Orthogonal bases of exponential functions for infinite convolutions

Abstract

Let $\mu$ denot the infinite convolution generated by $\{(N_k,B_k)\}_{k=1}^\infty$ given by $$ \mu =\delta_{{N_1}^{-1}B_1}\ast\delta_{(N_1N_2)^{-1}B_2}\ast\dots\ast\delta_{(N_1N_2\cdots N_k)^{-1}B_k} *\cdots. $$ where $B_k$ is a complete residue system for each integer $k>0$. We write $$ \nu_{>k}=\delta_{N_{k+1}^{-1} B_{k+1}} * \delta_{(N_{k+1} N_{k+2})^{-1} B_{k+2}} * \cdots. $$ Since the elements in $B_k$ 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 $B_k$ satisfies uniform discrete zero condition, and $\{\nu_{>k}\}_{k=1}^\infty$ is {\it tight}, then $\# B_k | N_k$ for all integers $k\geq 2$. For some special complete residue systems $\{B_k\}_{k=1}^\infty$, we provide the necessary and sufficient conditions for $\mu$ being a spectral measure.