Spectrality of random convolutions generated by finitely many Hadamard triples

TL;DR

This paper investigates the spectral properties of infinite convolutions generated by finitely many Hadamard triples, establishing conditions under which these convolutions are spectral measures with orthonormal bases.

Contribution

It proves spectrality of infinite convolutions from Hadamard triples under equi-positivity and gcd conditions, extending understanding of spectral measures in fractal analysis.

Findings

All such convolutions are spectral measures under gcd condition.

Existence of orthonormal bases for the associated L^2 spaces.

General spectrality results for infinite convolutions generated by Hadamard triples.

Abstract

Let $\{(N_j, B_j, L_j): 1 \le j \le m\}$ be finitely many Hadamard triples in $\mathbb{R}$. Given a sequence of positive integers $\{n_k\}_{k=1}^\infty$ and $\omega=(\omega_k)_{k=1}^\infty \in \{1,2,\cdots, m\}^\mathbb{N}$, let $\mu_{\omega,\{n_k\}}$ be the infinite convolution given by $$\mu_{\omega,\{n_k\}} = \delta_{N_{\omega_1}^{-n_1} B_{\omega_1}} * \delta_{N_{\omega_1}^{-n_1} N_{\omega_2}^{-n_2} B_{\omega_2}} * \cdots * \delta_{N_{\omega_1}^{-n_1} N_{\omega_2}^{-n_2} \cdots N_{\omega_k}^{-n_k} B_{\omega_k} }* \cdots. $$ In order to study the spectrality of $\mu_{\omega,\{ n_k\}}$, we first show the spectrality of general infinite convolutions generated by Hadamard triples under the equi-positivity condition. Then by using the integral periodic zero set of Fourier transform we show that if $\mathrm{gcd}(B_j - B_j)=1$ for $1 \le j \le m$, then all infinite convolutions $\mu_{\omega,\{n_k\}}$ are spectral measures. This implies that we may find a subset $\Lambda_{\omega,\{n_k\}}\subseteq \mathbb{R}$ such that $\big\{ e_\lambda(x) = e^{2\pi i \lambda x}: \lambda \in \Lambda_{\omega,\{n_k\}} \big\}$ forms an orthonormal basis for $L^2(\mu_{\omega,\{ n_k\}})$.