ORTHOGONAL EXPONENTIAL FUNCTIONS OF THE PLANAR SELF-AFFINE MEASURES WITH FOUR DIGITS

Fractals ◽  
2020 ◽  
Vol 28 (01) ◽  
pp. 2050016
Author(s):  
JUAN SU ◽  
ZHI-YONG WANG ◽  
MING-LIANG CHEN
Keyword(s):  
Complete Characterization ◽  
The Self ◽  
Exponential Functions ◽  
Digit Set ◽  
Matrix Formula ◽  
Orthogonal Set

For the self-affine measure [Formula: see text] generated by an expanding matrix [Formula: see text] and an integer digit set [Formula: see text] with [Formula: see text], Su et al. proved that if [Formula: see text], then [Formula: see text] contains an infinite orthogonal set of exponential functions if and only if [Formula: see text] [J. Su, Y. Liu and J. C. Liu, Non-spectrality of the planar self-affine measures with four-element digit sets, Fractals (2019), https://doi.org/10.1142/S0218348X19501159 ]. In this paper, we show that the above conclusion also holds for [Formula: see text]. So, a complete characterization of [Formula: see text] containing an infinite orthogonal set of exponential functions is given.

ORTHOGONAL EXPONENTIAL FUNCTIONS OF SELF-AFFINE MEASURES IN ℝn

Fractals ◽  
2019 ◽  
Vol 27 (03) ◽  
pp. 1950029 ◽  
Author(s):  
ZHI-MIN WANG ◽  
XIN-HAN DONG ◽  
ZHI-YONG WANG
Keyword(s):  
Positive Integer ◽  
The Self ◽  
Real Matrix ◽  
Exponential Functions ◽  
Digit Set ◽  
Matrix Formula ◽  
Orthogonal Set

For a positive integer [Formula: see text], let [Formula: see text]. Let the self-affine measure [Formula: see text] be generated by an expanding real matrix [Formula: see text] and a finite digit set [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text] is the [Formula: see text]th column of the [Formula: see text] identical matrix [Formula: see text], [Formula: see text]. In this paper, we prove that [Formula: see text] contains an infinite orthogonal set of exponential functions if and only if there exists [Formula: see text] such that [Formula: see text] for some [Formula: see text] with [Formula: see text] and [Formula: see text].

THE NON-SPECTRAL PROPERTY OF A CLASS OF PLANAR SELF-SIMILAR MEASURES

Fractals ◽  
2020 ◽  
Vol 28 (05) ◽  
pp. 2050091
Author(s):  
YANG-YANG XU ◽  
JING-CHENG LIU
Keyword(s):  
Spectral Property ◽  
The Self ◽  
Real Matrix ◽  
Exponential Functions ◽  
Similar Measure ◽  
Digit Set ◽  
Matrix Formula ◽  
Self Similar ◽  
Orthogonal Set

Let the self-similar measure [Formula: see text] be generated by an expanding real matrix [Formula: see text] and a digit set [Formula: see text] in space [Formula: see text]. In this paper, we only consider [Formula: see text] and the case [Formula: see text] is similar. We show that there exists an infinite orthogonal set of exponential functions in [Formula: see text] if and only if [Formula: see text] for some [Formula: see text] with [Formula: see text]. Furthermore, for the cases that [Formula: see text] does not admit any infinite orthogonal set of exponential functions, the exact cardinality of orthogonal exponential functions in [Formula: see text] is given.

Orthogonal exponentials of self-affine measures on ℝn

2020 ◽  
Vol 31 (08) ◽  
pp. 2050063
Author(s):  
Juan Su ◽  
Ming-Liang Chen
Keyword(s):  
The Self ◽  
Exponential Functions ◽  
Digit Set ◽  
Orthogonal Exponentials ◽  
Matrix Formula ◽  
Finite Integer ◽  
Orthogonal Set

Let the self-affine measure [Formula: see text] be generated by an expanding matrix [Formula: see text] and a finite integer digit set [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text]. In this paper, we show that if [Formula: see text] for an integer [Formula: see text], then [Formula: see text] admits an infinite orthogonal set of exponential functions if and only if there exists [Formula: see text] such that [Formula: see text] for some [Formula: see text] with [Formula: see text] and [Formula: see text].

NON-SPECTRALITY OF THE PLANAR SELF-AFFINE MEASURES WITH FOUR-ELEMENT DIGIT SETS

Fractals ◽  
2019 ◽  
Vol 27 (07) ◽  
pp. 1950115 ◽  
Author(s):  
JUAN SU ◽  
YAO LIU ◽  
JING-CHENG LIU
Keyword(s):  
Exponential Functions ◽  
Integer Matrix ◽  
Digit Set ◽  
Number Formula ◽  
Matrix Formula ◽  
Orthogonal Set

In this paper, we consider the non-spectrality of the planar self-affine measures [Formula: see text] generated by an expanding integer matrix [Formula: see text] and a four-element digit set [Formula: see text] We show that [Formula: see text] contains an infinite orthogonal set of exponential functions if and only if [Formula: see text]. Moreover, if [Formula: see text], then there exist at most [Formula: see text] mutually orthogonal exponential functions in [Formula: see text], and the number [Formula: see text] is the best.

Spectral property of the planar self-affine measures with three-element digit sets

2020 ◽  
Vol 32 (3) ◽  
pp. 673-681
Author(s):  
Ming-Liang Chen ◽  
Jing-Cheng Liu ◽  
Juan Su
Keyword(s):  
Spectral Property ◽  
Necessary Conditions ◽  
The Self ◽  
Real Matrix ◽  
Exponential Functions ◽  
Digit Set ◽  
Sufficient And Necessary Conditions ◽  
Orthogonal Set

AbstractLet the self-affine measure {\mu_{M,D}} be generated by an expanding real matrix {M=\operatorname{diag}(\rho_{1}^{-1},\rho_{2}^{-1})} and an integer digit set {D=\{(0,0)^{t},(\alpha_{1},\alpha_{2})^{t},(\beta_{1},\beta_{2})^{t}\}} with {\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\neq 0}. In this paper, the sufficient and necessary conditions for {L^{2}(\mu_{M,D})} to contain an infinite orthogonal set of exponential functions are given.

THE CARDINALITY OF ORTHOGONAL EXPONENTIAL FUNCTIONS ON THE SPATIAL SIERPINSKI GASKET

Fractals ◽  
2019 ◽  
Vol 27 (04) ◽  
pp. 1950056 ◽  
Author(s):  
JIA ZHENG ◽  
JING-CHENG LIU ◽  
MING-LIANG CHEN
Keyword(s):  
Sierpinski Gasket ◽  
Diagonal Matrix ◽  
The Self ◽  
The Other ◽  
Exponential Functions ◽  
Sierpiński Gasket ◽  
Digit Set ◽  
Orthogonal Exponentials ◽  
Matrix Formula

For the self-affine measures [Formula: see text] generated by a diagonal matrix [Formula: see text] with entries [Formula: see text] and the digit set [Formula: see text], Li showed that there exists an infinite orthogonal exponential functions set in [Formula: see text] if and only if at least two of the three numbers [Formula: see text] are even, and conjectured that there exist at most four mutually orthogonal exponential functions in [Formula: see text] for other cases [J.-L. Li, Non-spectrality of self-affine measures on the spatial Sierpinski gasket, J. Math. Anal. Appl. 432 (2015) 1005–1017]. This conjecture was disproved by Wang and Li through constructing a class of the eight-element orthogonal exponential functions [Q. Wang and J.-L. Li, There are eight-element orthogonal exponentials on the spatial Sierpinski gasket, Math. Nachr. 292 (2019) 211–226]. In this paper, we will show that there are any number of orthogonal exponential functions in [Formula: see text] if two of the three numbers [Formula: see text] are different odd and the other is even.

On the spectrality of self-affine measures with four digits on ℝ2

2021 ◽  
pp. 2150004
Author(s):  
Ming-Liang Chen ◽  
Zhi-Hui Yan
Keyword(s):  
Spectral Property ◽  
Orthonormal Basis ◽  
Zero Set ◽  
Real Matrix ◽  
Exponential Functions ◽  
Bernoulli Convolution ◽  
Digit Set ◽  
Similar Characterization ◽  
Equivalent Characterization ◽  
Matrix Formula

In this paper, we study the spectral property of the self-affine measure [Formula: see text] generated by an expanding real matrix [Formula: see text] and the four-element digit set [Formula: see text]. We show that [Formula: see text] is a spectral measure, i.e. there exists a discrete set [Formula: see text] such that the collection of exponential functions [Formula: see text] forms an orthonormal basis for [Formula: see text], if and only if [Formula: see text] for some [Formula: see text]. A similar characterization for Bernoulli convolution is provided by Dai [X.-R. Dai, When does a Bernoulli convolution admit a spectrum? Adv. Math. 231(3) (2012) 1681–1693], over which [Formula: see text]. Furthermore, we provide an equivalent characterization for the maximal bi-zero set of [Formula: see text] by extending the concept of tree-mapping in [X.-R. Dai, X.-G. He and C. K. Lai, Spectral property of Cantor measures with consecutive digits, Adv. Math. 242 (2013) 187–208]. We also extend these results to the more general self-affine measures.

scholarly journals Analysis of the limiting spectral measure of large random matrices of the separable covariance type

2014 ◽  
Vol 03 (04) ◽  
pp. 1450016 ◽  
Author(s):  
Romain Couillet ◽  
Walid Hachem
Keyword(s):  
Random Matrix ◽  
Spectral Measure ◽  
Statistical Estimation ◽  
Complete Characterization ◽  
Nonnegative Matrices ◽  
Spectral Measures ◽  
Estimation Problems ◽  
Matrix Formula ◽  
Limiting Spectral Measure

Consider the random matrix [Formula: see text] where D and [Formula: see text] are deterministic Hermitian nonnegative matrices with respective dimensions N × N and n × n, and where X is a random matrix with independent and identically distributed centered elements with variance 1/n. Assume that the dimensions N and n grow to infinity at the same pace, and that the spectral measures of D and [Formula: see text] converge as N, n → ∞ towards two probability measures. Then it is known that the spectral measure of ΣΣ* converges towards a probability measure μ characterized by its Stieltjes transform. In this paper, it is shown that μ has a density away from zero, this density is analytical wherever it is positive, and it behaves in most cases as [Formula: see text] near an edge a of its support. In addition, a complete characterization of the support of μ is provided. Aside from its mathematical interest, the analysis underlying these results finds important applications in a certain class of statistical estimation problems.

scholarly journals Self-adjoint Extensions of Schrödinger Operators with ?-magnetic Fields on Riemannian Manifolds

10.14311/1271 ◽  
2010 ◽  
Vol 50 (5) ◽  
Author(s):  
T. Mine
Keyword(s):  
Magnetic Field ◽  
Riemannian Manifold ◽  
Magnetic Fields ◽  
Boundary Conditions ◽  
Riemannian Manifolds ◽  
Complete Characterization ◽  
The Self ◽  
Minimal Operator ◽  
The Magnetic Field

We consider the magnetic Schr¨odinger operator on a Riemannian manifold M. We assume the magnetic field is given by the sum of a regular field and the Dirac δ measures supported on a discrete set Γ in M. We give a complete characterization of the self-adjoint extensions of the minimal operator, in terms of the boundary conditions. The result is an extension of the former results by Dabrowski-Šťoviček and Exner-Šťoviček-Vytřas.

Non-spectrality of self-affine measures on the three-dimensional Sierpinski gasket

2019 ◽  
Vol 31 (6) ◽  
pp. 1447-1455 ◽  
Author(s):  
Zheng-Yi Lu ◽  
Xin-Han Dong ◽  
Peng-Fei Zhang
Keyword(s):  
Sierpinski Gasket ◽  
Three Dimensional ◽  
Diagonal Matrix ◽  
Exponential Functions ◽  
Sierpiński Gasket ◽  
Digit Set ◽  
Orthogonal Set

AbstractLet {\mu_{M,D}} be a self-affine measure generated by an expanding diagonal matrix {M\in M_{3}(\mathbb{R})} with entries {\rho_{1},\rho_{2},\rho_{3}} and the digit set {D=\{(0,0,0)^{t},(1,0,0)^{t},(0,1,0)^{t},(0,0,1)^{t}\}}. In this paper, we prove that for any {\rho_{1},\rho_{2},\rho_{3}\in(1,\infty)}, if {\rho_{1},\rho_{2},\rho_{3}\in\{\pm x^{\frac{1}{r}}:x\in\mathbb{Q}^{+},r\in% \mathbb{Z}^{+}\}}, then {L^{2}(\mu_{M,D})} contains an infinite orthogonal set of exponential functions if and only if there exist two numbers of {\rho_{1},\rho_{2},\rho_{3}} that are in the set {\{\pm(\frac{p}{q})^{\frac{1}{r}}:p\in 2\mathbb{Z}^{+},q\in 2\mathbb{Z}^{+}-1% \text{ and }r\in\mathbb{Z}^{+}\}}. In particular, if {\rho_{1},\rho_{2},\rho_{3}\in\{\frac{p}{q}:p,q\in 2\mathbb{Z}+1\}}, then there exist at most 4 mutually orthogonal exponential functions in {L^{2}(\mu_{M,D})}, and the number 4 is the best possible.

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