Spectrality of convolution and product measures: a block decomposition approach

Abstract

We study the spectrality of convolution and product measures. Let~$μ$~be a compactly supported Borel probability measure on~$\Bbb{R}^d$~such that ~$\mathcal{Z}(\widehatμ)\cup\{0\}=T$~is an additive subgroup, and let~$ν$~be a Borel probability measure on~$\Bbb{R}^d$. We prove that~$μ*ν$~is spectral if and only if~$T$~is a spectrum of~$μ$, ~$ν$~is spectral, and~$suppν$~is contained in a translate of the annihilator of~$T$. Moreover, every spectrum of~$μ*ν$~decomposes into full cosets of~$T$, and every choice of one point from each coset is a spectrum of~$ν$. We also obtain a convolution-factor criterion which strengthens a result of Wu. As applications, we characterize the spectrality of~$L_{[0,1]}*ν$~and prove that the spectrality of~$L_{[0,1]^d}*ν$~implies that of~$ν$~without a compact support assumption. For product measures, we give necessary and sufficient conditions for an orthogonal set to be a spectrum by relating its block structure to spectra of the factors. No compact support assumptions are required.