Theory of a class of Wiener-Hopf equations of the first kind: Application to the Sommerfeld problem

A new theory of a class of Wiener-Hopf equations of the first kind in a space of distributions is presented. It is shown that the corresponding Wiener-Hopf operator is a Fredholm operator. This result is obtained by an appropriate modification of the standard Wiener-Hopf technique used for equations of the second kind. The nullity and defect numbers of the operator are determined from a factorization of the symbol. An application to the Sommerfeld problem is briefly considered.     

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.     

On an iterative method for a class of integral equations of the first kind

In this paper, we investigate an iterative method which has been proposed [1] for the numerical solution of a special class of integral equations of the first kind, where one of the essential assumptions is the positivity of the kernel and the given right-hand side. Integral equations of this special type occur in experimental physics, astronomy, medical tomography and other fields where density functions cannot be measured directly, but are related to observable functions via integral equations. In order to take into account the non-negativity of density functions, the proposed iterative scheme was defined in such a way that only non-negative solutions can be approximated. The first part of the paper presents a motivation for the iterative method and discusses its convergence. In particular, it is shown that there is a connection between the iterative scheme and a certain concave functional associated with integral equations of this type. This functional can be interpreted as a generalization of the log-likelihood function of a model from emission tomography. The second part of the paper investigates the convergence properties of the discrete analogue of the iterative method associated with the discretized equation. Sufficient conditions for local convergence are given; and it is shown that, in general, convergence is very slow. Two numerical examples are presented.     

On the injectivity of a class of Wiener-Hopf integral operators in a Hilbert space

A class of Wiener-Hopf integral operators, with kernels vanishing along the positive real axis, is obtained from considering weighted transaxial line-integrals of rotationally symmetric functions defined on ². An analysis of these operators is given when acting in, the Hilbert space L²(⁺). A necessary and sufficient condition for injectivity is established and inversion formulas are provided in some cases. A specific operator falling into this class, the so-called incomplete Abel transform., is presented and an inversion formula is given. This inversion formula makes precise a formal result previously established in Dallaset al. [J. Opt. Soc. Am. A4, 2039 (1987)] and it is also shown to be consistent with an inversion formula derived by Hansen [J. Opt. Soc. Am. A9, 2126 (1992)].This research was supported by NIH/NCI Grant R01 CA49261     

On an approximate solution of a class of boundary integral equations of the first kind

We construct a method for computing an approximate solution of the boundary integral equation of the first kind corresponding to the Dirichlet boundary value problems for the Helmholtz equation.     

On volterra equations of the first kind

The existence of a solution β of the equation $$\int_0^t {a(t - s)d\beta (s) = 1, t > 0} $$ is studied under fairly general assumptions on the function a. Sufficient conditions for the measure β to be absolutely continuous or satisfy some additional regularity properties are given. An extension to nonconvolution kernels is also considered.     

On the exact degree of complexity of a class of operator equations of the second kind in a Hilbert space

The exact exponent of complexity is found for approximate solutions of a certain class of operator equations in a Hilbert space. A method for information setup and the algorithm for realization of this optimal degree are presented. As a consequence, we find the exact exponent of complexity for approximate solutions of Fredholm integral equations of the second kind whose kernels and free terms include square integrable -derivatives.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 893–903, July, 1994.     

On the stability of the spline collocation method for a class of integral equations of the first kind

This paper is concerned with the stability of the spline collocation method for a class of integral equations of the first kind with logarithmic kernels. It is shown that a proper choice of the mesh size can be made in the numerical computation so that one will obtain an optimal rate of convergence for the approximate solutions.     

On a class of Riccati equations in a Hilbert space

We establish existence and uniqueness of solutions of a class of Riccati equations in Hilbert space ocurring in filtering problems for distributed parameter systems using point sensors.     

On linear Fredholm integro-differential equations of first kind

We study the question whether linear one-dimensional integro-differential equations with constant limits of integration (equations of Fredholm type) containing no free differential expression (equations of first kind) can be reduced to integral equations of first kind and to Fredholm integro-differential equations of second kind.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 20–27.     

A product integration method for a class of singular first kind Volterra equations

Summary Convergence of a midpoint product integration method for singular first kind Volterra equations with kernels of the formk(t, s)(t–s) ^– , 0<<1, wherek(t, s) is continuous, is examined. It is shown that convergence of order one holds if the solution of the Volterra equation has a Lipschitz continuous first derivative andk(t, s) is suitably smooth. In addition, convergence is shown to hold when the solution has only Lipschitz continuity and the same conditions onk(t, s) apply. An existence theorem of Kowalewski is used to relate these conditions on the solution to conditions on the data andk(t, s).     

On the regularization of a class of integral equations of the first kind whose kernels are discontinuous on the diagonals

For a class of integral equations of the first kind whose kernels are discontinuous on the diagonals, the convergence of the Lavrent??ev regularization method is proved by using methods of the spectral theory of integral operators. These methods lead to a special Dirac system, and finding the asymptotics of fundamental solutions is an important part of the proof.     

Fredholm equations of the first kind

A method is described for solving the Fredholm integral equation of the 1st kind by passage to the moment L-problem. The problem is reduced to a linear programming problem. A bound of the method for a particular normed space is derived. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 34–38.     

Research announcement on gaierkirfs method for a class of integra equations of the first kind

Let X and Y be locally convex spaces with K a closed convex cone in X Necessary and sufficient conditions are given for the image AK to be closed in Ywhen A:X→Y is a continuous linear map. This result is used to generalize a theorem of Abrams to infinite dimensional spaces and also to give sufficient conditions for the Hurwicz version of the Farkas lemma for locally convex spaces to hold.