Spectrality and non-spectrality of the Riesz product measures with three elements in digit sets

By Li-Xiang An, Liu He, Xinggang He. Journal of Functional Analysis, 2019

60 citations recorded by the source corpus.

Abstract Let μ be a Borel probability measure with compact support in R . The μ is called a spectral/Riesz spectral/frame spectral measure if there exists a set Λ ⊂ R such that the family of exponential functions E Λ = { e 2 π i λ x : λ ∈ Λ } forms an orthonormal basis/Riesz basis/frame for L 2 ( μ ) . In this paper we study the spectrality and the non-spectrality of a class of singular measures, which is one of the fundamental works for the analysis on L 2 ( μ ) . For a clear expression, we use the simplest setting in this paper although the most results of ours can be extended to more general cases, even to higher dimensions.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Signal Processing

View Paper: https://www.semanticscholar.org/paper/0abc8971459c9b6364bd2a244e87c5b3eff0a6f8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. J. Miao
2. Wenxia Li
3. Zhiqiang Wang
4. Li-Xiang An
5. Xiaoye Fu
6. Chun-Kit Lai
7. Xinggang He
8. Xin-Rong Dai

In Computer Vision and Pattern Recognition

1. Li-Xiang An
2. Chun-Kit Lai
3. Xiaoye Fu
4. J. Miao
5. Wenxia Li
6. Zhiqiang Wang

In Signal Processing

1. Li-Xiang An
2. Chun-Kit Lai
3. Xiaoye Fu

Citations in the corpus

• On spectral Cantor-Moran measures and a variant of Bourgain's sum of sine problem (2019, background): “ 2 and Bn = {0,3} except n= 1 which equals to {0,1}. From [2], it is known that µ is not a spectral measure even though (Nn,Bn,Ln) are Hadamard triples with Ln = {0,1}. In this example, ν>n = 1 3L|[0,3] for all n>1, the normalized Lebesgue measure supported on [0,3]. Hence, νd>n has a non-e”
• Weak convergence and spectrality of infinite convolutions (2022, background): “In [2], the authors conjectured that in R, if Ak ⊆ [0,+∞) and #Ak = 3 for every k ≥ 1, the assumption (1.”