Orthogonal bases of exponential functions for infinite convolutions

Let μ denot the infinite convolution generated by \(Nₖ,Bₖ)\ₖ₌₁^∞ given by μ =δ_N₁⁻¹B₁δ_(N₁N₂)⁻¹B₂δ_(N₁N₂⋯ Nₖ)⁻¹Bₖ *⋯. where Bₖ is a complete residue system for each integer $k>0$. We write ν>k=δ_Nₖ₊₁⁻¹ Bₖ₊₁ * δ_(Nₖ₊₁ Nₖ₊₂)⁻¹ Bₖ₊₂ * ⋯. Since the elements in Bₖ may have very large absolute values, the infinite convolution may not be compactly supported. In this paper, we study the necessary and sufficient conditions for such infinite convolutions being a spectral measure. Generally, for such infinite convolutions, the necessary conditions for spectrality mainly depend on the properties of the polynomials generated by the complete residue systems. The main result shows that if every Bₖ satisfies uniform discrete zero condition, and \ν>k\ₖ₌₁^∞ is { tight}, then # Bₖ | Nₖ for all integers k≥ 2. For some special complete residue systems ₖ\ₖ₌₁^∞, we provide the necessary and sufficient conditions for μ being a spectral measure.