三元数字集生成的Moran测度的谱性

Abstract

Abstract: Forn≥1,let p_n>1 and D_n={0,a_n,b_n}?Z, where 0 <a_n <b_n <p_n.In this paper we study the spectrality of the moran measure

$\mu:=\delta_{p_1^{-1}\{0,a_1,b_1\}} \ast \delta_{p_1^{-1}p_2^{-1}\{0,a_2,b_2\}} \ast \cdots \ast \delta_{p_1^{-1}p_2^{-1}\cdots p_n^{-1}\{0,a_n,b_n\}} \ast \cdots$ which is generated by the sequence of integers~

$\{p_n\}_{n=1}^\infty$ ~and the sequence of number sets~

$\{D_n\}_{n=1}^\infty.$ We show that when

$\{b_n\}_{n=1}^\infty$ is bounded,μ is a spectral measure if and only if the numbers of factor 3 in the sequence~

$\{\frac{p_1p_2\cdots p_n}{3gcd(a_n,b_n)}\}_{n=1}^\infty$ ~are different from each other and {a_n\gcd(a_n,b_n)},b_n\gcd(a_n,b_n)}≡{1,-1}(mod3)for all n≥1.

Key words: Exponential orthogonal basis, Moran measure, Spectral measure, Spectral

CLC Number:

O174.22

Ting XIONG. Spectrality of moran measures with three-element digit sets[J].Acta mathematica scientia,Series A, , (): 0-0.

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Spectrality of moran measures with three-element digit sets

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