case of a function f as simple as the square root of a polynomial of fourth order, the sequence of diagonal Padé approximants [n/n], n G N, can have spurious poles clustering everywhere in C (cf Theorem 6.6). Instead of the continued fractions (1.4) we study diagonal Padé approximants, which are the same in substance, and instead of the square root of a fourth order polynomial, we study the approximation of hyperelliptic functions (for a definition see (3.1) at the beginning of Section 3). A central place is taken by the investigation of spurious poles Their number and distribution is studied, and it is shown that in case of a hyperelliptic function f, diagonal Padé approximants can have only a finite number of them.
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