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1.
Approximation order provided by refinable function vectors 总被引:1,自引:0,他引:1
G. Plonka 《Constructive Approximation》1997,13(2):221-244
In this paper we considerL
p-approximation by integer translates of a finite set of functionsϕ
v (v=0, ...,r − 1) which are not necessarily compactly supported, but have a suitable decay rate. Assuming that the function vectorϕ=(ϕ
=0/
r−1 is refinable, necessary and sufficient conditions for the refinement mask are derived. In particular, if algebraic polynomials
can be exactly reproduced by integer translates ofϕ
v, then a factorization of the refinement mask ofϕ can be given. This result is a natural generalization of the result for a single functionϕ, where the refinement mask ofϕ contains the factor ((1 +e
−iu
)/2)
m
if approximation orderm is achieved.
Dedicated to Professor L. Berg on the occasion of his 65th birthday 相似文献
2.
Vladimir Protasov 《Journal of Fourier Analysis and Applications》2000,6(1):55-78
In this paper we analyze solutions of the n-scale functional equation Ф(x) = Σk∈ℤ
Pk Ф(nx−k), where n≥2 is an integer, the coefficients {Pk} are nonnegative and Σpk = 1. We construct a sharp criterion for the existence of absolutely continuous solutions of bounded
variation. This criterion implies several results concerning the problem of integrable solutions of n-scale refinement equations
and the problem of absolutely continuity of distribution function of one random series. Further we obtain a complete classification
of refinement equations with positive coefficients (in the case of finitely many terms) with respect to the existence of continuous
or integrable compactly supported solutions. 相似文献
3.
We study the large time behaviour of nonnegative solutions of the Cauchy problemu
t=Δu
m −u
p,u(x, 0)=φ(x). Specifically we study the influence of the rate of decay ofφ(x) for large |x|, and the competition between the diffusion and the absorption term. 相似文献
4.
Flávio Dickstein Miguel Loayza 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,59(1):1-23
We consider the Cauchy problem for the weakly coupled parabolic system ∂
t
w
λ−Δ w
λ = F(w
λ) in R
N
, where λ > 0, w
λ = (u
λ, v
λ), F(w
λ) = (v
λ
p
, u
λ
q
) for some p, q ≥ 1, pq > 1, and , for some nonnegative functions φ1, φ2
C
0(R
N
). If (p, q) is sub-critical or either φ1 or φ2 has slow decay at ∞, w
λ blows up for all λ > 0. Under these conditions, we study the blowup of w
λ for λ small.
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.
Flávio Dickstein Miguel Loayza 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,24(10):1-23
We consider the Cauchy problem for the weakly coupled parabolic system ∂
t
w
λ−Δ w
λ = F(w
λ) in R
N
, where λ > 0, w
λ = (u
λ, v
λ), F(w
λ) = (v
λ
p
, u
λ
q
) for some p, q ≥ 1, pq > 1, and
wl(0) = (lj1, l\fracq+1p+1j2)w_{\lambda}(0) = ({\lambda}{\varphi}_1, {\lambda}^{\frac{q+1}{p+1}}{\varphi}_2), for some nonnegative functions φ1, φ2
?\in
C
0(R
N
). If (p, q) is sub-critical or either φ1 or φ2 has slow decay at ∞, w
λ blows up for all λ > 0. Under these conditions, we study the blowup of w
λ for λ small. 相似文献
7.
David Walnut 《Journal of Fourier Analysis and Applications》1998,4(6):669-709
Explicit, compactly supported solutions, {vi, ϕ}
i=1
m
, to the deconvolution (or Bezout) equation
are computed where ϕ is a given function in C
c
∞
(Rd), and
, i=1, ..., m for some set of positive numbers {ri}
i=1
m
such that ri/rj is poorly approximated by rationals whenever i ≠ j. The novelty of the solution technique is that it uses new results in
the theory of sampling of bandlimited functions detailed in [13] to provide simple Fourier series representations for the
solutions, {vi, ϕ}
i=1
m
, which can be easily implemented numerically.
Several examples illustrating the use of sampling for solutions to variants of (0.1) are given, as well as some numerical
simulations.
Acknowledgements and Notes. The author gratefully acknowledges the support of the National Science Foundation, DMS-9500909, and Prof. K.J.R. Liu at
the Institute for Systems Research, University of Maryland, College Park, MD, 20742. 相似文献
((0.1)) |
8.
We consider the existence of distributional (or L
2
) solutions of the matrix refinement equation
where P is an r×r matrix with trigonometric polynomial entries.
One of the main results of this paper is that the above matrix refinement equation has a compactly supported distributional
solution if and only if the matrix P
(0) has an eigenvalue of the form 2
n
, . A characterization of the existence of L
2
-solutions of the above matrix refinement equation in terms of the mask is also given.
A concept of L
2
-weak stability of a (finite) sequence of function vectors is introduced. In the case when the function vectors are solutions
of a matrix refinement equation, we characterize this weak stability in terms of the mask.
August 1, 1996. Date revised: July 28, 1997. Date accepted: August 12, 1997. 相似文献
9.
Á. G. Horváth 《Monatshefte für Mathematik》2007,150(3):211-216
This paper presents a result concerning the connection between the parallel projection P
v,H
of a parallelotope P along the direction v (into a transversal hyperplane H) and the extension P + S(v), meaning the Minkowski sum of P and the segment S(v) = {λv | −1 ≤ λ ≤ 1}. A sublattice L
v
of the lattice of translations of P associated to the direction v is defined. It is proved that the extension P + S(v) is a parallelotope if and only if the parallel projection P
v,H
is a parallelotope with respect to the lattice of translations L
v,H
, which is the projection of the lattice L
v
along the direction v into the hyperplane H. 相似文献
10.
Futoshi Takahashi 《Calculus of Variations and Partial Differential Equations》2007,29(4):509-520
We continue to study the asymptotic behavior of least energy solutions to the following fourth order elliptic problem (E
p
): as p gets large, where Ω is a smooth bounded domain in R
4
. In our earlier paper (Takahashi in Osaka J. Math., 2006), we have shown that the least energy solutions remain bounded uniformly
in p and they have one or two “peaks” away form the boundary. In this note, following the arguments in Adimurthi and Grossi (Proc.
AMS 132(4):1013–1019, 2003) and Lin and Wei (Comm. Pure Appl. Math. 56:784–809, 2003), we will obtain more sharper estimates
of the upper bound of the least energy solutions and prove that the least energy solutions must develop single-point spiky
pattern, under the assumption that the domain is convex. 相似文献
11.
Summary LetX be the observed vector of thep-variate (p≧3) normal distribution with mean θ and covariance matrix equal to the identity matrix. Denotey
+=max{0,y} for any real numbery. We consider the confidence set estimator of θ of the formC
δa,φ={θ:|θ−δa,φ(X)}≦c}, whereδ
a,φ=[1−aφ({X})/{X}2]+X is the positive part of the Baranchik (1970,Ann. Math. Statist.,41, 642–645) estimator. We provide conditions on ϕ(•) anda which guarantee thatC
δa.φ has higher coverage probability than the usual one, {θ:|θ−X|≦c}. This dominance result will be shown to hold for spherically symmetric distributions, which include the normal distribution,t-distribution and double exponential distribution. The latter result generalizes that of Hwang and Chen (1983,Technical Report, Dept. of Math., Cornell University). 相似文献
12.
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
13.
Pascale Vitse 《Archiv der Mathematik》2005,85(4):374-385
For Banach space operators T satisfying the Tadmor-Ritt condition
a band limited H∞ calculus is established,
where
and a is at most of the order C(T)5. It follows that such a T allows a bounded Besov algebra B∞ 10 functional calculus,
These estimates are sharp in a convenient sense. Relevant embedding theorems for B∞ 10 are derived.
Received: 25 October 2004; revised: 31 January 2005 相似文献
14.
Factoring wavelet transforms into lifting steps 总被引:236,自引:0,他引:236
This article is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with
finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are
also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet
or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists (and expressed
by the formulaSL(n;R[z, z−1])=E(n;R[z, z−1])); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation,
building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering.
This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the
biorthogonal, i.e., non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces
the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining
a wavelet-like transform that maps integers to integers.
Research Tutorial
Acknowledgements and Notes. Page 264. 相似文献
15.
We consider the weighted Hardy integral operatorT:L
2(a, b) →L
2(a, b), −∞≤a<b≤∞, defined by
. In [EEH1] and [EEH2], under certain conditions onu andv, upper and lower estimates and asymptotic results were obtained for the approximation numbersa
n(T) ofT. In this paper, we show that under suitable conditions onu andv,
where ∥w∥p=(∫
a
b
|w(t)|p
dt)1/p.
Research supported by NSERC, grant A4021.
Research supported by grant No. 201/98/P017 of the Grant Agency of the Czech Republic. 相似文献
16.
Kh. Kh. Ruzimuradov 《Journal of Mathematical Sciences》1996,79(5):1320-1324
Let Λ be a unimodular lattice in ℝ2, μ a homogeneous minimum of Λ; let P(a,b)⊂ℝ2 be a rectangle with vertices at the points (a,0), ...(0,b), P(a, b)+X its image under the translation by a vector X ∈ ℝ2. We prove that there exists a sequence of positive numbers v1<v2<...<vk<... with
, such that for u>μ the rectangle P(u, vk)+X contains T=S(P)+R points of Λ, where |R|<5; here S(P) is the area of the rectangle. Bibliography: 4 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 82–89.
Translated by O. A. Ivanov. 相似文献
17.
Gabor frames, unimodularity, and window decay 总被引:4,自引:0,他引:4
We study time-continuous Gabor frame generating window functions g satisfying decay properties in time and/or frequency with
particular emphasis on rational time-frequency lattices. Specifically, we show under what conditions these decay properties
of g are inherited by its minimal dual γ0 and by generalized duals γ. We consider compactly supported, exponentially decaying, and faster than exponentially decaying
(i.e., decay like |g(t)|≤Ce−α|t|
1/α for some 1/2≤α<1) window functions. Particularly, we find that g and γ0 have better than exponential decay in both domains if and only if the associated Zibulski-Zeevi matrix is unimodular, i.e.,
its determinant is a constant. In the case of integer oversampling, unimodularity of the Zibulski-Zeevi matrix is equivalent
to tightness of the underlying Gabor frame. For arbitrary oversampling, we furthermore consider tight Gabor frames canonically
associated to window functions g satisfying certain decay properties. Here, we show under what conditions and to what extent
the canonically associated tight frame inherits decay properties of g. Our proofs rely on the Zak transform, on the Zibulski-Zeevi
representation of the Gabor frame operator, on a result by Jaffard, on a functional calculus for Gabor frame operators, on
results from the theory of entire functions, and on the theory of polynomial matrices. 相似文献
18.
We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|2/n
u = 0,x ∃ Rn,t } 0,u(0,x) = u0(x),x ∃ Rn, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β)≥ 0, θ = |∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|n-1nn/2 / ((n + 1)|α|2 + α2
n/2. Furthermore, we assume that the initial data u0 ∃ Lp are such that (1 + |x|)αu0 ∃ L1, with sufficiently small norm ∃ = (1 + |x|)α u0 1 + u0 p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L∞) ∩ C ([0, ∞); L1 ∩ Lp) satisfying the time decay estimates for allt0 u(t)||∞ Cɛt-n/2(1 + η log 〈t〉)-n/2, if hg = θ2/n 2π ℜδ(α, Β) 0; u(t)||∞ Cɛt-n/2(1 + Μ log 〈t〉)-n/4, if η = 0 and Μ = θ4/n 4π)2 (ℑδ(α, Β))2 ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)||∞ Cɛt-n/2(1 + κ log 〈t〉)-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31). 相似文献
19.
周泽华 《中国科学A辑(英文版)》2003,46(1):33-38
Let Un be the unit polydisc of Cn and φ= (φ1,...,φn? a holomorphic self-map of Un. Let 0≤α< 1. This paper shows that the composition operator Cφ, is bounded on the Lipschitz space Lipa(Un) if and only if there exists M > 0 such thatfor z∈Un. Moreover Cφ is compact on Lipa(Un) if and only if Cφ is bounded on Lipa(Un) and for every ε > 0, there exists a δ > 0 such that whenever dist(φ(z),σUn) <δ 相似文献
20.
An extension of a classical theorem of Rellich to the exterior of a closed proper convex cone is proved: Let Γ be a closed
convex proper cone inR
n and −Γ′ be the antipodes of the dual cone of Γ. Let
be a partial differential operator with constant coefficients inR
n, whereQ(ζ)≠0 onR
n−iΓ′ andP
i is an irreducible polynomial with real coefficients. Assume that the closure of each connected component of the set {ζ∈R
n−iΓ′;P
j(ζ)=0, gradP
j(ζ)≠0} contains some real point on which gradP
j≠0 and gradP
j∉Γ∪(−Γ). LetC be an open cone inR
n−Γ containing both normal directions at some such point, and intersecting each normal plane of every manifold contained in
{ξ∈R
n;P(ξ)=0}. Ifu∈ℒ′∩L
loc
2
(R
n−Γ) and the support ofP(−i∂/∂x)u is contained in Γ, then the condition
implies that the support ofu is contained in Γ. 相似文献