Fourier frames for singular measures and pure type phenomena

TL;DR

This paper investigates the conditions under which measures in Euclidean space admit exponential frames, revealing that mixed-dimensional singular measures can have frames, contrary to previous conjectures, especially when curvature conditions are considered.

Contribution

It disproves a conjecture by showing mixed-dimensional singular measures can admit exponential frames, and establishes curvature conditions that prevent such frames.

Findings

Sum of arc length and surface measure can have exponential frames.

Surface measure with non-zero Gaussian curvature cannot have exponential frames.

Pure components of measures may admit frames even if the combined measure does not.

Abstract

Let $\mu$ be a positive measure on $R^d$. It is known that if the space $L^2(\mu)$ has a frame of exponentials then the measure $\mu$ must be of "pure type": it is either discrete, absolutely continuous or singular continuous. It has been conjectured that a similar phenomenon should be true within the class of singular continuous measures, in the sense that $\mu$ cannot admit an exponential frame if it has components of different dimensions. We prove that this is not the case by showing that the sum of an arc length measure and a surface measure can have a frame of exponentials. On the other hand we prove that a measure of this form cannot have a frame of exponentials if the surface has a point of non-zero Gaussian curvature. This is in spite of the fact that each "pure" component of the measure separately may admit such a frame.