Fourier Frames for the Cantor-4 Set

TL;DR

This paper constructs Fourier frames for the Cantor-4 measure by extending it to a 2D fractal and using Cuntz algebra representations to generate frames through dilation and projection.

Contribution

It introduces a novel dilation-based method to create Fourier frames for fractal measures, expanding the Cantor-4 set to a higher-dimensional fractal.

Findings

Fourier frames are successfully constructed for the Cantor-4 measure.

A dilation and projection method is developed using Cuntz algebra representations.

The approach extends the known Fourier basis to a broader frame setting.

Abstract

The measure supported on the Cantor-4 set constructed by Jorgensen-Pedersen is known to have a Fourier basis, i.e. that it possess a sequence of exponentials which form an orthonormal basis. We construct Fourier frames for this measure via a dilation theory type construction. We expand the Cantor-4 set to a 2 dimensional fractal which admits a representation of a Cuntz algebra. Using the action of this algebra, an orthonormal set is generated on the larger fractal, which is then projected onto the Cantor-4 set to produce a Fourier frame.