Maximal properties of the normalized Cauchy transform

We study the normalized Cauchy transform in the unit disk. Our goal is to find an analog of the classical theorem by M. Riesz for the case of arbitrary weights.

Let be a positive finite measure on the unit circle of the complex plane and . Denote by and the Cauchy integrals of the measures and , respectively. The normalized Cauchy transform is defined as . We prove that is bounded as an operator in for but is unbounded (in general) for 2$">. The associated maximal non-tangential operator is bounded for and has weak type but is unbounded for 2$">.

     

具有弱衰减初值的不同速度的半线性Klein-Gordon方程组解的生命区间估计

本文讨论了具弱衰减Cauchy初值的不同速度半线性Klein-Gordon方程组解的生命区间估计问题.当初值具有尺度ε时,得到生命区间的下界估计ε-2|logε|-α。（当空间维数d≥3时。α=2,当d=2时α=3）.     

高阶非齐次GBBM方程

本文研究了高阶非齐次ＧＢＢＭ方程的Ｃａｕｃｈｙ问题和初边值问题。对任意的有界或无界光滑区域Ω，采用Ｂａｎａｃｈ不动点原理及一系列的积分估计，建立了高阶非齐次ＧＢＢＭ方程的Ｃａｕｃｈｙ问题和初边值问题在Ｗ＾２ｍ，ｐ（Ω）上整体强解的存在唯一性，这些结果改进并完善了ＢＢＭ方程的已有结果，与此同时，我们还讨论了强解的正则性。     

ON THE BOUNDED AND UNBOUNDED SOLUTIONS OF ONE DIMENSIONAL NONLINEAR REACTION－DIFFUSION PROBLEM

ＯＮＴＨＥＢＯＵＮＤＥＤＡＮＤＵＮＢＯＵＮＤＥＤＳＯＬＵＴＩＯＮＳＯＦＯＮＥＤＩＭＥＮＳＩＯＮＡＬＮＯＮＬＩＮＥＡＲＲＥＡＣＴＩＯＮ－ＤＩＦＦＵＳＩＯＮＰＲＯＢＬＥＭ￥ＧＥＷＥＩＧＡＯＲ．Ｏ．ＷＥＢＥＲＡｂｓｔｒａｃｔ：Ｔｈｅｅｘｉｓｔｅｎｃｅｏｆｂ...     

A remark on the application of interpolatory quadrature rules to the numerical solution of singular integral equations

A Cauchy type singular integral equation of the first or the second kind can be numerically solved either directly or after its reduction (by the usual regularization procedure) to an equivalent Fredholm integral equation of the second kind. The equivalence of these two methods (that is, the equivalence both of the systems of linear algebraic equations to which the singular integral equation is reduced and of the natural interpolation formulae) is proved in this paper for a class of Cauchy type singular integral equations of the first kind and of the second kind (but with constant coefficients) for general interpolatory quadrature rules under sufficiently mild assumptions. The present results constitute an extension of a series of previous results concerning only Gaussian quadrature rules, based on the corresponding orthogonal polynomials and their properties.     

Convergence and numbers

Our paper is devoted to ernbeddings of the rational numbers Q into exotic groups, linear spaces and fields, all of which carry a complete sequential convergence compatible with the algebraic structure. We enlarge the usual metric convergence on Q and study the impact on the categorical sequential group completion of Q. We compare the completion with the real line. In particular, we construct a convergence compatible with the group structure of Q such that the resulting completion is a Q-linear space and the real numbers are a proper subspace of it.     

On the distributions of boundary values of Cauchy integrals

We use new methods to give short proofs to some known results on the distributions of boundary values of Cauchy integrals. We also indicate some further generalizations.

     

Differentiability of solutions of second-order functional differential equations with unbounded delay

In this paper we study the differentiability of solutions of the second-order semilinear abstract retarded functional differential equation with unbounded delay, specially when the underlying space is reflexive or at least has the Radon–Nikodym property. We apply our results to characterize the infinitesimal generators of several strongly continuous semigroups of linear operators that arise in the theory of linear abstract retarded functional differential equations with unbounded delay on a phase space defined axiomatically.     

Parameter Identification for Nonlinear Abstract Cauchy Problems Using Quasilinearization

A quasilinearization approach to parameter identification in nonlinear abstract Cauchy problems in which the parameter appears in the nonlinear term, is presented. This approach has two main advantages over the classical one: it is much more intuitive and the derivation of the algorithm is done without need of the sensitivity equations on which classical quasilinearization is based. Sufficient conditions for the convergence of the algorithm are derived in terms of the regularity of the solutions with respect to the parameters. A comparison with the standard approach is presented and an application is included in which the nonphysical parameters in a mathematical model for shape memory alloys are estimated.     

A Lanczos‐type algorithm for the QR factorization of regular Cauchy matrices

We present a fast algorithm for computing the QR factorization of Cauchy matrices with real nodes. The algorithm works for almost any input matrix, does not require squaring the matrix, and fully exploits the displacement structure of Cauchy matrices. We prove that, if the determinant of a certain semiseparable matrix is non‐zero, a three term recurrence relation among the rows or columns of the factors exists. Copyright © 2002 John Wiley & Sons, Ltd.