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Non-spectral Problem for Some Self-similar Measures

Part of: Nontrigonometric harmonic analysis Classical measure theory Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  28 August 2019

Ye Wang ,
Xin-Han Dong  and
Yue-Ping Jiang
Ye Wang
Affiliation:
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, P.R. China Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA Email: hnsdwangye@163.com
Xin-Han Dong
Affiliation:
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, P.R. China Email: xhdonghnsd@163.comypjiang731@163.com
Yue-Ping Jiang
Affiliation:
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, P.R. China Email: xhdonghnsd@163.comypjiang731@163.com
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Abstract

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 $L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$.

Keywords

self-similar measurespectral measureorthogonal exponential functionFourier transform

MSC classification

Primary: 28A80: Fractals
Secondary: 42C05: Orthogonal functions and polynomials, general theory 46C05: Hilbert and pre-Hilbert spaces
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 2 , June 2020 , pp. 318 - 327
Copyright
© Canadian Mathematical Society 2019

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Footnotes

The research is supported in part by the NNSF of China (No. 11831007, No.11571099).

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