📝 SummarXiv

Abstract

Polynomial series approximations are a central theme in approximation theory due to their utility in an abundance of numerical applications. The two types of series, which are featured most prominently, are Taylor series expansions and expansions derived based on families of $L^{2}-$orthogonal polynomials on bounded intervals. Traditionally, however, the properties of these series are studied in an isolated manner and most results on the $L^{2}-$based approximations are restricted to the interval of approximation. In many applications, theoretical guarantees require global estimates. Since these often follow trivially from Taylor's remainder theorem, Taylor's series is often favored in these applications. Its suboptimal numerical performance on compact intervals, however, ultimately results in such numerical algorithms being suboptimal. We show that "Taylor-like" bounds valid outside of the interval of approximation may be derived for the important case of $e^{x}$ and Chebyshev polynomials on $[-1,1]$ and provide links to substantial numerical applications.