Spectral property of Cantor measures with consecutive digits
By Xin-Rong Dai, Xinggang He, Chun-Kit Lai. 2012
189 citations recorded by the source corpus.
Abstract We consider equally-weighted Cantor measures μ q , b arising from iterated function systems of the form b − 1 ( x + i ) , i = 0 , 1 , … , q − 1 , where q b . We classify the ( q , b ) so that they have infinitely many mutually orthogonal exponentials in L 2 ( μ q , b ) . In particular, if q divides b , the measures have a complete orthogonal exponential system and hence spectral measures. Improving the construction by Dutkay et al. (2009) [3] , we characterize all the maximal orthogonal sets Λ when q divides b via a maximal mapping on the q -adic tree in which all elements in Λ are represented uniquely in finite b -adic expansions and we can separate the maximal orthogonal sets into two types: regular and irregular sets. For a regular maximal orthogonal set, we show that its completeness in L 2 ( μ q , b ) is crucially determined by the certain growth rate of non-zero digits in the tail of the b -adic expansions of the elements. Furthermore, we exhibit complete orthogonal exponentials with zero Beurling dimensions. These examples show that the technical condition in Theorem 3.5 of Dutkay et al. (2011) [4] cannot be removed. For an irregular maximal orthogonal set, we show that under some conditions, its completeness is equivalent to that of the corresponding regularized mapping.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/2edce4ae1b81384f80f6ed18915782e6ddf0d168
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In Applied Mathematics
In Mathematical Physics
Citations in the corpus
- Spectra of Cantor measures (2014, methodology): “Spectra of the Cantor measure μb,q have their upper Beurling dimension less than log q/ logb [7], but unlike Fourier frames on the unit interval [23], they could be arbitrarily sparse a nd have zero upper Beurling dimension [3].”
- On spectral Cantor-Moran measures and a variant of Bourgain's sum of sine problem (2019, background): “te convolution, Fourier frames, and sum of sine functions. The research of Lixiang An and Xiaoye Fu is supported by NSFC grant 11601175, 11401205. 1 2 LIXIANG AN, XIAOYE FU, AND CHUN-KIT LAI measures [9, 10, 11], as well as the convergence properties of the associated Fourier series [38, 39, 13] and”
- Spectral measures with arbitrary Hausdorff dimensions (2014, methodology): “In fact, it is a very challenging problem to find appropriate su fficient conditions, see [4, 5, 9] and references therein for recent advances.”