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1.
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
2.
Wolfgang Lück 《Inventiones Mathematicae》2002,149(1):123-152
We prove a version of the L
2-index Theorem of Atiyah, which uses the universal center-valued trace instead of the standard trace. We construct for G-equivariant K-homology an equivariant Chern character, which is an isomorphism and lives over the ring ℤ⊂λ
G
⊂ℚ obtained from the integers by inverting the orders of all finite subgroups of G. We use these two results to show that the Baum-Connes Conjecture implies the modified Trace Conjecture, which says that
the image of the standard trace K
0(C
*
r
(G))→ℝ takes values in λ
G
. The original Trace Conjecture predicted that its image lies in the additive subgroup of ℝ generated by the inverses of all
the orders of the finite subgroups of G, and has been disproved by Roy [15].
Oblatum 10-IV-2001 & 18-X-2001?Published online: 15 April 2002 相似文献
3.
S. E. Kozlov 《Journal of Mathematical Sciences》2000,100(3):2239-2253
The paper deals with the properties of the exterior algebra ℝ(Λn) related to the Euclidean structure on ℝ(Λn) induced by the scalar product in ℝ(Λn). A geometric interpretation of inner multiplication for simple p-vectors is given. An invariant form of the Cartan criterion
for the simplicity of a p-vector is given. The Plücker model realizing the real Grassmann manifold as a submanifold of the
Euclidean space ℝ(Λn), and an isometry of this submanifold onto the classical Grassmann manifold with SO(n)-invariant metric are described. A
canonical decomposition of bivectors is given. Bibliography: 12 titles.
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 84–107.
Translated by N. Yu: Netsvetaev. 相似文献
4.
We define and study a class of summable processes, called additive summable processes, that is larger than the class used
by Dinculeanu and Brooks [D-B].
We relax the definition of a summable processesX:Ω×ℝ+→E⊂L(F, G) by asking for the associated measureI
X to have just an additive extension to the predictableσ-algebra ℘, such that each of the measures (I
X)
z
, forz∈(L
G
p
)*, beingσ-additive, rather than having aσ-additive extension. We define a stochastic integral with respect to such a process and we prove several properties of the
integral. After that we show that this class of summable processes contains all processesX:Ω×ℝ+→E⊂L(F, G) with integrable semivariation ifc
0 ∋G. 相似文献
5.
Guoxiang Chen Meiying Wang 《分析论及其应用》2007,23(3):266-273
For a continuous, increasing function ω: R → R \{0} of finite exponential type, this paper introduces the set Z(A, ω) of all x in a Banach space X for which the second order abstract differential equation (2) has a mild solution such that [ω(t)]-1u(t,x) is uniformly continues on R , and show that Z(A, ω) is a maximal Banach subspace continuously embedded in X, where A ∈ B(X) is closed. Moreover, A|z(A,ω) generates an O(ω(t))strongly continuous cosine operator function family. 相似文献
6.
A. Nemirovskii 《Mathematical Programming》1996,73(2):175-198
We establish polynomial time convergence of the method of analytic centers for the fractional programming problemt→min |x∈G, tB(x)−A(x)∈K, whereG ⊂ ℝ
n
is a closed and bounded convex domain,K ⊂ ℝ
m
is a closed convex cone andA(x):G → ℝ
n
,B(x):G→K are regular enough (say, affine) mappings.
This research was partly supported by grant #93-012-499 of the Fundamental Studies Foundation of Russian Academy of Sciences 相似文献
7.
I. Ginchev A. Guerraggio M. Rocca 《Journal of Optimization Theory and Applications》2009,143(1):87-105
The present paper studies the following constrained vector optimization problem: min
C
f(x), g(x)∈−K, h(x)=0, where f:ℝ
n
→ℝ
m
, g:ℝ
n
→ℝ
p
and h:ℝ
n
→ℝ
q
are locally Lipschitz functions and C⊂ℝ
m
, K⊂ℝ
p
are closed convex cones. In terms of the Dini set-valued directional derivative, first-order necessary and first-order sufficient
conditions are obtained for a point x
0 to be a w-minimizer (weakly efficient point) or an i-minimizer (isolated minimizer of order 1). It is shown that, under natural assumptions (given by a nonsmooth variant of the
implicit function theorem for the equality constraints), the obtained conditions improve some given by Clarke and Craven.
Further comparison is done with some recent results of Khanh, Tuan and of Jiiménez, Novo. 相似文献
8.
Marty Ross 《Journal of Geometric Analysis》1998,8(2):313-317
Let S ⊂ ℝn be a complete 2-dimensional areaminimizing mod 2 surface. Then S = x1 (M1) ∪ … ∪ xr (Mr) where each Mj is connected, xj: Mj → Vj is a classical minimal immersion into an affine subspace Vj of ℝn, and the subspaces V1,…, Vr are pairwise orthogonal. Here we prove that if Mj is orientable, then xj (Mj) is either aflat plane or, in suitable coordinates, a generalized complex hyperbola. 相似文献
9.
Joaquim Bruna 《Journal of Fourier Analysis and Applications》2006,12(1):71-82
We show that the discrete translation parameter sets Λ ⊂ ℝ for which some φ ∈ L1(ℝ) exists such that the translates φ(x − λ), λ ∈ Λ, span L1(ℝ) are exactly the uniqueness sets for certain quasianalytic classes, and give explicit constructions of such generators
φ. We also consider a similar situation for affine systems of the type φ(μx − λ), μ ∈ Γ, λ ∈ Λ. 相似文献
10.
O. P. Filatov 《Mathematical Notes》2000,67(3):365-371
For the class II(ℝ
m
) of continuous almost periodic functionsf: ℝ
m
→ ℝ, we consider the problem of the existence of the limit
where the least upper bound is taken over all solutions (in the sense of Carathéodory) of the generalized differential equation
{ie365-1} εG, γ(0)=a
0. We establish that if the compact setG ⊂ ℝ
m
is not contained in a subspace of ℝ
m
of dimensionm−1 (i.e., if it is nondegenerate), then the limit exists uniformly in the initial vectora
0 ε ℝ
m
. Conversely, if for any functionf ε π(ℝ
m
), the limit exists uniformly in the initial vectora
0 ε ℝ
m
, then the compact setG is nondegenerate. We also prove that there exists an extremal solution for which a limit of the maximal mean uniform in the
initial conditions is realized.
Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 433–440, March, 2000. 相似文献
(1) |
11.
Joe J. Perez 《Journal of Geometric Analysis》2009,19(1):87-106
Let G be a unimodular Lie group, X a compact manifold with boundary, and M be the total space of a principal bundle G→M→X so that M is also a strongly pseudoconvex complex manifold. In this work, we show that if G acts by holomorphic transformations in M, then the Laplacian
on M has the following properties: The kernel of □ restricted to the forms Λ
p,q
with q>0 is a closed, G-invariant subspace in L
2(M,Λ
p,q
) of finite G-dimension. Secondly, we show that if q>0, then the image of □ contains a closed, G-invariant subspace of finite G-codimension in L
2(M,Λ
p,q
). These two properties taken together amount to saying that □ is a G-Fredholm operator. It is a corollary of the first property mentioned that the reduced L
2-Dolbeault cohomology spaces
of M are finite G-dimensional for q>0. The boundary Laplacian □
b
has similar properties.
相似文献
12.
A finitely presented group G is hyperbolic iff H
(1)
1(G,ℝ)=0=(1)
2(G, ℝ), where H
(1)
* (resp. (1)
*) denotes the ℓ1-homology (resp. reduced ℓ1-homology). If Γ is a graph, then every ℓ1 1-cycle in Γ with real coefficients can be approximated by 1-cycles of compact support. A 1-relator group G is hyperbolic iff H
(1)
1(G,ℝ)=0.
Oblatum: 30-IV-1997 & 14-V-1998 / Published online: 14 January 1999 相似文献
13.
We study the spectrum of the boundary-value problem for the Laplace operator in a thin domain Ω(ε) obtained by small perturbation
of the cylinder Ω(ε)=ω×(-ε/2.ε/2) ⊂ ℝ3in a neighborhood of the lateral surface. The Dirichlet condition is imposed on the bases of the cylinder, and the Dirichlet
condition or the Neumann condition is imposed on the remaining part of ∂Ω(ε). We construct and justify asymptotic formulas
(as ε→+0) for eigenvalues and eigenfunctions. In view of a special form of the lateral surface, there are eigenfunctions of
boundary-layer type that exponentially decrease far from the lateral surface. For the mixed boundary-value problem such a
localization is possible in neighborhoods of local maxima of the curvature of the contour ∂ω. This property of eigenfunctions
is a characteristic feature of the first points of the spectrum (in particular, the first eigenvalue) and, under the passage
from Ω(h)() to Ω(h), the spectrum itself has perturbation O(h−2). Bibliography: 29 titles.
Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 105–149. 相似文献
14.
S. P. Ponomarev 《Mathematical Notes》1995,58(3):960-965
A functionf∶G→ℝ
n
, whereG is an open set in ℝ
n
, has theN
−1-property if for allE⊂ℝ
n
we have {|E|=0⇒|f
−1(E)|=0} (|·| is the Lebesgue measure). The article is concerned with the relations between theN
−1-property of functions, the maximal rank of derivatives, and the differentiability almost everywhere of composite functions.
Translated fromMatematicheskie Zametki, Vol. 58. No. 3, pp. 411–418, September, 1995. 相似文献
15.
Andreas Fr?hlich 《Annali dell'Universita di Ferrara》2000,46(1):11-19
We consider weights of Muckenhoupt classA
q, 1<q<∞. For a bounded Lipschitz domain Ω⊂ℝn we prove a compact embedding and a Poincaré inequality in weighted Sobolev spaces. These technical tools allow us to solve
the weak Neumann problem for the Laplace equation in weighted spaces on ℝn, ℝn
+, on bounded and on exterior domains Ω with boundary of classC
1, which will yield the Helmholtz decomposition ofL
ω
q(Ω)n for general ω∈A
q. This is done by transferring the method of Simader and Sohr [4] to the weighted case. Our result generalizes a result of
Farwig and Sohr [2] where the Helmholtz decomposition ofL
ω
p(Ω)n is proved for an exterior domain and weights of Muckenhoupt class without singularities or degeneracies in a neighbourhood
of ϖΩ.
Sunto In questo lavoro consideriamo dei pesi della classe di MuckenhouptA q, 1<q<∞. Per un dominio limitato lipschitziano Ω⊂ℝn, dimostriamo una immersione compatta ed una disuguaglianza di Poincaré in spazi di Sobolev con peso. Questa tecnica ci consente di risolvere il problema debole di Neumann per l’equazione di Laplace in spazi pesati in ℝn, ℝn + in domini limitati ed in domini esterni con frontiera di classeC 1, che conduce alla decomposizione di Helmholtz diL ω q(Ω)n per un qualsiasi ω∈A q. Il risultato è ottenuto trasferendo il metodo di Simader e Sohr [4] al caso pesato. Quello qui presente estende un risultato di Farwig e Sohr [2] dove la decomposizione di Helmholtz diL ω q(Ω)n è dimostrata per domini esterni e pesi della classe di Muckenhoupt privi di singolarità in un intorno di ϖΩ.相似文献
16.
Let G be a locally compact group with a weight function ω. Recently, we have shown that the Banach space L0∞ (G,1/ω) can be identified with the strong dual of L1(G, ω)equipped with some locally convex topologies τ. Here we use this duality to introduce an Arens multiplication on (L1(G, ω), τ)**, and prove that the topological center of (L1(G, ω), τ)** is (L1(G, ω); this enables us to conclude that (L1(G, ω), τ) is Arens regular if and only if G is discrete. We also give a characterization for Arens regularity of L0∞ (G, 1/ω)1.
Received: 8 March 2005 相似文献
17.
Y. Lacroix 《Israel Journal of Mathematics》2002,132(1):253-263
LetG denote the set of decreasingG: ℝ→ℝ withGэ1 on ]−∞,0], and ƒ
0
∞
G(t)dt⩽1. LetX be a compact metric space, andT: X→X a continuous map. Let μ denone aT-invariant ergodic probability measure onX, and assume (X, T, μ) to be aperiodic. LetU⊂X be such that μ(U)>0. Let τ
U
(x)=inf{k⩾1:T
k
xεU}, and defineG
U
(t)=1/u(U)u({xεU:u(U)τU(x)>t),tεℝ We prove that for μ-a.e.x∈X, there exists a sequence (U
n
)
n≥1
of neighbourhoods ofx such that {x}=∩
n
U
n
, and for anyG ∈G, there exists a subsequence (n
k
)
k≥1
withG
U
n
k
↑U weakly.
We also construct a uniquely ergodic Toeplitz flowO(x
∞,S, μ), the orbit closure of a Toeplitz sequencex
∞, such that the above conclusion still holds, with moreover the requirement that eachU
n
be a cylinder set.
In memory of Anzelm Iwanik 相似文献
18.
Martin Fuchs 《manuscripta mathematica》1991,72(1):131-140
Given a smooth domain Ω in ℝ
m+1 with compact closure and a smooth integrable functionh: ℝ
m+1→ℝ satisfyingh(x)≥H
∂Ω
(x) on ∂Ω whereH
∂ω denotes the mean curvature of ∂Ω calculated w.r.t. the interior unit normal we show that there is a setA⊂ℝ
m+1 with the properties
andH
∂A=h on ∂A. 相似文献
19.
Let X be a locally compact topological space and (X, E, Xω) be any triple consisting of a hyperfinite set X in a sufficiently saturated nonstandard universe, a monadic equivalence relation E on X, and an E-closed galactic set Xω ⊆ X, such that all internal subsets of Xω are relatively compact in the induced topology and X is homeomorphic to the quotient Xω/E. We will show that each regular complex Borel measure on X can be obtained by pushing down the Loeb measure induced by some internal function
X ? *\Bbb CX \rightarrow {}{^{\ast}{\Bbb C}}
. The construction gives rise to an isometric isomorphism of the Banach space M(X) of all regular complex Borel measures on X, normed by total variation, and the quotient
Mw(X)/M0(X){\cal M}_{\omega}(X)/{\cal M}_0(X)
, for certain external subspaces
M0(X), Mw(X){\cal M}_0(X), {\cal M}_{\omega}(X)
of the hyperfinite dimensional Banach space
*\Bbb CX{}{^{\ast}{\Bbb C}}^X
, with the norm ‖f‖1 = ∑x ∈ X |f(x)|. If additionally X = G is a hyperfinite group, Xω = Gω is a galactic subgroup of G, E is the equivalence corresponding to a normal monadic subgroup G0 of Gω, and G is isomorphic to the locally compact group Gω/G0, then the above Banach space isomorphism preserves the convolution, as well, i.e., M(G) and
Mw(G)/M0(G){\cal M}_{\omega}(G)/{\cal M}_0(G)
are isometrically isomorphic as Banach algebras. 相似文献
20.
Optimal in a certain sense sufficient conditions are given for the existence and uniqueness of ω-periodic solutions of the
nonautonomous ordinary differential equation u
(2m)
=f(t,u,...,u
(m-1)
), where the function f:ℝ×ℝ
m
→ℝ is periodic with respect to the first argument with period ω.
Received: December 21, 1999; in final form: August 12, 2000?Published online: October 2, 2001 相似文献