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.

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 Reducing Subspaces of Some Multiplication Operators on the Bergman Space over Polydisk

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Yanyue Shi ◽  
Na Zhou
Keyword(s):  
Bergman Space ◽  
Multiplication Operators ◽  
Direct Sum ◽  
Reducing Subspace ◽  
Reducing Subspaces

We consider the reducing subspaces ofMzNonAα2(Dk), wherek≥3,zN=z1N1⋯zkNk, andNi≠Njfori≠j. We prove that each reducing subspace ofMzNis a direct sum of some minimal reducing subspaces. We also characterize the minimal reducing subspaces in the cases thatα=0andα∈(-1,+∞)∖Q, respectively. Finally, we give a complete description of minimal reducing subspaces ofMzNonAα2(D3)withα>-1.

scholarly journals Resolution of the Identity of the Classical Hardy Space by Means of Barut-Girardello Coherent States

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Zouhaïr Mouayn
Keyword(s):  
Fourier Transform ◽  
Hardy Space ◽  
Real Line ◽  
Coherent States ◽  
Positive Real ◽  
Integrable Functions ◽  
Resolution Of The Identity ◽  
Square Integrable Functions ◽  
Complex Valued ◽  
Classical Hardy Space

We construct a one-parameter family of coherent states of Barut-Girdrardello type performing a resolution of the identity of the classical Hardy space of complex-valued square integrable functions on the real line, whose Fourier transform is supported by the positive real semiaxis.

scholarly journals An example of a non-exposed extreme function in the unit ball of H1

1994 ◽  
Vol 37 (1) ◽  
pp. 47-51 ◽  
Author(s):  
Jyunji Inoue
Keyword(s):  
Hardy Space ◽  
Unit Ball ◽  
Unit Disc ◽  
Inner Function ◽  
Classical Hardy Space

We construct a non-exposed extreme function f of the unit ball of H1, the classical Hardy space on the unit disc of the plane, which has the property: f(z)/(1−q(z))2 ∉ H1 for any nonconstant inner function q(z). This function constitutes a counterexample to a conjecture in D. Sarason [7].

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 On norms of composition operators on weighted hardy spaces

2018 ◽  
Vol 14 (1) ◽  
pp. 7424-7430
Author(s):  
Amenah Essa Shammaky ◽  
Sumitra Dalal
Keyword(s):  
Hardy Space ◽  
Composition Operator ◽  
Hardy Spaces ◽  
Composition Operators ◽  
Weighted Hardy Spaces ◽  
Classical Hardy Space

 The computation of composition operator on Hardy spaces is very hard. In this paper we propose  a  norm of a bounded composition operator on weighted Hardy spaces H2(b) induced by a disc  automorphism by embedding  the classical Hardy space . The estimate obtained is accurate in the sense that it provides the exact norm for particular instances of the sequence b. As an application of our results, an estimate for the norm of any bounded composition operator on H2(b) is obtained.

Weak nonhomogeneous wavelet dual frames for Walsh reducing subspace of L2(ℝ+)

2021 ◽  
pp. 2150040
Author(s):  
Yan Zhang ◽  
Yun-Zhang Li
Keyword(s):  
Extension Principle ◽  
Transform Domain ◽  
Wavelet Frames ◽  
Dual Frames ◽  
Reducing Subspace ◽  
Reducing Subspaces ◽  
Extension Principles ◽  
Dual Wavelet Frames ◽  
Dual Wavelet ◽  
Fourier Transform Domain

In wavelet analysis, refinable functions are the bases of extension principles for constructing (weak) dual wavelet frames for [Formula: see text] and its reducing subspaces. This paper addresses refinable function-based dual wavelet frames construction in Walsh reducing subspaces of [Formula: see text]. We obtain a Walsh–Fourier transform domain characterization for weak [Formula: see text]-adic nonhomogeneous dual wavelet frames; and present a mixed oblique extension principle for constructing weak [Formula: see text]-adic nonhomogeneous dual wavelet frames in Walsh reducing subspaces of [Formula: see text].

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.

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