Non-spectral fractal measures with Fourier frames

TL;DR

This paper introduces a new class of fractal measures that, despite lacking exponential orthonormal bases, still support Fourier frames, expanding understanding of harmonic analysis on fractals.

Contribution

It generalizes the compatible tower condition to almost-Parseval-frame towers and provides the first example of a singular fractal measure with finitely many orthogonal exponentials that admits Fourier frames.

Findings

Existence of non-trivial almost-Parseval-frame towers

First example of a singular fractal measure with finitely many orthogonal exponentials

Fractal measure admits Fourier frames despite lacking orthonormal bases

Abstract

We generalize the compatible tower condition given by Strichartz to the almost-Parseval-frame tower and show that non-trivial examples of almost-Parseval-frame tower exist. By doing so, we demonstrate the first singular fractal measure which has only finitely many mutually orthogonal exponentials (and hence it does not admit any exponential orthonormal bases), but it still admits Fourier frames.