共查询到20条相似文献,搜索用时 31 毫秒
1.
A. J. E. M. Janssen 《Journal of Fourier Analysis and Applications》1997,3(5):583-596
In this article we consider the question when one can generate a Weyl- Heisenberg frame for l
2
(ℤ) with shift parameters N, M
−1
(integer N, M) by sampling a Weyl-Heisenberg frame for L
2
(ℝ) with the same shift parameters at the integers. It is shown that this is possible when the window g ε L
2
(ℝ) generating the Weyl-Heisenberg frame satisfies an appropriate regularity condition at the integers. When, in addition,
the Tolimieri-Orr condition A is satisfied, the minimum energy dual window
o
γ ε L
2
(ℝ) can be sampled as well, and the two sampled windows continue to be related by duality and minimality. The results of this
article also provide a rigorous basis for the engineering practice of computing dual functions by writing the Wexler-Raz biorthogonality
condition in the time-domain as a collection of decoupled linear systems involving samples of g and
o
γ as knowns and unknowns, respectively. We briefly indicate when and how one can generate a Weyl-Heisenberg frame for the
space
of K-periodic sequences, where K=LCM (N, M), by periodization of a Weyl-Heisenberg frame for ℓ
2
ℤ with shift parameters N, M
−1
. 相似文献
2.
A Gabor frame multiplier is a bounded operator that maps normalized tight Gabor frame generators to normalized tight Gabor
frame generators. While characterization of such operators is still unknown, we give a complete characterization for the functional
Gabor frame multipliers. We prove that a L∞ -function h is a functional Gabor frame multiplier (for the time-frequency lattice aℤ × bℤ) if and only if it is unimodular
and
is a-periodic. Along the same line, we also characterize all the Gabor frame generators g (resp. frame wavelets ψ) for which
there is a function ∈ L∞(ℝ) such that {wgmn} (resp. ωψk,ℝ) is a normalized tight frame. 相似文献
3.
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
4.
Deguang Han 《Journal of Fourier Analysis and Applications》2009,15(2):201-217
Let
be a full rank time-frequency lattice in ℝ
d
×ℝ
d
. In this note we first prove that any dual Gabor frame pair for a Λ-shift invariant subspace M can be dilated to a dual Gabor frame pair for the whole space L
2(ℝ
d
) when the volume v(Λ) of the lattice Λ satisfies the condition v(Λ)≤1, and to a dual Gabor Riesz basis pair for a Λ-shift
invariant subspace containing M when v(Λ)>1. This generalizes the dilation result in Gabardo and Han (J. Fourier Anal. Appl. 7:419–433, [2001]) to both higher dimensions and dual subspace Gabor frame pairs. Secondly, for any fixed positive integer N, we investigate the problem whether any Bessel–Gabor family G(g,Λ) can be completed to a tight Gabor (multi-)frame G(g,Λ)∪(∪
j=1
N
G(g
j
,Λ)) for L
2(ℝ
d
). We show that this is true whenever v(Λ)≤N. In particular, when v(Λ)≤1, any Bessel–Gabor system is a subset of a tight Gabor frame G(g,Λ)∪G(h,Λ) for L
2(ℝ
d
). Related results for affine systems are also discussed.
Communicated by Chris Heil. 相似文献
5.
A refinable function φ(x):ℝn→ℝ or, more generally, a refinable function vector Φ(x)=[φ1(x),...,φr(x)]T is an L1 solution of a system of (vector-valued) refinement equations involving expansion by a dilation matrix A, which is an expanding
integer matrix. A refinable function vector is called orthogonal if {φj(x−α):α∈ℤn, 1≤j≤r form an orthogonal set of functions in L2(ℝn). Compactly supported orthogonal refinable functions and function vectors can be used to construct orthonormal wavelet and
multiwavelet bases of L2(ℝn). In this paper we give a comprehensive set of necessary and sufficient conditions for the orthogonality of compactly supported
refinable functions and refinable function vectors. 相似文献
6.
Characterization of Sobolev spaces of arbitrary smoothness using nonstationary tight wavelet frames 总被引:1,自引:0,他引:1
In this paper we shall characterize Sobolev spaces of an arbitrary order of smoothness using nonstationary tight wavelet frames
for L
2(ℝ). In particular, we show that a Sobolev space of an arbitrary fixed order of smoothness can be characterized in terms of
the weighted ℓ2-norm of the analysis wavelet coefficient sequences using a fixed compactly supported nonstationary tight wavelet frame in
L
2(ℝ) derived from masks of pseudosplines in [15]. This implies that any compactly supported nonstationary tight wavelet frame
of L
2(ℝ) in [15] can be properly normalized into a pair of dual frames in the corresponding pair of dual Sobolev spaces of an arbitrary
fixed order of smoothness.
Research supported in part by NSERC Canada under Grant RGP 228051.
Research supported in part by Grant R-146-000-060-112 at the National University of Singapore. 相似文献
7.
Demetrio Labate 《Journal of Geometric Analysis》2002,12(3):469-491
This article presents a general result from the study of shift-invariant spaces that characterizes tight frame and dual frame
generators for shift-invariant subspaces of L2(ℝn). A number of applications of this general result are then obtained, among which are the characterization of tight frames
and dual frames for Gabor and wavelet systems. 相似文献
8.
In this paper, we characterize the space of almost periodic (AP) functions in one variable using either a Weyl–Heisenberg (WH) system or an affine system. Our observation is that the sought-for characterization of the AP space is valid if and only if the given WH (respectively,
affine) system is an L
2(ℝ)-frame. Moreover, the frame bounds of the system are also the sharpest bounds in our characterization. This draws an intriguing
and quite unexpected connection between L
2(ℝ) representations and AP-representations.
相似文献
9.
H. G. Feichtinger H. Führ K. Gröchenig N. Kaiblinger 《Journal of Geometric Analysis》2006,16(1):53-67
We study the structure of operators from the Schwartz space S(ℝ
n
) into the tempered distributions S′(ℝ
n
) that commute with a discrete subgroup of translations. The formalism leads to simple derivations of recent results about
the frame operator of shift-invariant systems, Gabor, and wavelet frames. 相似文献
10.
A. B. Aleksandrov 《Journal of Mathematical Sciences》2008,148(6):785-794
We prove that each translation and dilation invariant subspace X ⊂ L
p
(ℝn), X ≠ L
p
(ℝn), is contained in a maximal translation and dilation invariant subspace of L
p
(ℝn). Moreover, we prove that the set of all maximal translation and dilation invariant subspaces of L
p
(ℝn) has the power of continuum. Bibliography: 6 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 345, 2007, pp. 5–21. 相似文献
11.
A. J. E. M. Janssen 《Journal of Fourier Analysis and Applications》1998,4(6):723-726
In this note we prove that the Wigner distribution of an f ∈ L2(ℝn) cannot be supported by a set of finite measure in ℝ2n unless f=0. We prove a corresponding statement for cross-ambiguity functions. As a strengthening of the conjecture we show
that for an f ∈ L2(ℝn) its Wigner distribution has a support of measure 0 or ∞ in any half-space of ℝ2n. 相似文献
12.
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
13.
Qingyue Zhang 《Numerical Functional Analysis & Optimization》2013,34(3):349-364
We study the structure of finitely generated shift-invariant subspaces with generators from the super Hilbert space L 2(? d )(N). We give a characterization for these subspaces. Moreover, we show that every finitely generated shift-invariant subspace possesses a tight frame. We also give a necessary and sufficient condition for such a space to be principal. Our results generalize similar ones for which generators are from L 2(? d ). 相似文献
14.
Marcinkiewicz Integrals with Non-Doubling Measures 总被引:2,自引:0,他引:2
Let μ be a positive Radon measure on
which may be non doubling. The only condition that μ must satisfy is μ(B(x, r)) ≤ Cr
n
for all
, r > 0 and some fixed constants C > 0 and n ∈ (0, d]. In this paper, we introduce the Marcinkiewicz integral related to a such measure with kernel satisfying some H?rmander-type
condition, and assume that it is bounded on L
2(μ). We then establish its boundedness, respectively, from the Lebesgue space L
1(μ) to the weak Lebesgue space L
1,∞(μ), from the Hardy space H
1(μ) to L
1(μ) and from the Lebesgue space L
∞(μ) to the space RBLO(μ). As a corollary, we obtain the boundedness of the Marcinkiewicz integral in the Lebesgue space L
p
(μ) with p ∈ (1,∞). Moreover, we establish the boundedness of the commutator generated by the RBMO(μ) function and the Marcinkiewicz integral with kernel satisfying certain slightly stronger H?rmander-type condition, respectively,
from L
p
(μ) with p ∈ (1,∞) to itself, from the space L log L(μ) to L
1,∞(μ) and from H
1(μ) to L
1,∞(μ). Some of the results are also new even for the classical Marcinkiewicz integral.
The third (corresponding) author was supported by National Science Foundation for Distinguished Young Scholars (No. 10425106)
and NCET (No. 04-0142) of China. 相似文献
15.
Biswaranjan Behera 《Proceedings Mathematical Sciences》2001,111(4):439-463
The orthonormal basis generated by a wavelet ofL
2(ℝ) has poor frequency localization. To overcome this disadvantage Coifman, Meyer, and Wickerhauser constructed wavelet packets.
We extend this concept to the higher dimensions where we consider arbitrary dilation matrices. The resulting basis ofL
2(ℝ
d
) is called the multiwavelet packet basis. The concept of wavelet frame packet is also generalized to this setting. Further,
we show how to construct various orthonormal bases ofL
2(ℝ
d
) from the multiwavelet packets. 相似文献
16.
A necessary and sufficient condition for dual Weyl-Heisenberg frames to be compactly supported 总被引:1,自引:0,他引:1
Helmut Bölcskei 《Journal of Fourier Analysis and Applications》1999,5(5):409-419
In this note we consider continuous-time Weyl-Heisenberg (Gabor) frame expansions with rational oversampling. We present a necessary and sufficient condition on a compactly supported function g(t) generating a Weyl-Heisenberg frame for L2 () for its minimal dual (Wexler-Razdual) 0 (t) to be compactly supported. We furthermore provide a necessary and sufficient condition for a band-limited function g(t) generating a Weyl-Heisenberg frame for L2 () to have a band-limited minimal dual 0 (t). As a consequence of these conditions, we show that in the cases of integer oversampling and critical sampling a compactly supported (band-limited) g(t) has a compactly supported (band-limited) minimal dual 0(t) if and only if the Weyl-Heisenberg frame operator is a multiplication operator in the time (frequency) domain. Our proofs rely on the Zak transform, on the Zibulski-Zeevi representation of the Weyl-Heisenberg frame operator, and on the theory of polynomial matrices.on leave from Department of Communications, Vienna University of TechnologyThis work was supported in part by FWF grants P10531-ÖPH, P12228-TEC, and J1629-TEC. 相似文献
17.
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
18.
LetM=(W, d) be a metric space. LetL
1 denote theL
1 metric. AnL
1-embedding ofM into Cartesiank-space ℝ
k
is a distance-preserving map from (W, d) into (ℝ
k
,L
1). Letc(k) be the smallest integer such that for every metric spaceM, M isL
1-embeddable inR
k iff everyc(k)-sized subspace ofM isL
1-embeddable inR
k. A special case of a theorem of Menger (see p. 94 of [5]) says thatc(1) exists and equals 4. We show thatc(2) exists and satisfies 6≦c(2)≦11. Whether or notc(k) exists for anyk≧3 is an open question.
The research of S. M. Malitz was partially supported by NSF Grant CCR-8909953. 相似文献
19.
N. A. Begun 《Vestnik St. Petersburg University: Mathematics》2010,43(2):74-81
The paper considers the problem of structural stability of systems under disturbance of coefficients having small L
2(ℝ)-norm. We derive conditions which guarantee that for every solution of the perturbed system there exists a solution of
the original system which is close to the former in L
2(ℝ)-norm. 相似文献
20.
Ilya A. Krishtal Benjamin D. Robinson Guido L. Weiss Edward N. Wilson 《Journal of Geometric Analysis》2007,17(1):87-96
An orthonormal wavelet system in ℝd, d ∈ ℕ, is a countable collection of functions {ψ
j,k
ℓ
}, j ∈ ℤ, k ∈ ℤd, ℓ = 1,..., L, of the form
that is an orthonormal basis for L2 (ℝd), where a ∈ GLd (ℝ) is an expanding matrix. The first such system to be discovered (almost 100 years ago) is the Haar system for which L
= d = 1, ψ1(x) = ψ(x) = κ[0,1/2)(x) − κ[l/2,1)
(x), a = 2. It is a natural problem to extend these systems to higher dimensions. A simple solution is found by taking appropriate
products Φ(x1, x2, ..., xd) = φ1 (x1)φ2(x2) ... φd(xd) of functions of one variable. The obtained wavelet system is not always convenient for applications. It is desirable to
find “nonseparable” examples. One encounters certain difficulties, however, when one tries to construct such MRA wavelet systems.
For example, if a = (
1-1
1 1
) is the quincunx dilation matrix, it is well-known (see, e.g., [5]) that one can construct nonseparable Haar-type scaling
functions which are characteristic functions of rather complicated fractal-like compact sets. In this work we shall construct
considerably simpler Haar-type wavelets if we use the ideas arising from “composite dilation” wavelets. These were developed
in [7] and involve dilations by matrices that are products of the form ajb, j ∈ ℤ, where a ∈ GLd(ℝ) has some “expanding” property and b belongs to a group of matrices in GLd(ℝ) having |det b| = 1. 相似文献