Spectrality of random convolutions generated by finitely many 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 $${\mu }_{\omega ,\{n_k\}}={\delta }_{N_{{\omega }_1}^{-n_1}{B}_{{\omega }_1}}\ast {\delta }_{N_{{\omega }_1}^{-n_1}{N}_{{\omega }_2}^{-n_2}{B}_{{\omega }_2}}\ast \cdots \ast {\delta }_{N_{{\omega }_1}^{-n_1}{N}_{{\omega }_2}^{-n_2}\cdots N_{{\omega }_k}^{-n_k}{B}_{{\omega }_k}}\ast \cdots .$$ 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 $\gcd (B_j-B_j)=1$ 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 $\{e_{\lambda }(x)=e^{2\pi i\lambda x}:\lambda \in {\Lambda }_{\omega ,\{n_k\}}\}$ forms an orthonormal basis for $L^2({\mu }_{\omega ,\{n_k\}})$.

21 pages; final version

References in corpus (2)

• Complex Hadamard matrices and the Spectral Set Conjecture
• On Fuglede's conjecture and the existence of universal spectra

Cited by in corpus (3)

• The generalized Fuglede's conjecture holds for a class of Cantor-Moran measures
• Spectrality of Infinite Convolutions and Random Convolutions
• The Spectrality of Infinite Convolutions in ${\mathbb{R}}^d$