共查询到20条相似文献,搜索用时 15 毫秒
1.
Kamil S. Kazimierski 《Computational Optimization and Applications》2011,48(2):309-324
For Tikhonov functionals of the form Ψ(x)=‖Ax−y‖
Y
r
+α‖x‖
X
q
we investigate a steepest descent method in the dual of the Banach space X. We show convergence rates for the proposed method and present numerical tests. 相似文献
2.
David Gilat 《Israel Journal of Mathematics》1988,63(3):270-280
For eachp>1, the supremum,S, of the absolute value of a martingale terminating at a random variableX inL
p, satisfiesES≦(Γ(q))1/q‖X‖p (q=p(p-1)-1).The maximum,M, of a mean-zero martingale which starts at zero and terminates atX, satisfiesES≦(Γ(q))1/q‖X‖p (q=p(p-1)-1), whereσ
q is the unique solution of the equationt = ‖Z −t ‖
q
for an exponentially distributed random variableZ with mean 1.σ
p has other characterizations and satisfies lim
p‖
q
− 1
σ
q =c withc determined byce
c+1 = 1. Equalities in (1) and (2) are attainable by appropriate martingales which can be realized as stopped segments of Brownian
motion. A presumably new property of the exponential distribution is obtained en route to inequality (2). 相似文献
3.
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
4.
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
5.
C. Alegre 《Acta Mathematica Hungarica》2009,122(4):357-372
If (X, p) and (Y, q) are two asymmetric normed spaces, the set LC(X, Y) of all continuous linear mappings from (X, p) to (Y, q) is not necessarily a linear space, it is a cone. If X and Y are two Banach lattices and p and q are, respectively, their associated asymmetric norms (p(x) = ‖+‖, q(y) = ‖y
+‖), we prove that the positive operators from X to Y are elements of the cone LC(X, Y). We also study the dual space of an asymmetric normed space and finally we give open mapping and closed graph type theorems
in the framework of asymmetric normed spaces. The classical results for normed spaces follow as particular cases.
The author acknowledges the support of the Ministerio de Educación y Ciencia of Spain and FEDER, under grant MTM2006-14925-C02-01
and Generalitat Valenciana under grant GV/2007/198. 相似文献
6.
We investigate the completeness and completions of the normed algebras (D
(1)(X), ‖ · ‖) for perfect, compact plane sets X. In particular, we construct a radially self-absorbing, compact plane set X such that the normed algebra (D
(1)(X), ‖ · ‖) is not complete. This solves a question of Bland and Feinstein. We also prove that there are several classes of
connected, compact plane sets X for which the completeness of (D
(1)(X), ‖ · ‖) is equivalent to the pointwise regularity of X. For example, this is true for all rectifiably connected, polynomially convex, compact plane sets with empty interior, for
all star-shaped, compact plane sets, and for all Jordan arcs in ℂ. 相似文献
7.
Let A and B be standard operator algebras on Banach spaces X and Y, respectively. The peripheral spectrum σπ (T) of T is defined by σπ (T) = z ∈ σ(T): |z| = maxw∈σ(T) |w|. If surjective (not necessarily linear nor continuous) maps φ, ϕ: A → B satisfy σπ (φ(S)ϕ(T)) = σπ (ST) for all S; T ∈ A, then φ and ϕ are either of the form φ(T) = A
1
TA
2
−1 and ϕ(T) = A
2
TA
1
−1 for some bijective bounded linear operators A
1; A
2 of X onto Y, or of the form φ(T) = B
1
T*B
2
−1 and ϕ(T) = B
2
T*B
−1 for some bijective bounded linear operators B
1;B
2 of X* onto Y.
相似文献
8.
Soon Mo JUNG 《数学学报(英文版)》2006,22(2):583-586
The stability problems of the exponential (functional) equation on a restricted domain will be investigated, and the results will be applied to the study of an asymptotic property of that equation. More precisely, the following asymptotic property is proved: Let X be a real (or complex) normed space. A mapping f : X → C is exponential if and only if f(x + y) - f(x)f(y) → 0 as ||x|| + ||y|| → ∞ under some suitable conditions. 相似文献
9.
Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let ‖.‖
X
and ‖.‖
Y
denote the supremum norms on X and Y, respectively. We first establish a result concerning a surjective map T between particular subsets of the uniform closures of A and B, preserving multiplicatively the norm, i.e. ‖Tf Tg‖
Y
= ‖fg‖
X
, for certain elements f and g in the domain. Then we show that if α ∈ ℂ {0} and T: A → B is a surjective, not necessarily linear, map satisfying ‖fg + α‖
X
= ‖Tf Tg + α‖
Y
, f,g ∈ A, then T is injective and there exist a homeomorphism φ: c(B) → c(A) between the Choquet boundaries of B and A, an invertible element η ∈ B with η(Y) ⊆ {1, −1} and a clopen subset K of c(B) such that for each f ∈ A,
$
Tf\left( y \right) = \left\{ \begin{gathered}
\eta \left( y \right)f\left( {\phi \left( y \right)} \right) y \in K, \hfill \\
- \frac{\alpha }
{{\left| \alpha \right|}}\eta \left( y \right)\overline {f\left( {\phi \left( y \right)} \right)} y \in c\left( B \right)\backslash K \hfill \\
\end{gathered} \right.
$
Tf\left( y \right) = \left\{ \begin{gathered}
\eta \left( y \right)f\left( {\phi \left( y \right)} \right) y \in K, \hfill \\
- \frac{\alpha }
{{\left| \alpha \right|}}\eta \left( y \right)\overline {f\left( {\phi \left( y \right)} \right)} y \in c\left( B \right)\backslash K \hfill \\
\end{gathered} \right.
相似文献
10.
Ralph deLaubenfels 《Israel Journal of Mathematics》1993,81(1-2):227-255
We show that, whenA generates aC-semigroup, then there existsY such that [M(C)] →Y →X, andA|
Y
, the restriction ofA toY, generates a strongly continuous semigroup, where ↪ means “is continuously embedded in” and ‖x‖[Im(C)]≡‖C
−1
x‖. There also existsW such that [C(W)] →X →W, and an operatorB such thatA=B|
X
andB generates a strongly continuous semigroup onW. If theC-semigroup is exponentially bounded, thenY andW may be chosen to be Banach spaces; in general,Y andW are Frechet spaces. If ρ(A) is nonempty, the converse is also true.
We construct fractional powers of generators of boundedC-semigroups.
We would like to thank R. Bürger for sending preprints, and the referee for pointing out reference [37]. This research was
supported by an Ohio University Research Grant. 相似文献
11.
W. T. Gowers 《Israel Journal of Mathematics》1990,69(2):129-151
We show that if 0<ε≦1, 1≦p<2 andx
1, …,x
n is a sequence of unit vectors in a normed spaceX such thatE ‖∑
l
n
εi
x
l‖≧n
1/p, then one can find a block basisy
1, …,y
m ofx
1, …,x
n which is (1+ε)-symmetric and has cardinality at leastγn
2/p-1(logn)−1, where γ depends on ε only. Two examples are given which show that this bound is close to being best possible. The first
is a sequencex
1, …,x
n satisfying the above conditions with no 2-symmetric block basis of cardinality exceeding 2n
2/p-1. This sequence is not linearly independent. The second example is a sequence which satisfies a lowerp-estimate but which has no 2-symmetric block basis of cardinality exceedingCn
2/p-1(logn)4/3, whereC is an absolute constant. This applies when 1≦p≦3/2. Finally, we obtain improvements of the lower bound when the spaceX containing the sequence satisfies certain type-condition. These results extend results of Amir and Milman in [1] and [2].
We include an appendix giving a simple counterexample to a question about norm-attaining operators. 相似文献
12.
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:
13.
Let (X, Xn; n ≥1) be a sequence of i.i.d, random variables taking values in a real separable Hilbert space (H, ||·||) with covariance operator ∑. Set Sn = X1 + X2 + ... + Xn, n≥ 1. We prove that, for b 〉 -1,
lim ε→0 ε^2(b+1) ∞ ∑n=1 (logn)^b/n^3/2 E{||Sn||-σε√nlogn}=σ^-2(b+1)/(2b+3)(b+1) B||Y|^2b+3 holds if EX=0,and E||X||^2(log||x||)^3bv(b+4)〈∞ where Y is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator ∑, and σ^2 denotes the largest eigenvalue of ∑. 相似文献 14.
Let X, Y be complex Banach spaces. Let G be a bounded balanced domain in X and B
Y
be the unit ball in Y. Assume that B
Y
is homogeneous. Let f: G → B
Y
be a holomorphic mapping. In this paper, we show that, if P = f(0), then we have Σ
k=0∞ ‖ D
φP
(P)[D
k
f(0)(z
k
)]‖/(k!‖D
φP
(P)‖) < 1 for z ∈ (1/3)G, where φP ∈ AutB
Y
) such that φP (P) = 0. Moreover, we show that the constant 1/3 is best possible, if B
Y
is the unit ball of a J*-algebra. The above result was proved by Liu and Wang in the case that G = B
Y
is one of the four classical domains in the sense of Hua. This result generalises a classical result of Bohr. 相似文献
15.
We prove that there exists a Lipschitz function froml
1 into ℝ2 which is Gateaux-differentiable at every point and such that for everyx, y εl
1, the norm off′(x) −f′(y) is bigger than 1. On the other hand, for every Lipschitz and Gateaux-differentiable function from an arbitrary Banach spaceX into ℝ and for everyε > 0, there always exist two pointsx, y εX such that ‖f′(x) −f′(y)‖ is less thanε. We also construct, in every infinite dimensional separable Banach space, a real valued functionf onX, which is Gateaux-differentiable at every point, has bounded non-empty support, and with the properties thatf′ is norm to weak* continuous andf′(X) has an isolated pointa, and that necessarilya ε 0.
This work has been initiated while the second-named author was visiting the University of Bordeaux. The second-named author
is supported by grant AV 1019003, A1 019 205, GA CR 201 01 1198. 相似文献
16.
Let B denote a separable Banach space with norm ‖⋅‖, and let μ be a probability measure on B for which linear functionals have mean zero and finite variance. Then there is a Hilbert space H
μ
determined by the covariance of μ such that H
μ
⊆B. Furthermore, for all ε>0 and x in the B-norm closure of H
μ
, there is a unique point, T
ε
(x), with minimum H
μ
-norm in the B-norm ball of radius ε>0 and center x. If X is a random variable in B with law μ, then in a variety of settings we obtain the central limit theorem (CLT) for T
ε
(X) and certain modifications of such a quantity, even when X itself fails the CLT. The motivation for the use of the mapping T
ε
(⋅) comes from the large deviation rates for the Gaussian measure γ determined by the covariance of X whenever γ exists. However, this is only motivation, and our results apply even when this Gaussian law fails to exist.
Research partially supported by NSA Grant H98230-06-1-0053. 相似文献
17.
LetX be a real linear normed space, (G, +) be a topological group, andK be a discrete normal subgroup ofG. We prove that if a continuous at a point or measurable (in the sense specified later) functionf:X →G fulfils the condition:f(x +y) -f(x) -f(y) ∈K whenever ‖x‖ = ‖y‖, then, under some additional assumptions onG,K, andX, there esists a continuous additive functionA :X →G such thatf(x) -A(x) ∈K. 相似文献
18.
M. Zippin 《Israel Journal of Mathematics》2000,115(1):253-268
A projectionP on a Banach spaceX with ‖P‖≤λ0 is called almost locally minimal if, for every α>0 small enough, the ballB(P,α) in the space of operatorsL(X) does not contain a projectionQ with ‖Q‖≤‖P‖(1–Dα2), whereD=D(λ0) is a constant independent of ‖P‖. It is shown that, for everyp≥1 and every compact abelian groupG, every translation invariant projection onL
p(G) is almost locally minimal. Orthogonal projections on ℓ
1
n
are investigated with respect to some weaker local minimality properties.
Participant in Workshop in Linear Analysis and Probability, Texas A&M University, College Station, Texas 1998. Partially supported
by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany). 相似文献
19.
Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type 总被引:17,自引:0,他引:17
W. A. Kirk 《Israel Journal of Mathematics》1974,17(4):339-346
LetX be a Banach space,K a nonempty, bounded, closed and convex subset ofX, and supposeT:K→K satisfies: for eachx∈K, lim sup
i→∞{sup
y∈K
‖t
ix−Tiy∼−‖x−y‖}≦0. IfT
N is continuous for some positive integerN, and if either (a)X is uniformly convex, or (b)K is compact, thenT has a fixed point inK. The former generalizes a theorem of Goebel and Kirk for asymptotically nonexpansive mappings. These are mappingsT:K→K satisfying, fori sufficiently large, ‖Tix−Tiy‖≦k
i‖x−y∼,x,y∈K, wherek
i→1 asi→∞. The precise assumption in (a) is somewhat weaker than uniform convexity, requiring only that Goebel’s characteristic of
convexity, ɛ0 (X), be less than one.
Research supported by National Science Foundation Grant GP 18045. 相似文献
20.
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
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