Orthogonal bases of exponential functions for infinite convolutions

TL;DR

This paper investigates conditions under which infinite convolutions generated by residue systems form spectral measures, revealing key properties related to the structure of the residue sets and their associated polynomials.

Contribution

It establishes necessary and sufficient conditions for the spectrality of infinite convolutions with residue systems, including the role of uniform zero conditions and tightness.

Findings

Spectrality depends on properties of polynomials generated by residue systems.

Uniform discrete zero condition is crucial for spectral measures.

Tightness of the tail measures is necessary for spectrality.

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.