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1.
Eugenio Hernandez Xihua Wang Guido Weiss 《Journal of Fourier Analysis and Applications》1996,3(1):23-41
The main purpose of this paper is to give a procedure to "mollify" the low-pass filters of a large number of Minimally Supported Frequency (MSF) wavelets so that the smoother functions obtained in this way are also low-pass filters for an MRA. Hence, we are able to approximate (in the L2-norm) MSF wavelets by wavelets with any desired degree of smoothness on the Fourier transform side. Although the MSF wavelets we consider are bandlimited, this may not be true for their smooth approximations. This phenomena is related to the invariant cycles under the transformation $x\mapsto 2x (\mbox{mod}2\pi).The main purpose of this paper is to give a procedure to “mollify” the low-pass filters of a large number ofMinimally Supported Frequency (MSF) wavelets so that the smoother functions obtained in this way are also low-pass filters for an MRA. Hence, we are able to approximate (in the L 2 -norm) MSF wavelets by wavelets with any desired degree of smoothness on the Fourier transform side. Although the MSF wavelets we consider are bandlimited, this may not be true for their smooth approximations. This phenomena is related to the invariant cycles under the transformation x ?2x (mod2π). We also give a characterization of all low-pass filters for MSF wavelets. Throughout the paper new and interesting examples of wavelets are described. 相似文献
2.
We introduce a new method to construct large classes of minimally supported frequency (MSF) wavelets of the Hardy space H
2
(ℝ)and symmetric MSF wavelets of L
2
(ℝ),and discuss the classification of such wavelets. As an application, we show that there are uncountably many such wavelet sets
of L
2
(ℝ)and H
2
(ℝ).We also enumerate some of the symmetric wavelet sets of L
2
(ℝ)and all wavelet sets of H
2
(ℝ)consisting of three intervals. Finally, we construct families of MSF wavelets of L
2
(ℝ)with Fourier transform even and not vanishing in any neighborhood of the origin. 相似文献
3.
We discuss how one can use certain filters
from signal processing to describe isomorphisms between certain projective
C(T
n
)-modules. Conversely, we show how cancellation properties for
finitely generated projective modules over C(T
n
) can often be used
to prove the existence of continuous high pass filters, of the kind needed for
multivariate wavelets, corresponding to a given continuous low-pass filter.
However, we also give an example of a continuous low-pass filter for which it
is impossible to find corresponding continuous high-pass filters. In this way
we give another approach to the solution of the matrix completion problem for
filters of the kind arising in wavelet theory. 相似文献
4.
Manos Papadakis 《Journal of Fourier Analysis and Applications》1998,4(2):199-214
This article provides classes of unitary operators of L2(R) contained in the commutant of the Shift operator, such that for any pair of multiresolution analyses of L2(R) there exists a unitary operator in one of these classes, which maps all the scaling functions of the first multiresolution
analysis to scaling functions of the other. We use these unitary operators to provide an interesting class of scaling functions.
We show that the Dai-Larson unitary parametrization of orthonormal wavelets is not suitable for the study of scaling functions.
These operators give an interesting relation between low-pass filters corresponding to scaling functions, which is implemented
by a special class of unitary operators acting on L2([−π, π)), which we characterize. Using this characterization we recapture Daubechies' orthonormal wavelets bypassing the
spectral factorization process.
Acknowledgements and Notes. Partially supported by NSF Grant DMS-9157512, and Linear Analysis and Probability Workshop, Texas A&M University Dedicated
to the memory of Professor Emeritus Vassilis Metaxas. 相似文献
5.
We study biorthogonal bases of compactly supported wavelets constructed from box splines in ℝ
N
with any integer dilation factor. For a suitable class of box splines we write explicitly dual low-pass filters of arbitrarily
high regularity and indicate how to construct the corresponding high-pass filters (primal and dual).
Received: August 23, 2000; in final form: March 10, 2001?Published online: May 29, 2002 相似文献
6.
Maciej Paluszyński Hrvoje Šikić Guido Weiss Shaoliang Xiao 《Journal of Geometric Analysis》2001,11(2):311-342
A tight frame wavelet ψ is an L
2(ℝ) function such that {ψ jk(x)} = {2j/2
ψ(2
j
x −k), j, k ∈ ℤ},is a tight frame for L
2 (ℝ).We introduce a class of “generalized low pass filters” that allows us to define (and construct) the subclass of MRA tight
frame wavelets. This leads us to an associated class of “generalized scaling functions” that are not necessarily obtained
from a multiresolution analysis. We study several properties of these classes of “generalized” wavelets, scaling functions
and filters (such as their multipliers and their connectivity). We also compare our approach with those recently obtained
by other authors. 相似文献
7.
A. Calogero 《Journal of Geometric Analysis》2000,10(4):597-622
In the context of a general lattice Γ in Rn and a strictly expanding map M which preserves the lattice, we characterize all the wavelet families. This result generalizes
the characterization of Frazier, Garrigós, Wang, and Weis about the wavelet families with Γ = Zn and M = 21. In the second part of the paper, we characterize all the MSF wavelets. Moreover, we give a constructive method
for the support of the Fourier transform of an MSF wavelet and apply this method by giving examples with particular attention
to the quincunx lattice. 相似文献
8.
Eugen J. Ionascu David R. Larson Carl M. Pearcy 《Journal of Fourier Analysis and Applications》1998,4(6):711-721
It is proved that associated with every wavelet set is a closely related “regularized” wavelet set which has very nice properties.
Then it is shown that for many (and perhaps all) pairs E, F, of wavelet sets, the corresponding MSF wavelets can be connected
by a continuous path in L2(ℝ) of MSF wavelets for which the Fourier transform has support contained in E ∪ F. Our technique applies, in particular,
to the Shannon and Journe wavelet sets. 相似文献
9.
Iain Raeburn 《Acta Appl Math》2009,108(3):509-514
In recent joint work with Nadia Larsen, we gave a new proof of a theorem of Mallat which describes how to construct wavelets
from quadrature mirror filters. Our main innovation was to show how the scaling function associated to the filter can be used
to identify a particular direct limit of Hilbert spaces with L
2(ℝ). Here we show that wavelet-packet bases for L
2(ℝ) also fit naturally into the same direct-limit framework. 相似文献
10.
Refinable functions with exponential decay arise from applications such as the Butterworth filters in signal processing. Refinable
functions with exponential decay also play an important role in the study of Riesz bases of wavelets generated from multiresolution
analysis. A fundamental problem is whether the standard solution of a refinement equation with an exponentially decaying mask
has exponential decay. We investigate this fundamental problem by considering cascade algorithms in weighted L
p
spaces (1≤p≤∞). We give some sufficient conditions for the cascade algorithm associated with an exponentially decaying mask to converge
in weighted L
p
spaces. Consequently, we prove that the refinable functions associated with the Butterworth filters are continuous functions
with exponential decay. By analyzing spectral properties of the transition operator associated with an exponentially decaying
mask, we find a characterization for the corresponding refinable function to lie in weighted L
2 spaces. The general theory is applied to an interesting example of bivariate refinable functions with exponential decay,
which can be viewed as an extension of the Butterworth filters. 相似文献
11.
Damir Baki Ilya Krishtal Edward N. Wilson 《Applied and Computational Harmonic Analysis》2005,19(3):386
We study Parseval frame wavelets in with matrix dilations of the form , where A is an arbitrary expanding n×n matrix with integer coefficients, such that |detA|=2. We show that each A-MRA admits either Parseval frame wavelets, or Parseval frame bi-wavelets. The minimal number of generators for a Parseval frame associated with an A-MRA (i.e. 1 or 2) is determined in terms of a scaling function. All Parseval frame (bi)wavelets associated with A-MRA's are described. We then introduce new classes of filter induced wavelets and bi-wavelets. It is proved that these new classes strictly contain the classes of all A-MRA Parseval frame wavelets and bi-wavelets, respectively. Finally, we demonstrate a method of constructing all filter induced Parseval frame (bi)wavelets from generalized low-pass filters. 相似文献
12.
Zhongyan Li Xingde Dai Yuanan Diao Jianguo Xin 《Journal of Fourier Analysis and Applications》2010,16(2):155-176
Let A be any 2×2 real expansive matrix. For any A-dilation wavelet ψ, let
[^(y)]\widehat{\psi}
be its Fourier transform. A measurable function f is called an A-dilation wavelet multiplier if the inverse Fourier transform of
(f[^(y)])(f\widehat{\psi})
is an A-dilation wavelet for any A-dilation wavelet ψ. In this paper, we give a complete characterization of all A-dilation wavelet multipliers under the condition that A is a 2×2 matrix with integer entries and |{det }(A)|=2. Using this result, we are able to characterize the phases of A-dilation wavelets and prove that the set of all A-dilation MRA wavelets is path-connected under the L
2(ℝ2) norm topology for any such matrix A. 相似文献
13.
Abderrazek Karoui 《Central European Journal of Mathematics》2008,6(4):504-525
The construction of nonseparable and compactly supported orthonormal wavelet bases of L
2(R
n
); n ≥ 2, is still a challenging and an open research problem. In this paper, we provide a special method for the construction
of such wavelet bases. The wavelets constructed by this method are dyadic wavelets. Also, we show that our proposed method
can be adapted for an eventual construction of multidimensional orthogonal multiwavelet matrix masks, candidates for generating
multidimensional multiwavelet bases.
相似文献
14.
A series of admissible wavelets is fixed, which forms an orthonormal basis for the Hilbert space of all the quaternion-valued
admissible wavelets. It turns out that their corresponding admissible wavelet transforms give an orthogonal decomposition
of L
2(IG(2), ℍ).
相似文献
15.
Maciej Paluszyński Hrvoje Šikić Guido Weiss Shaoliang Xiao 《Advances in Computational Mathematics》2003,18(2-4):297-327
This is a continuation of our study of generalized low pass filters and MRA frame wavelets. In this first study we concentrated on the construction of such functions. Here we are particularly interested in the role played by the dimension function. In particular we characterize all semi-orthogonal Tight Frame Wavelets (TFW) by showing that they correspond precisely to those for which the dimension function is non-negative integer-valued. We also show that a TFW arises from our MRA construction if and only if the dimension of a particular linear space is either zero or one. We present many examples. In addition we obtain a result concerning the connectivity of TFW's that are MSF tight frame wavelets. 相似文献
16.
17.
Ante Mimica 《Potential Analysis》2010,32(3):275-303
In this paper we prove Harnack inequality for nonnegative functions which are harmonic with respect to random walks in ℝ
d
. We give several examples when the scale invariant Harnack inequality does not hold. For any α ∈ (0,2) we also prove the Harnack inequality for nonnegative harmonic functions with respect to a symmetric Lévy process
in ℝ
d
with a Lévy density given by $c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}$c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}, where 0 ≤ j(r) ≤ cr
− d − α
, ∀ r > 1, for some constant c. Finally, we establish the Harnack inequality for nonnegative harmonic functions with respect to a subordinate Brownian motion
with subordinator with Laplace exponent ϕ(λ) = λ
α/2ℓ(λ), λ > 0, where ℓ is a slowly varying function at infinity and α ∈ (0,2). 相似文献
18.
In this paper we consider a convolution operator Tf=p.v. Ω * f with Ω(x)=K(x)×eiλh(x), λ>0, where K(x) is a weak Calderón-Zygmund kernel and h(x) is a real-valued differentiable function. We give a boundedness
criterion for such an operator to map the Besov space B
1
0.1
(Rn) into itself.
This research was partially supported by NNSF and NEC in P. R. China. 相似文献
19.
Q. X. Yang 《Proceedings Mathematical Sciences》2005,115(3):347-368
In this paper, we characterize the symbol in Hormander symbol classS
ρ
m
,δ (m ∈ R, ρ, δ ≥ 0) by its wavelet coefficients. Consequently, we analyse the kerneldistribution property for the symbol in the symbol classS
ρ
m
,δ (m ∈R, ρ > 0, δ≥ 0) which is more general than known results ; for non-regular symbol operators, we establish sharp L2-continuity which is better than Calderón and Vaillancourt’s result, and establishL
p
(1 ≤p ≤∞) continuity which is new and sharp. Our new idea is to analyse the symbol operators in phase space with relative wavelets,
and to establish the kernel distribution property and the operator’s continuity on the basis of the wavelets coefficients
in phase space. 相似文献
20.
We present a construction of anisotropic multiresolution and anisotropic wavelet frames based on multilevel ellipsoid covers
(dilations) of ℝ
n
. The wavelets we construct are C
∞ functions, can have any prescribed number of vanishing moments and fast decay with respect to the anisotropic quasi-distance
induced by the cover. The dual wavelets are also C
∞, with the same number of vanishing moments, but with only mild decay with respect to the quasi-distance. An alternative construction
yields a meshless frame whose elements do not have vanishing moments, but do have fast anisotropic decay. 相似文献