共查询到20条相似文献,搜索用时 31 毫秒
1.
Roger D. Nussbaum 《Israel Journal of Mathematics》1991,76(3):345-380
Suppose thatE is a finite-dimensional Banach space with a polyhedral norm ‖·‖, i.e., a norm such that the unit ball inE is a polyhedron. ℝ
n
with the sup norm or ℝ
n
with thel
1-norm are important examples. IfD is a bounded set inE andT:D→D is a map such that ‖T(y)−T(z)‖≤ ‖y−z‖ for ally andz inE, thenT is called nonexpansive with respect to ‖·‖, and it is known that for eachx ∈D there is an integerp=p(x) such that lim
j→∞
T
jp
(x) exists. Furthermore, there exists an integerN, depending only on the dimension ofE and the polyhedral norm onE, such thatp(x)≤N: see [1,12,18,19] and the references to the literature there. In [15], Scheutzow has raised a question about the optimal
choice ofN whenE=ℝ
n
,D=K
n
, the set of nonnegative vectors in ℝ
n
, and the norm is thel
1-norm. We provide here a reasonably sharp answer to Scheutzow’s question, and in fact we provide a systematic way to generate
examples and use this approach to prove that our estimates are optimal forn≤24. See Theorem 2.1, Table 2.1 and the examples in Section 3. As we show in Corollary 2.3, these results also provide information
about the caseD=ℝ
n
, i.e.,T:ℝ
n
→ℝ
n
isl
1-nonexpansive. In addition, it is conjectured in [12] thatN=2
n
whenE=ℝ
n
and the norm is the sup norm, and such a result is optimal, if true. Our theorems here show that a sharper result is true
for an important subclass of nonexpansive mapsT:(ℝ
n
,‖ · ‖∞)→(ℝ
n
,‖ · ‖∞).
Partially supported by NSF DMS89-03018. 相似文献
2.
Yehoram Gordon 《Israel Journal of Mathematics》1969,7(2):151-163
Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ
p(X) as inf{Σ
i
=1/m
|x*(x
i)|p
p Σ
i
=1/m
‖x
i‖p
p]1
p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x
1,x
2, …,x
m} ⊂X such that Σ
i
=1/m
‖x
i‖>0. It follows immediately from [2] thatμ
p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ
p(X) for various spaces, and obtain some asymptotic estimates ofμ
p(X) for general finite dimensional Banach spaces.
This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof.
A. Dvoretzky and Prof. J. Lindenstrauss. 相似文献
3.
Reinhard Wolf 《Israel Journal of Mathematics》1999,110(1):125-151
The average distance theorem of Gross implies that for each realN-dimensional Banach space (N≥2) there is a unique positive real numberr(E) with the following property: For each positive integern and for all (not necessarily distinct)x
1,x
2, …,x
n inE with ‖x
1‖=‖x
2‖=…=‖x
n‖=1, there exists anx inE with ‖x‖=1 such that
The main result of this paper shows, thatr(E)≤2−1/N for each realN-dimensional Banach spaceE (N≥2) with the so-called quasihypermetric property (which is equivalent toE isL
1-embeddable). Moreover, equality holds if and only ifE is isometrically isomorphic to ℝ
N
equipped with the usual 1-norm. 相似文献
4.
It is well known that in the case of the Luxemburg normE
Φ (resp.h
Φ) is anM ideal inL
Φ (resp.l
Φ), see [1], [9], [15] and [6]; [17] and [18]. It is proved in this paper that in the case of the Orlicz normE
Φ (resph
Φ) is anM-ideal inL
Φ (resp.l
Φ) iff Φ satisfies the suitable Δ2 or Φ*(a(Φ*)), where a(Φ*) is linear on the interval [0,u]} and Φ* denotes the function complementary to Φ in the sense of Young. It is also proved that any linear continuous regular (i.e.
order continuous) functional ξυ overE
Φ (resp.h
Φ) generated byv∈ L(h
Φ*) (resp.v∈ L(h
Φ*)) which attains its norm on the unit sphereS(E
Φ) (resp.S(h
Φ)), has a unique norm-preserving extension toL
Φ (resp.l
Φ). Finally, it is proved thatL
Φ (resp.l
Φ) has the property that any linear continuous regular functional ξυ overE
Φ (resp.h
Φ) has a unique norm-preserving extension toL
Φ (resp.l
Φ) iff Φ orE
Φ satisfies the suitable Δ2 and in the second case Φ* attains the value 1. 相似文献
5.
M. Zippin 《Israel Journal of Mathematics》1981,39(4):349-358
It is proved that there exists a positive function Φ(∈) defined for sufficiently small ∈ 〉 0 and satisfying limt→0 Φ(∈) =0 such that for any integersn 〉>0, ifQ is a projection ofl
1
n
onto ak-dimensional subspaceE with ‖|Q‖|≦1+∈ then there is an integerh〉=k(1−Φ(∈)) and anh-dimensional subspaceF ofE withd(F,l
1
h
) 〈= 1+Φ (∈) whered(X, Y) denotes the Banach-Mazur distance between the Banach spacesX andY. Moreover, there is a projectionP ofl
1
n
ontoF with ‖|P‖| ≦1+Φ(∈).
Author was partially supported by the N.S.F. Grant MCS 79-03042. 相似文献
6.
Thierry De Pauw 《Journal of Geometric Analysis》2002,12(1):29-61
A concentrated (ξ, m) almost monotone measure inR
n
is a Radon measure Φ satisfying the two following conditions: (1) Θ
m
(Φ,x)≥1 for every x ∈spt (Φ) and (2) for everyx ∈R
n
the ratioexp [ξ(r)]r−mΦ(B(x,r)) is increasing as a function of r>0. Here ξ is an increasing function such thatlim
r→0-ξ(r)=0. We prove that there is a relatively open dense setReg (Φ) ∋spt (Φ) such that at each x∈Reg(Φ) the support of Φ has the following regularity property: given ε>0 and λ>0 there is an m dimensional spaceW ⊂R
n
and a λ-Lipschitz function f from x+W into x+W‖ so that (100-ε)% ofspt(Φ) ∩B (x, r) coincides with the graph of f, at some scale r>0 depending on x, ε, and λ. 相似文献
7.
G. S. Watson 《Annals of the Institute of Statistical Mathematics》1986,38(1):263-275
Summary We consider distributions with densities of the formf(μ′x) andf(‖x
v
‖) where μ andx are unit vectors inR
q
and ‖x
v
‖ is the norm of the part ofx in somes dimensional subspaceV ofR
q
. For several loss functions, optimal Bayesian and Pitman estimators of μ andV are given. When uniform priors are used, these estimators are identical. Then the infinitesimal robustness characteristics
of several special cases of these estimators are calculated. 相似文献
8.
Jie-Cheng Chen 《Israel Journal of Mathematics》1993,81(1-2):193-202
In this paper, we get a necessary and sufficient condition on the weights (μ,v) for the Poisson integral operator to be bounded fromL
Φ(R
n, v(x)dx) to weak-L
Φ(R
+
n+1
,dμ), where Φ is anN-function satisfying the Δ2-condition. We also find a necessary and sufficient condition on the weights (μ,v) for the Poisson integral operator to be bounded fromL
Φ(R
n,v(x)dx) toL
Φ(R
+
n+1
,dμ) under some additional condition.
Partially supported by NNSF of P.R. China 相似文献
9.
YANG Shouzhi & PENG Lizhong Department of Mathematics Shantou University Shantou China LMAM School of Mathematical Sciences Peking University Beijing China 《中国科学A辑(英文版)》2006,49(1):86-97
An algorithm is presented for raising an approximation order of any given orthogonal multiscaling function with the dilation factor a. Let φ(x) = [φ1(x),φ2(x),…,φr(x)]T be an orthogonal multiscaling function with the dilation factor a and the approximation order m. We can construct a new orthogonal multiscaling function φnew(x) = [ φT(x). f3r 1(x),φr 2(x),…,φr s(x)}T with the approximation order m L(L ∈ Z ). In other words, we raise the approximation order of multiscaling function φ(x) by increasing its multiplicity. In addition, we discuss an especial setting. That is, if given an orthogonal multiscaling function φ(x) = [φ1 (x), φ2(x), …, φr(x)]T is symmetric, then the new orthogonal multiscaling function φnew(x) not only raise the approximation order but also preserve symmetry. Finally, some examples are given. 相似文献
10.
Let X be a normed space that satisfies the Johnson–Lindenstrauss lemma (J–L lemma, in short) in the sense that for any integer
n and any x
1,…,x
n
∈X, there exists a linear mapping L:X→F, where F⊆X is a linear subspace of dimension O(log n), such that ‖x
i
−x
j
‖≤‖L(x
i
)−L(x
j
)‖≤O(1)⋅‖x
i
−x
j
‖ for all i,j∈{1,…,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-dimensional subspace of X embeds into Hilbert space with distortion
22O(log*n)2^{2^{O(\log^{*}n)}}
. On the other hand, we show that there exists a normed space Y which satisfies the J–L lemma, but for every n, there exists an n-dimensional subspace E
n
⊆Y whose Euclidean distortion is at least 2Ω(α(n)), where α is the inverse Ackermann function. 相似文献
11.
Let E be a Banach lattice and L1(μ, E) be the space of E-valued Bochner integrable functions. Some order properties of L1(μ, E) are given. It is shown that Ls∞(μ, Z(E)) is the ideal centre of L1(μ, E) and it is obtained a Radon-Nikodym type theorem for B -integrable functions.
相似文献
12.
Suppose that(T
t
)t>0 is aC
0 semi-group of contractions on a Banach spaceX, such that there exists a vectorx∈X, ‖x‖=1 verifyingJ
−1(Jx)={x}, whereJ is the duality mapping fromX toP(X
*). If |<T
t
x,f>|→1, whent→+∞ for somef∈X
*, ‖f‖≤1 thenx is an eigenvector of the generatorA, associated with a purcly imaginary eigenvalue. Because of Lin's example [L], the hypothesis onx∈X is the best possible.
If the hypothesisJ
−1(Jx)={x} is not verified, we can prove that ifJx is a singleton and ifJ
−1(Jx) is weakly compact, then if |<T
t
x, f>|→1, whent→+∞ for somef∈X
*, ‖f‖≤1, there existsy∈J
−1(Jx) such thaty is an eigenvector of the generatorA, associated with a purely imaginary eigenvalue. We give also a counter-example in the case whereX is one of the spaces ℓ1 orL
1. 相似文献
13.
Jean-René Licois 《Journal d'Analyse Mathématique》1995,66(1):1-36
LetM be a compact riemannian manifold,h an odd function such thath(r)/r is non-decreasing with limit 0 at 0. Letf(r)=h(r)-γr and assume there exist non-negative constantsA andB and a realp>1 such thatf(r)>Ar
P-B. We prove that any non-negative solutionu ofu
tt+Δgu=f(u) onM x ℝ+ satisfying Dirichlet or Neumann boundary conditions on ϖM converges to a (stationary) solution of Δ
g
Φ=f(Φ) onM with exponential decay of ‖u-Φ‖C
2(M). For solutions with non-constant sign, we prove an homogenisation result for sufficiently small λ; further, we show that
for every λ the map (u(0,·),u
t(0,·))→(u(t,·), u
t(t,·)) defines a dynamical system onW
1/2(M)⊂C(M)×L
2(M) which possesses a compact maximal attractor.
相似文献
14.
G. Schlüchtermann 《manuscripta mathematica》1991,73(1):397-409
A sufficient condition is given when a subspaceL⊂L
1(μ,X) of the space of Bochner integrable function, defined on a finite and positive measure space (S, Φ, μ) with values in a Banach spaceX, is locally uniformly convex renormable in terms of the integrable evaluations {∫
A
fdμ;f∈L}. This shows the lifting property thatL
1(μ,X) is renormable if and only ifX is, and indicates a large class of renormable subspaces even ifX does not admit and equivalent locally uniformly convex norm. 相似文献
15.
Rainer Wittmann 《Israel Journal of Mathematics》1987,59(1):8-28
LetT be a positive linear contraction inL
p (1≦p<∞), then we show that lim ‖T
pf −T
n+1
f‖
p
≦(1 − ε)21/p
(f∈L
p
+
, ε>0 independent off) implies already limn
n→∞ ‖T
nf −T
n+1
n+1f ‖p
p=0. Several other related results as well as uniform variants of these are also given. Finally some similar results inLsu/t8 andC(X) are shown. 相似文献
16.
Robert S. Strichartz 《Journal of Geometric Analysis》1991,1(3):269-289
Let μ be a measure on ℝn that satisfies the estimate μ(B
r(x))≤cr
α for allx ∈ ℝn and allr ≤ 1 (B
r(x) denotes the ball of radius r centered atx. Let ϕ
j,k
(ɛ)
(x)=2
nj2ϕ(ɛ)(2
j
x-k) be a wavelet basis forj ∈ ℤ, κ ∈ ℤn, and ∈ ∈E, a finite set, and letP
j
(T)=Σɛ,k
<T,ϕ
j,k
(ɛ)
>ϕ
j,k
(ɛ)
denote the associated projection operators at levelj (T is a suitable measure or distribution). Iff ∈Ls
p(dμ) for 1 ≤p ≤ ∞, we show thatP
j(f dμ) ∈ Lp(dx) and ||P
j
(fdμ)||L
p(dx)≤c2
j((n-α)/p′))||f||L
p(dμ) for allj ≥ 0. We also obtain estimates for the limsup and liminf of ||P
j
(fdμ)||L
p(dx) under more restrictive hypotheses.
Communicated by Guido Weiss 相似文献
17.
Let (Ω,Σ,μ) be a measure space and letP be an operator onL
2(Ω,Σ,μ) with ‖P‖≦1,Pf≧0 a.e. wheneverf≧0. If the subspaceK is defined byK={x| ||P
n
x||=||P
*n
x||=||x||,n=1,2,...} thenK=L
2(Ω,Σ1,μ), where Σ1 ⊂ Σ and onK the operatorP is “essentially” a measure preserving transformation. Thus the eigenvalues ofP of modulus one, form a group under multiplication.
This last result was proved by Rota for finiteμ here finiteness is not assumed) and is a generalization of a theorem of Frobenius and Perron on positive matrices.
The research reported in this document has been sponsored in part by Air Force Office of Scientific Research, OAR through
the European Office, Aerospace Research, United States Air Force. 相似文献
18.
S. J. Montgomery-Smith 《Israel Journal of Mathematics》1989,67(1):123-128
We show that the canonical embeddingC(K) →L
Φ(μ) has Gaussian cotypep, where μ is a Radon probability measure onK, and Φ is an Orlicz function equivalent tot
p(logt)
p/2 for larget. 相似文献
19.
LetX be a Banach space and letA be the infinitesimal generator of a differentiable semigroup {T(t) |t ≥ 0}, i.e. aC
0-semigroup such thatt ↦T(t)x is differentiable on (0, ∞) for everyx εX. LetB be a bounded linear operator onX and let {S(t) |t ≥ 0} be the semigroup generated byA +B. Renardy recently gave an example which shows that {S(t) |t ≥ 0} need not be differentiable. In this paper we give a condition on the growth of ‖T′(t)‖ ast ↓ 0 which is sufficient to ensure that {S(t) |t ≥ 0} is differentiable. Moreover, we use Renardy’s example to study the optimality of our growth condition. Our results can
be summarized roughly as follows:
We also show that if lim sup
t→0+t
p ‖T′(t)‖<∞ for a givenp ε [1, ∞), then lim sup
t→0+t
p‖S′(t)‖<∞; it was known previously that if limsup
t→0+t
p‖T′(t)‖<∞, then {S(t) |t ≥ 0} is differentiable and limsup
t→0+t
2p–1‖S′(t)‖<∞. 相似文献
(i) | If lim sup t→0+t log‖T′(t)‖/log(1/2) = 0 then {S(t) |t ≥ 0} is differentiable. |
(ii) | If 0<L=lim sup t→0+t log‖T′(t)‖/log(1/2)<∞ thent ↦S(t ) is differentiable on (L, ∞) in the uniform operator topology, but need not be differentiable near zero |
(iii) | For each function α: (0, 1) → (0, ∞) with α(t)/log(1/t) → ∞ ast ↓ 0, Renardy’s example can be adjusted so that limsup t→0+t log‖T′(t)‖/α(t) = 0 andt →S(t) is nowhere differentiable on (0, ∞). |
20.
Ulrich Krengel 《Israel Journal of Mathematics》1987,58(2):193-197
It is shown that the pointwise ergodic theorem for Markov operators inL
1, having a finite invariant measure, fails to extend to the case of nonlinear operators.
Recall thatT is called nonexpansive inL
p
if ‖Tf – Tg ‖
p
≦‖f – g‖
p
holds for allf andg. 相似文献