On Spectral Cantor Measures

By Izabella Laba, Yang Wang. 2001

302 citations recorded by the source corpus.

A probability measure in Rd is called a spectral measure if it has an orthonormal basis consisting of exponentials. In this paper, we study spectral Cantor measures. We establish a large class of such measures, and give a necessary and sufficient condition on the spectrum of a spectral Cantor measure. These results extend the studies by Jorgensen and Pedersen (J. Anal. Math.75 (1998), 185–228) and Strichartz (J. D'Analyse Math.81 (2000), 209–238).

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f25b84d54dc191f74979a8e346458ad79b61d69e

How researchers see this paper

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In Applied Mathematics

1. Xin-Rong Dai
2. Xinggang He
3. Chun-Kit Lai
4. Li-Xiang An
5. Qiyu Sun
6. Xiaoye Fu
7. Ka-Sing Lau
8. Dorin Ervin Dutkay

In Computational Theory and Mathematics

1. Jian-Lin Li
2. Xinggang He
3. Li-Xiang An
4. Chun-Kit Lai
5. Ka-Sing Lau
6. Xiaoye Fu
7. J. Miao
8. Qiyu Sun

In Mathematical Physics

1. Xin-Rong Dai
2. Xinggang He
3. Chun-Kit Lai
4. Li-Xiang An
5. Qiyu Sun
6. Ka-Sing Lau
7. Xiaoye Fu
8. Dorin Ervin Dutkay

Citations in the corpus

• Spectral measures with arbitrary Hausdorff dimensions (2014, background): “Since then, various singular spectral measures on self-similar/self-affine fractal sets have been found, see for instance [3–6,9,11,15–17,19,21,22,24,26,27,30,34].”
• Spectral property of the Bernoulli convolutions (2008, unknown): “MSC: primary 42C05, 42A65; secondary 28A78, 28A80 Keywords: Bernoulli convolution; Minimal polynomial; Orthogonality; Spectral set; Self-similarity”
• Spectra of a class of self-affine measures (2011, background): “1 holds in the dimension n = 1 [14], in the case when |D| = |S| = |det(M)| [17] and in the case when |det(M)| is a prime number [16], it also holds in higher dimensions with additional conditions [1,4].”
• On Fourier frame of absolutely continuous measures (2011, background): “It is also conjectured that there are orthonormal complex exponentials if and only if λ = 1/2n [15].”
• Weak convergence and spectrality of infinite convolutions (2022, result): “[42] Y. Liu, Y. Wang, The uniformity of non-uniform Gabor bases, Adv.”