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.

scholarly journals Gramian analysis of multivariate frame multiresolution analyses

2002 ◽  
Vol 66 (2) ◽  
pp. 291-300 ◽  
Author(s):  
Jae Kun Lim
Keyword(s):  
Multiresolution Analysis ◽  
Wavelet Set ◽  
Wavelet Space ◽  
Frame Bounds ◽  
Frame Multiresolution Analysis

We perform a Gramian analysis of a frame multiresolution analysis to give a condition for it to admit a minimal wavelet set and to show that the frame bounds of the natural generator for the wavelet space of a degenerate frame multiresolution analysis shrink.

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.

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.

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.

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.

On wavelet induced isomorphisms for reducing subspaces

2016 ◽  
Vol 14 (03) ◽  
pp. 1650009 ◽  
Author(s):  
Swati Srivastava ◽  
G. C. S. Yadav
Keyword(s):  
Hardy Space ◽  
Wavelet Sets ◽  
Reducing Subspace ◽  
Wavelet Set ◽  
Michigan Math ◽  
Reducing Subspaces ◽  
Real Anal ◽  
New Construction ◽  
Classical Hardy Space

In this paper, we adapt the notion of a wavelet induced isomorphism of [Formula: see text] associated with a wavelet set, introduced in [E. J. Ionascu, A new construction of wavelet sets, Real Anal. Exchange 28(2) (2002/03) 593–610], to the case of an [Formula: see text]-wavelet set, where [Formula: see text] is a reducing subspace [X. Dai and S. Lu, Wavelets in subspaces, Michigan Math. J. 43 (1996) 81–98]. We characterize all these wavelet induced isomorphisms similar to those given in Ionascu paper and provide specific examples of this theory in the case of symmetric [Formula: see text]-wavelet sets. Examples when [Formula: see text] is the classical Hardy space are also considered.

CONSTRUCTING NON-MSF WAVELETS FROM GENERALIZED JOURNÉ WAVELET SETS

2011 ◽  
Vol 09 (02) ◽  
pp. 225-233 ◽  
Author(s):  
N. K. SHUKLA ◽  
G. C. S. YADAV
Keyword(s):  
Wavelet Sets ◽  
Wavelet Set ◽  
The One

Dai and Larson [Mem. Amer. Math. Soc.134 (1998), no. 640] obtained a family of wavelet sets using the Journé wavelet set. In this paper, we expand this family and call its members to be generalized Journé wavelet sets. Furthermore, with the help of these wavelet sets, we provide a class of non-MSF wavelets which includes the one constructed by Vyas [Bull. Polish Acad. Sci. Math.57 (2009) 33–40]. Most of these non-MSF wavelets are found to be non-MRA.

On fixed point sets of wavelet induced isomorphisms and frame induced monomorphisms

2016 ◽  
Vol 14 (03) ◽  
pp. 1650016
Author(s):  
Swati Srivastava ◽  
G. C. S. Yadav
Keyword(s):  
Fixed Point ◽  
Point Sets ◽  
Wavelet Sets ◽  
Wavelet Set ◽  
Fixed Point Sets

In this paper, we study fixed point sets of wavelet induced isomorphisms for symmetric wavelet sets. Also, introducing the notion of frame induced monomorphism for a frame wavelet set, as a generalization of the wavelet induced isomorphism for a wavelet set, we provide a construction of frame wavelet sets in [Formula: see text]. We also study fixed point sets of these maps.

scholarly journals Vector valued nonuniform nonstationary wavelets and associated MRA on local fields

2021 ◽  
Vol 17 (2) ◽  
pp. 19-46
Author(s):  
O. Ahmad ◽  
A. H. Wani ◽  
N. A. Sheikh ◽  
M. Ahmad
Keyword(s):  
Multiresolution Analysis ◽  
Complete Characterization ◽  
Local Fields ◽  
Dimension Function ◽  
Vector Valued

Abstract In this paper we study nonstationary wavelets associated with vector valued nonuniform multiresolution analysis on local fields. By virtue of dimension function a complete characterization of vector valued nonuniform nonstationary wavelets is obtained.

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