Spectrality of random convolutions generated by finitely many Hadamard triples

Abstract

Abstract Let { ( N j , B j , L j ) : 1 ⩽ j ⩽ m } be finitely many Hadamard triples in R . Given a sequence of positive integers { n k } k = 1 ∞ and ω = ( ω k ) k = 1 ∞ ∈ { 1 , 2 , … , m } N , let μ ω , { n k } be the infinite convolution given by μ ω , n k = δ N ω 1 − n 1 B ω 1 ∗ δ N ω 1 − n 1 N ω 2 − n 2 B ω 2 ∗ ⋯ ∗ δ N ω 1 − n 1 N ω 2 − n 2 ⋯ N ω k − n k B ω k ∗ ⋯ . In order to study the spectrality of μ ω , { 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 g c d ( B j − B j ) = 1 for 1 ⩽ j ⩽ m , then all infinite convolutions μ ω , { n k } are spectral measures. This implies that we may find a subset Λ ω , { n k } ⊆ R such that { e λ ( x ) = e 2 π i λ x : λ ∈ Λ ω , { n k } } forms an orthonormal basis for L 2 ( μ ω , { n k } ) .