The error of Chebyshev approximations on shrinking domains

TL;DR

This paper analyzes the asymptotic error and interpolation properties of rational Chebyshev approximants on shrinking domains, revealing their convergence behavior and relation to Chebyshev and Padé approximants.

Contribution

It provides new insights into the asymptotic error, interpolation properties, and convergence of rational Chebyshev approximants on shrinking domains, extending previous results.

Findings

Point-wise error approaches a Chebyshev polynomial times Padé error term

Approximants attain nodes approaching scaled Chebyshev nodes

Results apply to complex and real Chebyshev approximations and exponential functions

Abstract

Previous works show convergence of rational Chebyshev approximants to the Pad\'e approximant as the underlying domain of approximation shrinks to the origin. In the present work, the asymptotic error and interpolation properties of rational Chebyshev approximants are studied in such settings. Namely, the point-wise error of Chebyshev approximants is shown to approach a Chebyshev polynomial multiplied by the asymptotically leading order term of the error of the Pad\'e approximant, and similar results hold true for the uniform error and Chebyshev constants. Moreover, rational Chebyshev approximants are shown to attain interpolation nodes which approach scaled Chebyshev nodes in the limit. Main results are formulated for interpolatory best approximations and apply for complex Chebyshev approximation as well as real Chebyshev approximation to real functions and unitary best approximation to the exponential function.