Continuous and Discrete Fourier Frames for Fractal Measures

TL;DR

This paper explores the existence and construction of Fourier frame measures for fractal and self-similar measures, introducing new methods and properties related to Bessel and frame measures, and their invariance under convolution and discretization.

Contribution

It introduces a general framework for constructing Bessel and frame measures for measures on ^d, including new results on their properties and examples for self-similar measures.

Findings

Frame measures are not always available for all measures.

Convolution with a probability measure preserves Beurling dimension.

Existence of atomic and weighted Fourier frames for measures with frame measures.

Abstract

Motivated by the existence problem of Fourier frames on fractal measures, we introduce Bessel and frame measures for a given finite measure on $\br^d$, as extensions of the notions of Bessel and frame spectra that correspond to bases of exponential functions. Not every finite compactly supported Borel measure admits frame measures. We present a general way of constructing Bessel/frame measures for a given measure. The idea is that if a convolution of two measures admits a Bessel measure then one can use the Fourier transform of one of the measures in the convolution as a weight for the Bessel measure to obtain a Bessel measure for the other measure in the convolution. The same is true for frame measures, but with certain restrictions. We investigate some general properties of frame measures and their Beurling dimensions. In particular we show that the Beurling dimension is invariant under convolution (with a probability measure) and under a certain type of discretization. Moreover, if a measure admits a frame measure then it admits an atomic one, and hence a weighted Fourier frame. We also construct some examples of frame measures for self-similar measures.