Consistent rational approximations of power series, trigonometric series and series of Chebyshev polynomials

TL;DR

This paper develops new methods for approximating trigonometric and Chebyshev polynomial series using Hermite-Padé and Hermite-Jacobi techniques, establishing criteria for their existence and explicit forms.

Contribution

It introduces novel approximation methods for series of special functions, providing existence criteria and explicit formulas for these approximations.

Findings

Established criteria for existence and uniqueness of trigonometric Hermite-Padé polynomials.

Derived explicit forms of these polynomials.

Provided examples illustrating differences between approximation types.

Abstract

For trigonometric series and series of Chebyshev polynomials, we defined trigonometric Hermite-Pad\'e and Hermite-Jacobi approximations, linear and nonlinear Hermite-Chebyshev approximations. We established criterion of the existence and uniqueness of trigonometric Hermite-Pad\'e polynomials, associated with an arbitrary set of $k$ trigonometric series, and we found explicit form of these polynomials. Similar results were obtained for linear Hermite-Chebyshev approximations. We made examples of systems of functions for which trigonometrical Hermite-Jacobi approximations are existed but aren't the same as trigonometric Hermite-Pad\'e approximations. Similar examples were made for linear and nonlinear Hermite-Chebyshev approximations.