Topological stability of continuous functions with respect to averaging by measures with locally constant densities

TL;DR

This paper investigates the topological stability of continuous functions under averaging by measures with locally constant densities, establishing new conditions that relate the stability of the whole function to its local extrema.

Contribution

It proves that topological stability of a function's averagings implies stability of its germs at local extrema and extends stability conditions to measures with locally constant densities.

Findings

Proves the converse of previous stability results for functions with finitely many local extrema.

Establishes sufficient conditions for stability with measures having locally constant densities.

Extends stability analysis to measures with locally continuous densities.

Abstract

Let $\mu$ be a measure on $[-1,1]$. Then for every continuous function $f:\mathbb{R}\to\mathbb{R}$ and $\alpha>0$ one can define its averaging $f_{\alpha}:\mathbb{R}\to\mathbb{R}$ by the formula: \[ f_{\alpha}(x) = \int_{-1}^{1} f(x+t\alpha)d\mu. \] In arXiv:1509.06064 the authors studied the problem when $f_{\alpha}$ is topologically equivalent to $f$ for all $\alpha>0$ and call this property a topological stability of $f$ under averagings with respect to measure $\mu$. Similarly one can define topological stability of a germ of $f$ at some point $x\in\mathbb{R}$. It was shown that for a continuous function $f:\mathbb{R}\to\mathbb{R}$ having only finitely many local extremes and any measure $\mu$ topological stability of averagings of germs of $f$ at local extremes implies topological stability of averagings of $f$. In the present paper we prove the converse statement: topological stability of averagings of $f$ implies topological stability of averagings of its germs at local extremes. Moreover, in arXiv:1509.06064 it was also extablished a sufficient condition for topological stability of averagings of local extremes with respect to measures with finite supports. In the present paper we obtain similar sufficient conditions for measures with locally continuous and in particular with locally constant densities.