Small into isomorphisms onL ∞ spaces

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.     

An algorithm for best approximate solutions ofAx=b with a smooth strictly convex norm

Summary In this paper, overdetermined systems ofm linear equations inn unknowns are considered. With ^m equipped with a smooth strictly convex norm, ·, an iterative algorithm for finding the best approximate solution of the linear system which minimizes the ·-error is given. The convergence of the algorithm is established and numerical results are presented for the case when · is anl _p norm, 1<p<.Portions of this paper are taken from the author's Ph.D. thesis at Michigan State University     

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.     

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     

Discretely normed Abelian groups

A discrete norm on an Abelian groupA is a non-negative function · A which satisfies the triangle inequality, is homogenous with respect to scaling ofA by and is bounded away from 0 onA/{0}.A countable Abelian group is discretely normed if and only if the group is free.     

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.     

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.     

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     

On some variational problems for 2-dimensional Hermitian metrics

Summary Let (M, J, g) be a compact complex 2-dimensional Hermitian manifold with the Kähler form , and the torsion 1-form defined by d = . In this note we obtain the Euler-Lagrange equations for the variational functionals defined by ² and d², whereg runs in the space of all the Hermitian metrics onM. In the first case, the extremals are precisely the Kähler metrics [Gd]. In the second case, we also write down a formula for the second variation.Communicated by J. Szenthe     

The functional equation of multiplicative derivation is superstable on standard operator algebras

LetX be a real or complex infinite dimensional Banach space andA a standard operator algebra onX. Denote byB(X) the algebra of all bounded linear operators onX. Let : _{+ +} be a function with the property lim _t (t)t ^–1=0. Assume that a mappingD:A B(X) satisfies D(AB)–AD(B)–D(A)B<(A B) for all operatorsA, B D (no linearity or continuity ofD is assumed). ThenD is of the formD(A)=AT–TA for someTB(X).This work was supported by the Research Council of Slovenia     

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.     

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.     

Numerical stability of descent methods for solving linear equations

Summary In this paper we perform a round-off error analysis of descent methods for solving a liner systemAx=b, whereA is supposed to be symmetric and positive definite. This leads to a general result on the attainable accuracy of the computed sequence {x _i } when the method is performed in floating point arithmetic. The general theory is applied to the Gauss-Southwell method and the gradient method. Both methods appear to be well-behaved which means that these methods compute an approximationx _i to the exact solutionA ^–1 b which is the exact solution of a slightly perturbed linear system, i.e. (A+A)x _i =b, A of order A, where is the relative machine precision and · denotes the spectral norm.     

Numerical stability of GMRES

The Generalized Minimal Residual Method (GMRES) is one of the significant methods for solving linear algebraic systems with nonsymmetric matrices. It minimizes the norm of the residual on the linear variety determined by the initial residual and then-th Krylov residual subspace and is therefore optimal, with respect to the size of the residual, in the class of Krylov subspace methods. One possible way of computing the GMRES approximations is based on constructing the orthonormal basis of the Krylov subspaces (Arnoldi basis) and then solving the transformed least squares problem. This paper studies the numerical stability of such formulations of GMRES. Our approach is based on the Arnoldi recurrence for the actually, i.e., in finite precision arithmetic, computed quantities. We consider the Householder (HHA), iterated modified Gram-Schmidt (IMGSA), and iterated classical Gram-Schmidt (ICGSA) implementations. Under the obvious assumption on the numerical nonsingularity of the system matrix, the HHA implementation of GMRES is proved backward stable in the normwise sense. That is, the backward error for the approximation is proportional to machine precision . Additionally, it is shown that in most cases the norm of the residual computed from the transformed least squares problem (Arnoldi residual) gives a good estimate of the true residual norm, until the true residual norm has reached the level A x.This work was supported by NSF contract Int921824.     

Comparison theorems for nonlinear evolution equations and their application to the investigation of the dependence of the solutions of the equation ut=δϕ(u) ON ϕ and on the boundary conditions

One obtains estimates of the form, whereu. are generalized solutions of the equationsdu/dt=Au, du/dt=Bu whileA, B are non-linear,m-dissipative operators in a Banach space, and there exists an operatorP:D(A)D(B), such thatPw · W+BPw –Aw, uniformly on some setw. These results are applied to the investigation of the dependence of the solutions of the Cauchy, Dirichlet problems and of the problem with the boundary condition –du/dn=(u) for the equation u₁=(u) on the continuous nondecreasing functions and.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 183–198, 1984.The author is sincerely grateful to O. A. Ladyzhenskaya and N. N. Ural'tseva for their interest in this paper and for useful discussions.     

Upper bounds for theL p -norms of martingales

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     

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 Note on the Optimal L 2-Estimate of the Finite Volume Element Method

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 0}Ch ²|ln h|^1/2f_1,1 and u–u _{h 0}Ch ²f_1,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.     

Perturbation analysis of the orthogonal procrustes problem

Given two arbitrary real matricesA andB of the same size, the orthogonal Procrustes problem is to find an orthogonal matrixM such that the Frobenius norm MA – B is minimized. This paper treats the common case when the orthogonal matrixM is required to have a positive determinant. The stability of the problem is studied and supremum results for the perturbation bounds are derived.     

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.