The measure of a translated ball in uniformly convex spaces

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.     

Stability inL 1 of some integral operators

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.     

Round-off error analysis of iterations for large linear systems

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     

The possibility of integral formulas in R n

We consider the integral We solve the problem of determination of necessary and sufficient conditions in order that (u) be independent of the values of u(x) inside a bounded domain . These conditions are written in the form of a set of differential equations for the functions f(x,u,¯p,T^ij) on the set m{x; u+¯p+ T^ij<}. For such functions (u) is represented in the form of a boundary integral.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 52, pp. 35–51, 1975.     

On Interpolation of the Fourier Maximal Operator in Orlicz Spaces

Let and be positive increasing convex functions defined on [0, ). Suppose satisfies the 2-condition, that is, (t)2 (C1t) for sufficiently large t, and has some nice properties. If -1(u)log(u+1) C2-1(u) for sufficiently large uthen we have S*(f) L CfL for all f L ([-, ])where S*(f) is the majorant function of partial sums of trigonometric Fourier series and fL is the Orlicz norm of f. This result is sharp.     

A fixed point theorem for local pseudo-contractions in uniformly convex spaces

It is proved that if D is a bounded open subset of a uniformly convex Banach space X and is a continuous mapping which is a local pseudo-contraction (e.g., locally nonexpansive) on D, then T has a fixed point in D if there exists xD such that z–Tz相似文献   

Representations approchees d'un groupe dans une algebre de banach

Let T be a continuous map from a compact group G to the group of invertible bounded linear operators on a Hilbert space H, the latter being endowed with the norm topology. If the norms T(gh)-T(g)T(h) are small enough (g,h G), we show that T is a small perturbation of some norm continuous representation of G on H.     

On the norm of polynomials of two adjoint projections

Letf(X, Y) be a polynomial of two non-commuting variables and letP be an arbitrary nontrivial projection operator in Hilbert space. The class of all polynomialsf(X, Y) for which f(P, P ^*) depends only onf and P are described. In the case when such a dependence exists the explicit formula is obtained. Some applications to singular integral operators are given.     

A note on operator norm inequalities

If P is a positive operator on a Hilbert space H whose range is dense, then a theorem of Foias, Ong, and Rosenthal says that: [(P)]^–1T[(P)]<-12 max {T, P^–1TP} for any bounded operator T on H, where is a continuous, concave, nonnegative, nondecreasing function on [0, P]. This inequality is extended to the class of normal operators with dense range to obtain the inequality [(N)]^–1T[(N)]<-12c² max {tT, N^–1TN} where is a complex valued function in a class of functions called vase-like, and c is a constant which is associated with by the definition of vase-like. As a corollary, it is shown that the reflexive lattice of operator ranges generated by the range NH of a normal operator N consists of the ranges of all operators of the form (N), where is vase-like. Similar results are obtained for scalar-type spectral operators on a Hilbert space.This author gratefully acknowledges the support of Central Michigan University in the form of a Research Professorship.     

A superfast algorithm for multi-dimensional Padé systems

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.     

Characterizations of logA≥logB and normaloid operators via Heinz inequality

In 1951, Heinz showed the following useful norm inequality:If A, B0and XB(H), then AXB ^r X^1–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.     

Convergence of random products of compact contractions in Hilbert space

Let {T₁, ..., T_N} 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 S_n=T_r(n)...T_r(1) which could be described as a random product of the T_i's. Roughly, the S_n 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): x_n1, Tx_n1 implies (I-T)x_n0 weakly, (S): x_n1, Tx_n1 implies (I-T)x_n0 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 T_i's are compact contractions. First, if r assumes each value infinitely often, then S_n converges uniformly to the projection Q on the subspace _i= ₁ ^N [x|T_ix=x]. Secondly we prove that for such compact contractions, the three conditions (W), (S), and (K) are equivalent. Finally if S=S(T₁, ..., T_N) denotes the algebraic semigroup generated by the T_i's, then there exists a fixed positive constant K such that each element in S satisfies (K) with that K.     

Numerical stability of the Chebyshev method for the solution of large linear systems

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     

On the stability of semi-implicit methods for ordinary differential equations

The aim of this paper is to analyze the stability properties of semi-implicit methods (such as Rosenbrock methods,W-methods, and semi-implicit extrapolation methods) for nonlinear stiff systems of differential equations. First it is shown that the numerical solution satisfies y ₁ (h)y ₀, if the method is applied with stepsizeh to the systemy =Ay ( denotes the logarithmic norm ofA). Properties of the function(x) are studied. Further, conditions for the parameters of a semi-implicit method are given, which imply that the method produces contractive numerical solutions over a large class of nonlinear problems for sufficiently smallh. The restriction on the stepsize, however, does not depend on the stiffness of the differential equation. Finally, the presented theory is applied to the extrapolation method based on the semi-implicit mid-point rule.     

Continuity of Approximation by Neural Networks in Lp Spaces

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.     

On the Monotone Error Rule for Parameter Choice in Iterative and Continuous Regularization Methods

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.     

Parsimonious Least Norm Approximation

A theoretically justifiable fast finite successive linear approximation algorithm is proposed for obtaining a parsimonious solutionto a corrupted linear system Ax=b+p, where the corruption p is due to noise or error in measurement. The proposedlinear-programming-based algorithm finds a solution x by parametrically minimizing the number of nonzeroelements in x and the error Ax-b-p₁.Numerical tests on a signal-processing-based exampleindicate that the proposed method is comparable to a method that parametrically minimizesthe 1-norm of the solution x and the error Ax-b-p₁, and that both methods are superior, byorders of magnitude, to solutions obtained by least squares as well by combinatorially choosing an optimal solution with a specific number of nonzero elements.     

An extremal property of outer functions

Our main result is the following: iff (z) is in the space H², and F(z) is its outer part, then F⁽ⁿ⁾_H2F⁽ⁿ⁾_H2(n=1,2,...), the left side being finite if the right side is finite. Under certain essential restrictions, this. inequality was proved by B. I. Korenblyum and V. S. Korolevich [1].Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 53–56, July, 1971.     

Perron-frobenius theory for Banach spaces with a hyperbolic cone

If X is a real Banach space, then the inequality x defines so-called hyperbolic cone in E=X. We develop a relevant version of Perron-Frobenius-Krein-Rutman theory.     

Extreme values and the law of the iterated logarithm

Summary IfX takes values in a Banach spaceB and is in the domain of attraction of a Gaussian law onB, thenX satisfies the compact law of the iterated logarithm (LIL) with respect to a regular normalizing sequence { _n } iffX satisfies a certain integrability condition. The integrability condition is equivalent to the fact that the maximal term of the sample {X ₁, X ₂,..., X _n} does not dominate the partial sums {S _n}, and here we examine the precise influence of these maximal terms and its relation to the compactLIL. In particular, it is shown that if one deletes enough of the maximal terms there is always a compactLIL with non-trivial limit set.Supported in part by NSF Grant MCS-8219742Work done while visiting the University of Wisconsin, Madison, with partial support by NSF Grant MCS-8219742