On the orthogonality of measures of different spectral type with respect to twisted Eberlein convolution

TL;DR

This paper demonstrates that under certain conditions, the twisted Eberlein convolution of measures with pure point and continuous spectra results in zero, simplifying the diffraction spectrum of their linear combinations.

Contribution

It establishes conditions under which the twisted Eberlein convolution of measures with different spectral types is zero, clarifying spectral interactions.

Findings

Twisted Eberlein convolution of measures with different spectral types can be zero.

Diffraction spectrum of linear combinations is a simple sum of individual spectra.

Provides conditions on Fourier--Bohr coefficients for spectral orthogonality.

Abstract

In this paper we show that under suitable conditions on their Fourier--Bohr coefficients, the twisted Eberlein convolution of a measure with pure point diffraction spectra and a measure with continuous diffraction spectra is zero. In particular, the diffraction spectrum of a linear combinations of the two measures is simply the linear combinations of the two diffraction spectra with absolute value square coefficients.