2026/07/17 by Xin Yang, Wei-Jie Wang
#math.FA
Previously, An \citeAL01 showed that the self-similar measure μ generated by a product-form Hadamard triple is a spectral measure. In this paper, we study its spectral eigenvalue problem. A set A⊂\mathbb R is called a spectral eigenvalue set of μ if there exists a spectrum Λ of μ such that aΛ is a spectrum of μ for every a∈ A. We introduce the Product-form Hadamard multiplier set T_*, and prove that for any s∈ [0,(log #D)/(log N)], the spectral eigensubspace V(s)(μN,D,T_*):=\Λ:t Λ is a spectrum of μ for all t \inT_* and dimBe(Λ)=s\ has the cardinality of the continuum. This result allows us to show that for the four-digit self-similar measures, a real number t is a spectral eigenvalue if and only if t ∈ \(u)/(v):u,v∈ 2ℤ+1\. And for any subset S of ℝ is a spectral eigenvalue set if and only if S ⊂ t-1 (2ℤ+1) for some t∈ 2ℤ+1.