Approximation of functions and their derivatives by analytic maps on certain Banach spaces

TL;DR

This paper proves that on certain Banach spaces, any bounded, Lipschitz, and differentiable function with a uniformly continuous derivative can be uniformly approximated by analytic functions, including their derivatives.

Contribution

It establishes the approximation of functions and their derivatives by analytic maps on Banach spaces admitting a separating polynomial, extending approximation theory in infinite-dimensional spaces.

Findings

Analytic approximation is possible on certain Banach spaces.

Approximate functions can be made arbitrarily close in both value and derivative.

Results apply to separable Hilbert spaces and similar Banach spaces.

Abstract

Let X be a separable Banach space which admits a separating polynomial; in particular X a separable Hilbert space. Let $f:X \rightarrow R$ be bounded, Lipschitz, and $C^1$ with uniformly continuous derivative. Then for each {\epsilon}>0, there exists an analytic function $g:X \rightarrow R$ with $|g-f|<\epsilon$ and $||g'-f'||<\epsilon$.