On C*-Algebras of Toeplitz Operators on the Harmonic Bergman Space

Commutative algebras of Toeplitz operators acting on the Bergman space on the unit disk have been completely classified in terms of geometric properties of the symbol class. The question when two Toeplitz operators acting on the harmonic Bergman space commute is still open. In some papers, conditions on the symbols have been given in order to have commutativity of two Toeplitz operators. In this paper, we describe three different algebras of Toeplitz operators acting on the harmonic Bergman space: The C*-algebra generated by Toeplitz operators with radial symbols, in the elliptic case; the C*-algebra generated by Toeplitz operators with piecewise continuous symbols, in the parabolic and hyperbolic cases. We prove that the Calkin algebra of the first two algebras are commutative, like in the case of the Bergman space, while the last one is not.     

On the convergence of the spline collocation with discontinuous data

In this article we study the convergence of the collocation method in the case where the smoothest splines are used as trial functions. The given data is allowed to be piecewise continuous. Our model problem is stated by means of an explicit Fourier representation in the space of periodic functions. Thus the results are applicable e.g. to differential operators and to classical integral operators of the convolutional type. Error estimates are given for a class of Sobolev norms. An application to the single layer potential is discussed.     

An Approximation Theory for Operators Generated by Shifts

This paper develops a theory about the applicability of approximation methods to operators that can be represented as a function of a shift. An axiomatic approach is used, in which a small set of conditions that involve the operator and the method are proved to guarantee the applicability. All concrete methods we know for the underlying operators are subjected to this approach. We also study the behavior of the singular values and related questions. New results concerning Galerkin approximation methods for Mellin operators with piecewise continuous symbol are given to illustrate the application of the theory.     

Spline Approximation Methods with Uniform Meshes in Algebras of Multiplication and Convolution Operators

The purpose of this paper is to obtain necessary and sufficient conditions for maximum defect spline approximation methods with uniform meshes to be stable. The methods are applied to operators belonging to the closed subalgebra of ℒ︁ (L² (ℝ)) generated by operators of multiplication by piecewise continuous functions on ℝ and convolution operators also with piecewise continuousgenerating functions. To that purpose, a C*-algebra of sequences is introduced, which contains the special sequences of approximating operators we are interested in. There is a direct relationship between the applicability of the approximation method to a given operator and invertibility of the corresponding sequence in this C*-algebra. Exploring this relationship, applicability criteria are derived by the use of C*-algebra and Banach algebra techniques (essentialization, localization andidentification of the local algebras by means of construction of locally equivalent representations). Finally, examples are presented, including explicit conditions for the applicability of spline Galerkin methods to Wiener-Hopf operators with piecewise continuous symbols.     

Algebra of singular integral operators with a Carleman backward slowly oscillating shift

In this paper we study the Banach algebra of singular integral operators with piecewise continuous coefficients and a Carleman orientation-reversing slowly oscillating shift on the Lebesgue space with a power weight on the unit circle. The slow oscillation of the shift derivative, in contrast to the classic assumption on its piecewise continuity, leads to the appearance of massive local spectra for the considered operators. Applying localization techniques and the theory of Mellin pseudodifferential and associated limit operators, we construct a symbol calculus for the above-mentioned operator algebra and find a Fredholm criterion and an index formula for the operators in this algebra in terms of their symbols.Partially supported by CONaCYT grant, Cátedra Patrimonial, No. 990017-EX and by CONACYT project 32726-E, México.Partially supported by F. C. T. grant Praxis XXI/2/2.1/MAT/441/94, Portugal.     

Automorphisms on the Toeplitz Algebra with Piecewise Continuous Symbol on the Unit Ball

In this paper, we obtain a Fredholm index formula for Toeplitz operators whose symbols are certain piecewise continuous function matrices on the unit ball. Moreover, using this formula, we discuss the automorphisms on the corresponding Toeplitz algebra     

有界多连通域的具分段连续符号的Toeplitz算子

探讨多连通域的Bergman空间上的具有分段连续符号的Toeplitz算子,刻画了它们的本质谱和Fredholm指标.     

On the properties of the radiosity equation near corners

The radiosity equation is an integral equation of the second kind which describes the energy exchange by radiation between surfaces in R³. It is assumed that all surfaces are Lambertian reflectors and that all emitters are diffusive emitters. The radiosity equation plays an important role for the calculation of photo realistic images with the help of computers. Many surfaces which are used in practical calculations are only piecewise smooth and contain edges or corners. In this contribution we present regularity results for the solution of the radiosity equation in the vicinity of corners. The space of piecewise continuous functions is not suitable for this equation and we construct a new function space which contains the solution of the radiosity equation.     

One-dimensional nonlinear Laplacians under a 3-point boundary condition

We consider a three-point boundary value problem for operators such as the one-dimensional p-Laplacian, and show when they have solutions or not, and how many. The inverse operators are given by various formulas involving zeros of a real-valued function. They are shown to be orderpreserving, for some parameter values, and non-singleton valued for others. The operators are shown to be m-dissipative in the space of continuous functions.     

GLOBAL DISCONTINUOUS SOLUTIONS TO A CLASS OF DISCONTINUOUS INITIAL VALUE PROBLEMS FOR THE SYSTEM OF ISENTROPIC FLOW AND APPLICATIONS~(***)

In this paper, the authors discuss a kind of discontinuous initial value problems for the system of one-dimensional isentropic flow and prove the existence or nonexistence of global discontinuous solutions only containing one shock in a class of piecewise continuous and piecewise smooth functions. As applications, various interaction problems of a typical shock with a rarefaction wave are considered.     

Generalized factorization for a class of non-rational 2×2 matrix functions

In this paper the generalized factorization for a class of 2×2 piecewise continuous matrix functions on is studied. Using a space transformation the problem is reduced to the generalized factorization of a scalar piecewise continuous function on a contour in the complex plane. Both canonical and non-canonical generalized factorization of the original matrix function are studied.Sponsored by J.N.I.C.T. (Portugal) under grant no. 87422/MATM     

On algebras of singular integral operators with shift in Hölder weighted spaces

We consider algebras of singular integral operators with shift and piecewise Hölder coefficients in a Hölder weighted space on a Lyapunov contour. For this algebra, we construct the similarity isomorphism to the algebra of singular integral operators with piecewise Hölder coefficients in a Hölder space with “canonical” weight on the circle. We construct the symbol calculus, formulate necessary and sufficient conditions for the Fredholm property, and give the formula for the index of Fredholm operators.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 9, Suzdal Conference-3, 2003.     

Approximation by means of piecewise linear functions

In the present paper we use piecewise linear functions in order to obtain representations and estimates for the remainder in approximating continuous functions by positive linear operators. Applications of these results for Bernstein and Stancu’s operators are also presented. In addition, we give some partial results concerning the best constant problem for Bernstein operators with respect to the second order modulus of continuity.     

Integro-Differential Equations over a Closed Circuit with Gaussian Function in the Kernel

Integro-differential equations with kernels including hypergeometric Gaussian function that depends on the arguments ratio are studied over a closed curve in the complex plane. Special cases of the equations considered are the special integro-differential equation with Cauchy kernel, equations with power and logarithmic kernels. By means of the curvilinear convolution operator with the kernel of special kind, the equations with derivatives are reduced to the equations without derivatives. We find out the connection between special cases of the above-mentioned convolution operator and the known integral representations of piecewise analytical functions applied in the study of boundary value problems of the Riemann type. The correct statement of Noetherian property for the investigated class of equations is given. In this case, the operators corresponding to the equations are considered acting from the space of summable functions into the space of fractional integrals of the curvilinear convolution type. Examples of integro-differential equations solvable in a closed form are given.

     

THE DEFECT ITERATION OF THE FINITE ELEMENT FOR ELLIPTIC BOUNDARY VALUE PROBLEMS AND PETROV-GALERKIN APPROXIMATION

1.IntroductionFranketc.of.[l]establishedtheiterateddefectcorrectionschemeforfiniteelemelltofellipticboundaryproblems.FOrlinearellipticboundaryvalueproblem[2--5]havediscllssedtheefficiencyoftheschemebyusillgsuperconvergenceandasymptoticexpansion"lidertheco…     

Singular Integral Operators with a Generalized Cauchy Kernel

Doklady Mathematics - Singular integral operators with piecewise continuous matrix coefficients are considered on a piecewise smooth curve in weighted Lebesgue spaces. In contrast to the classical...     

Characterizations of the polynomial numerical hull of degree k

Six characterizations of the polynomial numerical hull of degree k are established for bounded linear operators on a Hilbert space. It is shown how these characterizations provide a natural distinction between interior and boundary points. One of the characterizations is used to prove that the polynomial numerical hull of any fixed degree k for a Toeplitz matrix whose symbol is piecewise continuous approaches all or most of that of the infinite-dimensional Toeplitz operator, as the matrix size goes to infinity.     

Degenerate Evolution Equations in Weighted Continuous Function Spaces, Markov Processes and the Black-Scholes Equation — Part I

The present paper is mainly devoted to the study of initial boundary problems associated with a wide class of degenerate second-order differential operators on real intervals, in the framework of weighted continuous function spaces. Such operators are of particular interest, since they often occur, for instance, while building up theoretical models in Mathematical Finance. In order to develop our approach, essentially based on semigroup theory, we provide here some general tools which, perhaps, cover an interest on their own, being concerned with the generation of positive strongly continuous semigroups and their deep connection with Markov processes, in the setting of weighted spaces of continuous functions on a locally compact space. Due to its length, the paper is split up into two parts; the second part will appear in this same journal.     

Fredholmness of singular integral operators with discrete subexponentialgroups of shifts on Lebesgue spaces

The paper is devoted to an application of a general local method of studying the Fredholmness of nonlocal bounded linear operators to Banach algebras of singular integral operators with piecewise continuous coefficients and discrete subexponential groups of piecewise smooth shifts acting topologically freely on composed contours. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convolution equations of the first kind on a finite interval in Sobolev spaces

The study of a class of operators associated with convolution equations of the first kind on a finite interval is reduced to the study of Wiener-Hopf operators with piecewise continuous symbol on R. Fredholm properties and invertibility conditions for this class of operators are investigated. An example from diffraction theory is considered.Sponsored by J.N.I.C.T. (Portugal) under grant n ^o 87422/MATM.