Pade interpolation by F-polynomials and transfinite diameter

TL;DR

This paper introduces a novel approach to Pade interpolation using F-polynomials, providing explicit approximations with sharp coefficient estimates and linking optimal frequencies to Fekete points, with results related to transfinite diameter.

Contribution

It develops a new interpolation method using F-polynomials, connecting optimal frequencies to Fekete points and analyzing coefficient bounds via transfinite diameter.

Findings

Optimal frequencies for interpolation are similar to Fekete points.

Coefficient norms of interpolating polynomials are bounded by transfinite diameter.

Conditions are provided for boundedness of coefficients in Laplace transform cases.

Abstract

We define $F$-polynomials as linear combinations of dilations by some frequencies of an entire function $F$. In this paper we use Pade interpolation of holomorphic functions in the unit disk by $F$-polynomials to obtain explicitly approximating $F$-polynomials with sharp estimates on their coefficients. We show that when frequencies lie in a compact set $K\subset\mathbb C$ then optimal choices for the frequencies of interpolating polynomials are similar to Fekete points. Moreover, the minimal norms of the interpolating operators form a sequence whose rate of growth is determined by the transfinite diameter of $K$. In case of the Laplace transforms of measures on $K$, we show that the coefficients of interpolating polynomials stay bounded provided that the frequencies are Fekete points. Finally, we give a sufficient condition for measures on the unit circle which ensures that the sums of the absolute values of the coefficients of interpolating polynomials stay bounded.