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Let { S d } d = 0 2 N − 1 be an iterated function system defined by S d (x) = (− 1) d ρ (x + d) , x ∈ R , where 0 < ρ < 1 and N is a positive integer. Let μ ρ , D 2 N be a self-similar measure defined by μ ρ , D 2 N (⋅) = 1 2 N ∑ d = 0 2 N − 1 μ ρ , D 2 N (S d − 1 (⋅)). We show that μ ρ , D 2 N is a spectral measure if and only if ρ = p − 1 for some nature number p and 2 N divides p. Moreover, we introduce a useful way to determine the spectrality of the convolution of two infinitely supported measures. [ABSTRACT FROM AUTHOR]

Abstract: For and an integer, let μ be the self-similar measure defined by . We prove that has an exponential orthonormal basis if and only if for some and N divides q. The special case is the Cantor measure with and [16], which was proved recently to be the only spectral measure among the Bernoulli convolutions with [4]. [Copyright &y& Elsevier]