Geodesic
Non-spectral Problem for Some Self-similar Measures
Canadian mathematical bulletin, Tome 63 (2020) no. 2, pp. 318-327

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DOI

Suppose that $0<|\unicode[STIX]{x1D70C}|<1$ and $m\geqslant 2$ is an integer. Let $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}$ be the self-similar measure defined by $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\cdot )=\frac{1}{m}\sum _{j=0}^{m-1}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\unicode[STIX]{x1D70C}^{-1}(\cdot )-j)$. Assume that $\unicode[STIX]{x1D70C}=\pm (q/p)^{1/r}$ for some $p,q,r\in \mathbb{N}^{+}$ with $(p,q)=1$ and $(p,m)=1$. We prove that if $(q,m)=1$, then there are at most $m$ mutually orthogonal exponential functions in $L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$ and $m$ is the best possible. If $(q,m)>1$, then there are any number of orthogonal exponential functions in $L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$.
DOI : 10.4153/S0008439519000304
Mots-clés : self-similar measure, spectral measure, orthogonal exponential function, Fourier transform 
Wang, Ye; Dong, Xin-Han; Jiang, Yue-Ping. Non-spectral Problem for Some Self-similar Measures. Canadian mathematical bulletin, Tome 63 (2020) no. 2, pp. 318-327. doi: 10.4153/S0008439519000304
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     author = {Wang, Ye and Dong, Xin-Han and Jiang, Yue-Ping},
     title = {Non-spectral {Problem} for {Some} {Self-similar} {Measures}},
     journal = {Canadian mathematical bulletin},
     pages = {318--327},
     year = {2020},
     volume = {63},
     number = {2},
     doi = {10.4153/S0008439519000304},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439519000304/}
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