Note on a result of Kerman and Weit

TL;DR

This paper discusses a general phenomenon in harmonic analysis where certain continuous functions on the circle, with specific extremal properties, generate dense translation-invariant subspaces, extending previous results.

Contribution

It generalizes a known result by showing that functions with unique maxima or minima can generate dense subspaces, broadening the scope of the original theorem.

Findings

Functions with unique maxima or minima generate dense subspaces

The phenomenon applies broadly in harmonic analysis

Extension of previous specific cases to more general functions

Abstract

A result in \cite{Ker-Weit} states that a real valued continuous function $f$ on the circle and its nonnegative integral powers can generate a dense translation invariant subspace in the space of all continuous functions on the circle if $f$ has a unique maximum or a unique minimum. In this note we endeavour to show that this is quite a general phenomenon in harmonic analysis.