Bessel sequences of exponentials on fractal measures

TL;DR

This paper demonstrates that fractal measures derived from affine iterated function systems contain Bessel sequences of complex exponentials with positive Beurling dimension, extending understanding of spectral properties of fractal measures.

Contribution

It proves the existence of Bessel sequences of complex exponentials with positive Beurling dimension in a broad class of fractal measures, generalizing previous orthogonality results.

Findings

Existence of Bessel sequences with positive Beurling dimension

Extension of spectral properties to affine fractal measures

Contrasts with orthonormal set limitations in fractal measures

Abstract

Jorgensen and Pedersen have proven that a certain fractal measure $\nu$ has no infinite set of complex exponentials which form an orthonormal set in $L^2(\nu)$. We prove that any fractal measure $\mu$ obtained from an affine iterated function system possesses a sequence of complex exponentials which forms a Riesz basic sequence, or more generally a Bessel sequence, in $L^2(\mu)$ such that the frequencies have positive Beurling dimension.