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 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 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].

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.

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 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.

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.

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.

scholarly journals Oscillation of Second-Order Differential Equations with Multiple and Mixed Delays under a Canonical Operator

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1323
Author(s):  
Shyam Sundar Santra ◽  
Rami Ahmad El-Nabulsi ◽  
Khaled Mohamed Khedher
Keyword(s):  
Differential Equations ◽  
Necessary Conditions ◽  
Second Order ◽  
Multiple Delays ◽  
Mixed Delays ◽  
Neutral Differential Equations ◽  
Canonical Operator ◽  
Second Order Differential Equations ◽  
The Future ◽  
Sufficient And Necessary Conditions

In this work, we obtained new sufficient and necessary conditions for the oscillation of second-order differential equations with mixed and multiple delays under a canonical operator. Our methods could be applicable to find the sufficient and necessary conditions for any neutral differential equations. Furthermore, we proved the validity of the obtained results via particular examples. At the end of the paper, we provide the future scope of this study.

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