共查询到20条相似文献,搜索用时 31 毫秒
1.
Let (E, ¦·¦) be a uniformly convex Banach space with the modulus of uniform convexity of power type. Let be the convolution of the distribution of a random series inE with independent one-dimensional components and an arbitrary probability measure onE. Under some assumptions about the components and the smoothness of the norm we show that there exists a constant such that |{·<t}–{·+r<t}|r
q
, whereq depends on the properties of the norm. We specify it in the case ofL
spaces, >1. 相似文献
2.
H. Woźniakowski 《Numerische Mathematik》1977,28(2):191-209
Summary This paper contains the rounding error analysis for the Chebyshev method for the solution of large linear systemsAx+g=0 whereA=A
* is positive definite. We prove that the Chebyshev method in floating point arithmetic is numerically stable, which means that the computed sequence {x
k} approximates the solution such that
x
k
– is of order AA
–1 where is the relative computer precision.We also point out that in general the Chebyshev method is not well-behaved, which means that the computed residualsr
k=Ax
k+g are of order A2A
–1.This work was supported in part by the Office of Naval Research under Contract N0014-67-0314-0010, NR 044-422 and by the National Science Foundation under Grant GJ32111 相似文献
3.
H. Woźniakowski 《Numerische Mathematik》1978,30(3):301-314
Summary We deal with the rounding error analysis of successive approximation iterations for the solution of large linear systemsA
x
=b. We prove that Jacobi, Richardson, Gauss-Seidel and SOR iterations arenumerically stable wheneverA=A
*>0 andA has PropertyA. This means that the computed resultx
k
approximates the exact solution with relative error of order A·A
–1 where is the relative computer precision. However with the exception of Gauss-Seidel iteration the residual vector Ax
k
–b is of order A2 A
–1 and hence the remaining three iterations arenot well-behaved.This work was partly done during the author's visit at Carnegie-Mellon University and it was supported in part by the Office of Naval Research under Contract N00014-76-C-0370; NR 044-422 and by the National Science Foundation under Grant MCS75-222-55 相似文献
4.
Takeaki Yamazaki 《Integral Equations and Operator Theory》2002,43(2):237-247
In 1951, Heinz showed the following useful norm inequality:If A, B0and XB(H), then AXB
r
X1–r
A
r
XB
r
holds for r [0, 1]. In this paper, we shall show the following two applications of this inequality:Firstly, by using Furuta inequality, we shall show an extension of Cordes inequality. And we shall show a characterization of chaotic order (i.e., logAlogB) by a norm inequality.Secondly, we shall study the condition under which
, where
is Aluthge transformation ofT. Moreover we shall show a characterization of normaloid operators (i.e.,r(T)=T) via Aluthge transformation. 相似文献
5.
We consider in Hilbert spaces linear ill-posed problems Ax = y with noisy data y
satisfying y
–y. Regularized approximations x
r
to the minimum-norm solution x
of Ax = y are constructed by continuous regularization methods or by iterative methods. For the choice of the regularization parameter r (the stopping index n in iterative methods) the following monotone error rule (ME rule) is used: we choose r = r
ME (n = n
ME) as the largest r-value with the guaranteed monotonical decrease of the error x
r
– x
for r [0, r
ME] (x
n
– x
<#60; x
n–1
– x
for n = 1, 2, ..., n
ME). Main attention is paid to iterative methods of gradient type and to nonstationary implicit iteration methods. As shown, the ME rule leads for many methods to order optimal error bounds. Comparisons with other rules for the choice of the stopping index are made and numerical examples are given.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
6.
For a vector ofk+1 matrix power series, a superfast algorithm is given for the computation of multi-dimensional Padé systems. The algorithm provides a method for obtaining matrix Padé, matrix Hermite Padé and matrix simultaneous Padé approximants. When the matrix power series is normal or perfect, the algorithm is shown to calculate multi-dimensional matrix Padé systems of type (n
0,...,n
k
) inO(n · log2n) block-matrix operations, where n=n
0+...+n
k
. Whenk=1 and the power series is scalar, this is the same complexity as that of other superfast algorithms for computing Padé systems. Whenk>1, the fastest methods presently compute these matrix Padé approximants with a complexity ofO(n2). The algorithm succeeds also in the non-normal and non-perfect case, but with a possibility of an increase in the cost complexity.Supported in part by NSERC grant No. A8035.Partially supported by NSERC operating grant No. 6194. 相似文献
7.
Devices such as neural networks typically approximate the elements of some function space X by elements of a nontrivial finite union M of finite-dimensional spaces. It is shown that if X=L
p
() (1<p< and R
d
), then for any positive constant and any continuous function from X to M, f–(f)>f–M+ for some f in X. Thus, no continuous finite neural network approximation can be within any positive constant of a best approximation in the L
p
-norm. 相似文献
8.
Ryszard Rudnicki 《Integral Equations and Operator Theory》1996,24(3):320-327
A class of Markov operators appearing in biomathematics is investigated. It is proved that these operators are asymptotic stable inL
1, i.e. lim
n
P
n f=0 forfL
1 and f(x) dx=0. 相似文献
9.
John Dye 《Integral Equations and Operator Theory》1989,12(1):12-22
Let {T1, ..., TN} be a finite set of linear contraction mappings of a Hilbert space H into itself, and let r be a mapping from the natural numbers N to {1, ..., N}. One can form Sn=Tr(n)...Tr(1) which could be described as a random product of the Ti's. Roughly, the Sn converge strongly in the mean, but additional side conditions are necessary to ensure uniform, strong or weak convergence. We examine contractions with three such conditions. (W): xn1, Txn1 implies (I-T)xn0 weakly, (S): xn1, Txn1 implies (I-T)xn0 strongly, and (K): there exists a constant K>0 such that for all x, (I-T)x2K(x2–Tx2).We have three main results in the event that the Ti's are compact contractions. First, if r assumes each value infinitely often, then Sn converges uniformly to the projection Q on the subspace i=
1
N
[x|Tix=x]. Secondly we prove that for such compact contractions, the three conditions (W), (S), and (K) are equivalent. Finally if S=S(T1, ..., TN) denotes the algebraic semigroup generated by the Ti's, then there exists a fixed positive constant K such that each element in S satisfies (K) with that K. 相似文献
10.
Peter Volkmann 《Mathematische Annalen》1978,234(2):139-144
Zusammenfassung In vorliegender Note wird ein Satz von Kato [7] über die Störung eines abgeschlossenen, normal auflösbaren OperatorsT mit endlichem Null-defekt (T) durch einen streng singulären Operator verallgemeinert. Zu diesem Zweck wird für jedes 0 mit Hilfe des Kuratowskischen Nichtkompaktheitsmaßes eine KlasseC
von beschränkten, linearen Operatoren eingeführt, welche sowohl die streng singulären Operatoren als auch die OperatorenS mit S enthält.Das erzielte Resultat steht in engem Zusammenhang mit den Untersuchungen von Gol'denteinn, Gohberg und Markus [5] und von Gol'denteienn und Markus [6]. 相似文献
11.
Guanggui Ding 《数学学报(英文版)》1995,11(4):417-421
IfT is an isomorphism ofL
(A, ) intoL
(B, ) which satisfies the condition T T
–11+, where (A, ) is a -finite measure space, thenT/T is close to an isometry with an error less than 4. 相似文献
12.
In this note, the optimal L
2-error estimate of the finite volume element method (FVE) for elliptic boundary value problem is discussed. It is shown that u–u
h
0Ch
2|ln h|1/2f1,1 and u–u
h
0Ch
2f1,p
, p>1, where u is the solution of the variational problem of the second order elliptic partial differential equation, u
h
is the solution of the FVE scheme for solving the problem, and f is the given function in the right-hand side of the equation. 相似文献
13.
Gerald Beer 《Monatshefte für Mathematik》1993,115(4):281-290
LetX be a Banach space, and let {f
i:iI} be a family of proper lower semicontinuous convex functions defined onX, each of whose epigraphs meets a fixed bound subset ofX×. We say that {f
i:iI} is uniformly linearly minorized if there exists a positive scalar such that for alliI andxX, we havef
i(x)–(1+x). We present two very different characterizations of uniform linear minorization for such a family. Using one of these, we show that either strong or weak epi-convergence of a sequence of convex functions at some point in the effective domain of the limit implies, uniform linear minorization for the entire sequence.With 1 Figure 相似文献
14.
Summary Ann×n complex matrixB is calledparacontracting if B21 and 0x[N(I-B)]Bx2<x2. We show that a productB=B
k
B
k–1
...B
1 ofk paracontracting matrices is semiconvergent and give upper bounds on the subdominant eigenvalue ofB in terms of the subdominant singular values of theB
i
's and in terms of the angles between certain subspaces. Our results here extend earlier results due to Halperin and due to Smith, Solomon and Wagner. We also determine necessary and sufficient conditions forn numbers in the interval [0, 1] to form the spectrum of a product of two orthogonal projections and hence characterize the subdominant eigenvalue of such a product. In the final part of the paper we apply the upper bounds mentioned earlier to provide an estimate on the subdominant eigenvalue of the SOR iteration matrix associated with ann×n hermitian positive semidefinite matrixA none of whose diagonal entries vanish.The work of this author was supported in part by NSF Research Grant No. MCS-8400879 相似文献
15.
B. T. Polyak 《Set-Valued Analysis》2001,9(1-2):159-168
Let f: XY be a nonlinear differentiable map, X,Y are Hilbert spaces, B(a,r) is a ball in X with a center a and radius r. Suppose f
(x) is Lipschitz in B(a,r) with Lipschitz constant L and f
(a) is a surjection: f
(a)X=Y; this implies the existence of >0 such that f
(a)*
yy, yY. Then, if r,/(2L), the image F=f(B(a,)) of the ball B(a,) is convex. This result has numerous applications in optimization and control. First, duality theory holds for nonconvex mathematical programming problems with extra constraint x–a. Special effective algorithms for such optimization problems can be constructed as well. Second, the reachability set for small power control is convex. This leads to various results in optimal control. 相似文献
16.
The Komlós-Révész theorem states: For r.v.s.X
n
with X
n
1M there exists a subsequenceX
k
n
and a r.v.X with X1M such that
相似文献
17.
B. P. Duggal 《Monatshefte für Mathematik》1991,112(4):265-270
LetB (H) denote the algebra of operators on the separable Hilbert spaceH. LetC
2 denote the (Hilbert) space of Hilbert-Schmidt operators onH, with norm .2 defined by S
2
2
=(S,S)=tr(SS
*). GivenA, B B (H), define the derivationC (A, B):B(H)B(H) byC(A, B)X=AX-XB. We show that C(A,B)X+S
2
2
=C(A,B)X
2
2
+S
2
2
holds for allXB(H) and for everySC
2 such thatC(A, B)S=0 if and only if
reducesA, ker
S reducesB, andA |
S and B| ker
S are unitarily equivalent normal operators. We also show that ifA, BB(H) are contractions andR(A, B)B(H)B(H) is defined byR(A, B)X=AXB-X, thenSC
2 andR(A, B)S=0 imply R(A,B)X+S
2
2
=R(A,B)X
2
2
+S
2
2
for allXB(H). 相似文献
18.
A. S. Markus 《Mathematical Notes》1969,5(4):277-281
We shall establish certain characteristic properties of Bari* bases of subspaces. We shall show that a complete sequence of finite-dimensional subspaces {N
j}1
is a Bari basis if and only if each sequence {j{1
(j€N
j, j=1) is a Bari basis of its own closed linear hull.Translated from Matematicheskie Zametki, Vol. 5, No. 4, pp. 461–469, April, 1969. 相似文献
19.
Paweŀ Hitczenko 《Probability Theory and Related Fields》1990,86(2):225-238
Summary Let (f
n
) be a martingale. We establish a relationship between exponential bounds for the probabilities of the typeP(|f
n
|>·T(f
n
)) and the size of the constantC
p
appearing in the inequality f
*
p
C
p
T
*(f)
p
, for some quasi-linear operators acting on martingales.This research was supported in part by NSF Grant, no. DMS-8902418On leave from Academy of Physical Education, Warsaw, Poland 相似文献
20.
LetX be a complex Banach space andA: D(A)X a densely defined closed linear operator whose resolvent set contains the real line and for which (–A)–1 is bounded onR. We give a necessary and sufficient condition, in terms of the complex powers ofA and –A, for the existence of a decompositionX=X
+X
–, whereX
± are closed subspaces, invariant forA, the spectra of the reduced operatorsA
± are {(A);Im>0} and {(A);Im<0} respectively, and (–A
±)–1 is bounded forIm0.Finally we give an example of an operator in anL
p-type space for which the decomposition exists if 1<p<+ and does not exist ifp=1. 相似文献
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