Abstract

The set $M(\T)$ of regular Borel measures on the circle equipped with its usual addition and convolution as multiplication is a Banach algebra. The spectrum of a measure $\mu \in M(\T)$ contains all of the Fourier(-Stieltjes) coefficients of $\mu$ and if it is essentially no larger then we say that $\mu$ has natural spectrum.

The Wiener-Pitt phenomenon is the fact that not all measures have natural spectrum. We are interested in the other direction: It is a short exercise to see that any measure whose Fourier coefficients are a subset of a finite set has natural spectrum. We shall discuss the infinite sets $K$ such that if the Fourier coefficients of $\mu$ are in $K$ then $\mu$ has natural spectrum.

No expertise in Banach algebras will be assumed (either on the part of the speaker or the audience); the focus will be on the discrete analysis.

This is joint work with Ohrysko and Wojciechowski.