The Existences of Frame Wavelet Set

2013 ◽  
Vol 347-350 ◽  
pp. 2841-2845
Author(s):  
Wan She Li ◽  
Hong Xia Zhao
Keyword(s):  
Equivalent Condition ◽  
Wavelet Sets ◽  
Wavelet Set ◽  
Expansive Matrix ◽  
Equivalent Description

In this paper, Let be a real expansive matrix, it mainly discusses the existences of frame wavelet set, we discuss the characterization of frame wavelet sets in, and several examples are presented, in order to deepen the understanding of frame wavelet set, which gives the two related theorems; we try to use an equivalent condition to describe frame scale sets, and give an equivalent description about a normalized frame scale set.

Scaling sets and generalized scaling sets on Cantor dyadic group

2020 ◽  
Vol 18 (04) ◽  
pp. 2050019
Author(s):  
Prasadini Mahapatra ◽  
Divya Singh
Keyword(s):  
Abelian Groups ◽  
Real Case ◽  
Locally Compact ◽  
Wavelet Sets ◽  
Locally Compact Abelian Groups ◽  
Dyadic Group ◽  
Compact Abelian Groups ◽  
Cantor Dyadic Group

Scaling and generalized scaling sets determine wavelet sets and hence wavelets. In real case, wavelet sets were proved to be an important tool for the construction of MRA as well as non-MRA wavelets. However, any result related to scaling/generalized scaling sets is not available in case of locally compact abelian groups. This paper gives a characterization of scaling sets and its generalized version along with relevant examples in dual Cantor dyadic group [Formula: see text]. These results can further be generalized to arbitrary locally compact abelian groups.

scholarly journals A characterization of topologically completely positive entropy for shifts of finite type

2013 ◽  
Vol 34 (6) ◽  
pp. 2054-2065 ◽  
Author(s):  
RONNIE PAVLOV
Keyword(s):  
Dynamical System ◽  
Finite Type ◽  
American Mathematical Society ◽  
Equivalent Condition ◽  
Positive Entropy ◽  
Completely Positive ◽  
Shift Of Finite Type ◽  
Formula Group ◽  
Zero Entropy

AbstractA topological dynamical system was defined by Blanchard [Fully Positive Topological Entropy and Topological Mixing (Symbolic Dynamics and Applications (in honor of R. L. Adler), 135). American Mathematical Society Contemporary Mathematics, Providence, RI, 1992, pp. 95–105] to have topologically completely positive entropy (or TCPE) if its only zero entropy factor is the dynamical system consisting of a single fixed point. For ${ \mathbb{Z} }^{d} $ shifts of finite type, we give a simple condition equivalent to having TCPE. We use our characterization to derive a similar equivalent condition to TCPE for the subclass of ${ \mathbb{Z} }^{d} $ group shifts, which was proved by Lind and Schmidt in the abelian case [Homoclinic points of algebraic ${ \mathbb{Z} }^{d} $-actions. J. Amer. Math. Soc. 12(4) (1999), 953–980] and by Boyle and Schraudner in the general case [${ \mathbb{Z} }^{d} $ group shifts and Bernoulli factors. Ergod. Th. & Dynam. Sys. 28(2) (2008), 367–387]. We also give an example of a ${ \mathbb{Z} }^{2} $ shift of finite type which has TCPE but is not even topologically transitive, and prove a result about block gluing ${ \mathbb{Z} }^{d} $ SFTs motivated by our characterization of TCPE.

ON WAVELET INDUCED ISOMORPHISMS

2010 ◽  
Vol 08 (03) ◽  
pp. 359-371 ◽  
Author(s):  
DIVYA SINGH
Keyword(s):  
Fixed Point ◽  
Point Sets ◽  
Wavelet Sets ◽  
One Dimensional ◽  
Dyadic Wavelet ◽  
Wavelet Set ◽  
Fixed Point Sets

While constructing a dyadic wavelet set through an approach which is purely set-theoretic, Ionascu observed that a dyadic one-dimensional wavelet set W gives rise to a specific measurable, bijective, piecewise increasing selfmap [Formula: see text] on [0, 1) and termed it to be a wavelet induced isomorphism. Further, he found that such maps provide wavelet sets which, in turn, characterize wavelet sets. In this paper, we consider two-interval, three-interval and symmetric four-interval wavelet sets and determine their wavelet induced isomorphisms. Also, fixed point sets of [Formula: see text] are determined for these wavelet sets.

scholarly journals The Wold-Type Decomposition for m-Isometries

2021 ◽  
Author(s):  
Jakub Kośmider
Keyword(s):  
Composition Operators ◽  
Directed Graphs ◽  
Equivalent Condition ◽  
Weighted Shifts ◽  
Type Decomposition

AbstractThe aim of this paper is to study the Wold-type decomposition in the class of m-isometries. One of our main results establishes an equivalent condition for an analytic m-isometry to admit the Wold-type decomposition for $$m\ge 2$$ m ≥ 2 . In particular, we introduce the k-kernel condition which we use to characterize analytic m-isometric operators which are unitarily equivalent to unilateral operator valued weighted shifts for $$m\ge 2$$ m ≥ 2 . As a result, we also show that m-isometric composition operators on directed graphs with one circuit containing only one element are not unitarily equivalent to unilateral weighted shifts. We also provide a characterization of m-isometric unilateral operator valued weighted shifts with positive and commuting weights.

scholarly journals Molecular Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2216
Author(s):  
Jun Liu ◽  
Long Huang ◽  
Chenlong Yue
Keyword(s):  
Hardy Space ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Lebesgue Space ◽  
Linear Operators ◽  
Mixed Norm ◽  
Expansive Matrix

Let p→∈(0,∞)n be an exponent vector and A be a general expansive matrix on Rn. Let HAp→(Rn) be the anisotropic mixed-norm Hardy spaces associated with A defined via the non-tangential grand maximal function. In this article, using the known atomic characterization of HAp→(Rn), the authors characterize this Hardy space via molecules with the best possible known decay. As an application, the authors establish a criterion on the boundedness of linear operators from HAp→(Rn) to itself, which is used to explore the boundedness of anisotropic Calderón–Zygmund operators on HAp→(Rn). In addition, the boundedness of anisotropic Calderón–Zygmund operators from HAp→(Rn) to the mixed-norm Lebesgue space Lp→(Rn) is also presented. The obtained boundedness of these operators positively answers a question mentioned by Cleanthous et al. All of these results are new, even for isotropic mixed-norm Hardy spaces on Rn.

NON-MSF A-WAVELETS FROM A-WAVELET SETS

2013 ◽  
Vol 11 (01) ◽  
pp. 1350002
Author(s):  
NIRAJ K. SHUKLA
Keyword(s):  
Approximation Theory ◽  
Multiresolution Analysis ◽  
Dimension Function ◽  
Wavelet Sets ◽  
Wavelet Set ◽  
Vanderbilt University ◽  
Theory X

Generalizing the result of Bownik and Speegle [Approximation Theory X: Wavelets, Splines and Applications, Vanderbilt University Press, pp. 63–85, 2002], we provide plenty of non-MSF A-wavelets with the help of a given A-wavelet set. Further, by showing that the dimension function of the non-MSF A-wavelet constructed through an A-wavelet set W coincides with the dimension function of W, we conclude that the non-MSF A-wavelet and the A-wavelet set through which it is constructed possess the same nature as far as the multiresolution analysis is concerned. Some examples of non-MSF d-wavelets and non-MSF A-wavelets are also provided. As an illustration we exhibit a pathwise connected class of non-MSF non-MRA wavelets sharing the same wavelet dimension function.

SMOOTH PARSEVAL FRAMES FOR L2(ℝ) AND GENERALIZATIONS TO L2(ℝd)

2013 ◽  
Vol 11 (06) ◽  
pp. 1350047
Author(s):  
EMILY J. KING
Keyword(s):  
Higher Dimensions ◽  
Wavelet Sets ◽  
Parseval Frame ◽  
Wavelet Set ◽  
Parseval Frames ◽  
Schwartz Class ◽  
Frame Wavelets ◽  
Class Of Functions ◽  
Parseval Frame Wavelet ◽  
Provided Examples

Wavelet set wavelets were the first examples of wavelets that may not have associated multiresolution analyses. Furthermore, they provided examples of complete orthonormal wavelet systems in L2(ℝd) which only require a single generating wavelet. Although work had been done to smooth these wavelets, which are by definition discontinuous on the frequency domain, nothing had been explicitly done over ℝd, d > 1. This paper, along with another one cowritten by the author, finally addresses this issue. Smoothing does not work as expected in higher dimensions. For example, Bin Han's proof of existence of Schwartz class functions which are Parseval frame wavelets and approximate Parseval frame wavelet set wavelets does not easily generalize to higher dimensions. However, a construction of wavelet sets in [Formula: see text] which may be smoothed is presented. Finally, it is shown that a commonly used class of functions cannot be the result of convolutional smoothing of a wavelet set wavelet.

On wavelet induced isomorphisms for joint (d, -d)-dilation wavelet and multiwavelet sets

2015 ◽  
Vol 13 (05) ◽  
pp. 1550034
Author(s):  
Pooja Singh ◽  
Dania Masood
Keyword(s):  
Fixed Point ◽  
Point Sets ◽  
Wavelet Sets ◽  
Dyadic Wavelet ◽  
Wavelet Set ◽  
Almost Everywhere ◽  
Real Anal ◽  
New Construction ◽  
Fixed Point Sets

For a dyadic wavelet set W, Ionascu [A new construction of wavelet sets, Real Anal. Exchange28 (2002) 593–610] obtained a measurable self-bijection on the interval [0, 1), called the wavelet induced isomorphism of [0, 1), denoted by [Formula: see text]. Extending the result for a d-dilation wavelet set, we characterize a joint (d, -d)-dilation wavelet set, where |d| is an integer greater than 1, in terms of wavelet induced isomorphisms. Its analogue for a joint (d, -d)-dilation multiwavelet set has also been provided. In addition, denoting by [Formula: see text], the wavelet induced isomorphism associated with a d-dilation wavelet set W, we show that for a joint (d, -d)-dilation wavelet set W, the measures of the fixed point sets of [Formula: see text] and [Formula: see text] are equal almost everywhere.

scholarly journals The Alternating Sign Matrix Polytope

10.37236/130 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Jessica Striker
Keyword(s):  
Convex Hull ◽  
Complete Characterization ◽  
Stochastic Matrices ◽  
Von Neumann ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Alternating Sign Matrix ◽  
Sign Matrix ◽  
Equivalent Description

We define the alternating sign matrix polytope as the convex hull of $n\times n$ alternating sign matrices and prove its equivalent description in terms of inequalities. This is analogous to the well known result of Birkhoff and von Neumann that the convex hull of the permutation matrices equals the set of all nonnegative doubly stochastic matrices. We count the facets and vertices of the alternating sign matrix polytope and describe its projection to the permutohedron as well as give a complete characterization of its face lattice in terms of modified square ice configurations. Furthermore we prove that the dimension of any face can be easily determined from this characterization.

scholarly journals Minimally Supported Frequency (MSF) d-Dilation Wavelets

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1284
Author(s):  
Aparna Vyas ◽  
Gibak Kim
Keyword(s):  
Infinite Number ◽  
Geometric Construction ◽  
The Other ◽  
Wavelet Sets ◽  
Number Of Components ◽  
Wavelet Set

In this paper, we provide a geometric construction of a symmetric 2n-interval minimally supported frequency (MSF) d-dilation wavelet set with d∈(1,∞) and characterize all symmetric d-dilation wavelet sets. We also provide two special kinds of symmetric d-dilation wavelet sets, one of which has 4m-intervals whereas the other has (4m+2)-intervals, for m∈N. In addition, we construct a family of d-dilation wavelet sets that has an infinite number of components.

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