On uniform boundedness principles and banach - steinhaus theorems with rates

Abstract

This paper combines Banach-Steinhaus theorems with rates of Butzer-Scherer-Westphal (1973) with two versions of a uniform boundedness principle (UBP) with rates. This leads to equivalence assertions which, as in the classical situation without rates, do not only cover tests for convergence but also tests for nonconvergence, each time with rates. The method of proof of the UBPs consists in the familiar gliding hump method, but now equipped with rates. The present approach considerably unifies and extends results concerning the sharpness of error bounds in various areas of analysis. Explicit applications are given to numerical quadrature, interpolation, multiplier theory, and to difference schemes for initial value problems.