ON THE SPECTRA OF A CANTOR MEASURE

By Dorin Ervin Dutkay, Deguang Han, Qiyu Sun. 2008

173 citations recorded by the source corpus.

Abstract We analyze all orthonormal bases of exponentials on the Cantor set defined by Jorgensen and Pedersen in J. Anal. Math. 75 (1998) 185–228. A complete characterization for all maximal sets of orthogonal exponentials is obtained by establishing a one-to-one correspondence with the spectral labelings of the infinite binary tree. With the help of this characterization we obtain a sufficient condition for a spectral labeling to generate a spectrum (an orthonormal basis). This result not only provides us an easy and efficient way to construct various of new spectra for the Cantor measure but also extends many previous results in the literature. In fact, most known examples of orthonormal bases of exponentials correspond to spectral labelings satisfying this sufficient condition. We also obtain two new conditions for a labeling tree to generate a spectrum when other digits (digits not necessarily in { 0 , 1 , 2 , 3 } ) are used in the base 4 expansion of integers and when bad branches are allowed in the spectral labeling. These new conditions yield new examples of spectra and in particular lead to a surprizing example which shows that a maximal set of orthogonal exponentials is not necessarily an orthonormal basis.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/19f659102276d2a35b4bf6c4bf12771f74707f5c

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In Applied Mathematics

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

In Mathematical Physics

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

In Statistical and Nonlinear Physics

1. Xinggang He
2. Ka-Sing Lau
3. Xin-Rong Dai
4. De-Jun Feng

Citations in the corpus

• Spectrality of Self-Similar Tiles (2013, background): “The problem also led to the study of spectral property of discrete integer tiles [12,33,40] and the Cantor-type singularly continuous measures [1,4–6,8,9,18,23,29,41].”
• When does a Bernoulli convolution admit a spectrum (2012, methodology): “We remark that different labeling with similar flavor is used in [5] to study spectra for the Bernoulli convolution whose contraction rate is the reciprocal of an even integer.”
• Spectra of Cantor measures (2014, background): “The fi rst attempt to classify those maximal orthogonal sets and spectra was made in [5], wh ere certain tree structure of a maximal orthogonal set of the Cantor measure μ2,4 was discovered.”
• Spectra of a class of self-affine measures (2011, unknown): “The other results concerning the spectral property of the Bernoulli convolutions have been obtained in [12,9,7,6,13].”