Iterative approximations of exponential bases on fractal measures

TL;DR

This paper investigates the existence and characterization of spectral sets for fractal measures generated by iterated function systems, providing new results on spectra existence, Bessel spectrum characterization, and frame spectrum conditions.

Contribution

It introduces new existence results for spectra related to Hadamard pairs and characterizes Bessel spectra via finite matrices for affine IFS measures.

Findings

Existence results for spectra associated with Hadamard pairs

Characterization of Bessel spectrum in terms of finite matrices

Sufficient condition for frame spectrum when affine IFS has no overlap

Abstract

For some fractal measures it is a very difficult problem in general to prove the existence of spectrum (respectively, frame, Riesz and Bessel spectrum). In fact there are examples of extremely sparse sets that are not even Bessel spectra. In this paper we investigate this problem for general fractal measures induced by iterated function systems (IFS). We prove some existence results of spectra associated with Hadamard pairs. We also obtain some characterizations of Bessel spectrum in terms of finite matrices for affine IFS measures, and one sufficient condition of frame spectrum in the case that the affine IFS has no overlap.