Let An∈M2(ℤ)be integral matrices such that the infinite convolution of Dirac measures with equal weightsμ{A_(n),n≥1}δA_(1)^(-1)D*δA_(1)^(-1)A_(2)^(-2)D*…is a probability measure with compact support,where D={(0,...Let An∈M2(ℤ)be integral matrices such that the infinite convolution of Dirac measures with equal weightsμ{A_(n),n≥1}δA_(1)^(-1)D*δA_(1)^(-1)A_(2)^(-2)D*…is a probability measure with compact support,where D={(0,0)^(t),(1,0)^(t),(0,1)^(t)}is the Sierpinski digit.We prove that there exists a setΛ⊂ℝ2 such that the family{e2πi〈λ,x〉:λ∈Λ} is an orthonormal basis of L^(2)(μ{A_(n),n≥1})if and only if 1/3(1,-1)A_(n)∈Z^(2)for n≥2 under some metric conditions on A_(n).展开更多
The problems of determining the spectrality or non-spectrality of a measure have been received much attention in recent years. One of the non-spectral problems on <span style="white-space:nowrap;"><...The problems of determining the spectrality or non-spectrality of a measure have been received much attention in recent years. One of the non-spectral problems on <span style="white-space:nowrap;"><em>μ<sub>M,D</sub></em></span><sub> </sub>is to estimate the number of orthogonal exponentials in <em>L</em><sup>2</sup><span style="white-space:normal;">(</span><em>μ<sub>M,D<span style="white-space:normal;">)</span></sub></em>. In the present paper, we establish some relations inside the zero set <img src="Edit_2196df81-d10f-4105-a2a9-779f454a56c3.png" width="55" height="23" alt="" /> by the Fourier transform of the self-affine measure <em>μ<sub>M,D</sub></em>. Based on these facts, we show that <em>μ<sub>M,D</sub></em> is a non-spectral measure<em><sub> </sub></em>and there exist at most 4 mutually orthogonal exponential functions in <em style="white-space:normal;"><em style="white-space:normal;">L</em><sup style="white-space:normal;">2</sup><span style="white-space:normal;">(</span><span style="white-space:normal;"></span><em style="white-space:normal;">μ<sub>M,D)</sub></em></em>, where the number 4 is the best possible. This extends several known conclusions.展开更多
The self-affine measure associated with an expanding matrix and a finite digit set is uniquely determined by the self-affine identity with equal weight.The spectral and non-spectral problems on the selfaffine measures...The self-affine measure associated with an expanding matrix and a finite digit set is uniquely determined by the self-affine identity with equal weight.The spectral and non-spectral problems on the selfaffine measures have some surprising connections with a number of areas in mathematics,and have been received much attention in recent years.In the present paper,we shall determine the spectrality and non-spectrality of a class of self-affine measures with decomposable digit sets.We present a method to deal with such case,and clarify the spectrality and non-spectrality of a class of self-affine measures by applying this method.展开更多
基金supported by the National Natural Science Foundation of China (Grant Nos. 12371087, 11971109,11971194, 11672074 and 12271185)supported by the program for Probability and Statistics:Theory and Application (Grant No. IRTL1704)+1 种基金the program for Innovative Research Team in Science and Technology in Fujian Province University (Grant No. IRTSTFJ)supported by Guangdong NSFC (Grant No. 2022A1515011124)
文摘Let An∈M2(ℤ)be integral matrices such that the infinite convolution of Dirac measures with equal weightsμ{A_(n),n≥1}δA_(1)^(-1)D*δA_(1)^(-1)A_(2)^(-2)D*…is a probability measure with compact support,where D={(0,0)^(t),(1,0)^(t),(0,1)^(t)}is the Sierpinski digit.We prove that there exists a setΛ⊂ℝ2 such that the family{e2πi〈λ,x〉:λ∈Λ} is an orthonormal basis of L^(2)(μ{A_(n),n≥1})if and only if 1/3(1,-1)A_(n)∈Z^(2)for n≥2 under some metric conditions on A_(n).
文摘The problems of determining the spectrality or non-spectrality of a measure have been received much attention in recent years. One of the non-spectral problems on <span style="white-space:nowrap;"><em>μ<sub>M,D</sub></em></span><sub> </sub>is to estimate the number of orthogonal exponentials in <em>L</em><sup>2</sup><span style="white-space:normal;">(</span><em>μ<sub>M,D<span style="white-space:normal;">)</span></sub></em>. In the present paper, we establish some relations inside the zero set <img src="Edit_2196df81-d10f-4105-a2a9-779f454a56c3.png" width="55" height="23" alt="" /> by the Fourier transform of the self-affine measure <em>μ<sub>M,D</sub></em>. Based on these facts, we show that <em>μ<sub>M,D</sub></em> is a non-spectral measure<em><sub> </sub></em>and there exist at most 4 mutually orthogonal exponential functions in <em style="white-space:normal;"><em style="white-space:normal;">L</em><sup style="white-space:normal;">2</sup><span style="white-space:normal;">(</span><span style="white-space:normal;"></span><em style="white-space:normal;">μ<sub>M,D)</sub></em></em>, where the number 4 is the best possible. This extends several known conclusions.
基金supported by National Natural Science Foundation of China (Grant Nos.10871123,11171201)
文摘The self-affine measure associated with an expanding matrix and a finite digit set is uniquely determined by the self-affine identity with equal weight.The spectral and non-spectral problems on the selfaffine measures have some surprising connections with a number of areas in mathematics,and have been received much attention in recent years.In the present paper,we shall determine the spectrality and non-spectrality of a class of self-affine measures with decomposable digit sets.We present a method to deal with such case,and clarify the spectrality and non-spectrality of a class of self-affine measures by applying this method.
