A note on first kind convolution equations on a finite interval

This note deals with a class of convolution operators of the first kind on a finite interval. Necessary and sufficient conditions for such an operator to be Fredholm are given. The argument is based on a process of reduction of convolution-type operators on a finite interval to operators of the same type on the half line.Research supported by the Netherlands organization for scientific research (NWO).     

Convolutional equations of the first kind with periodic kernel on a finite interval

We find, in a closed form, the solvability and well-posedness conditions for the equations in question, as well as all solutions. The results of the article are applicable in practice.     

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.     

Analysis of a convolution integral equation of the second kind with periodic kernel on a finite interval

There are found in closed forms the solvability and well-posedness conditions as well as all solutions to the equation under consideration. The results are applicable in practice.     

Necessary and sufficient well-posedness conditions for a convolution equation of the second kind with even kernel on a finite interval

We derive some necessary and sufficient conditions for the well-posedness of a convolution equation of the second kind with even kernel on a finite interval. In order to check these conditions it suffices to compute a one-dimensional integral (of a given function) with precision less than 0.5. As an intermediate result we give a strengthening of the Fredholm alternative for the equation in question with an arbitrary kernel in L ₁.     

Inversion of convolution equations on a finite interval and realization triples

This paper develops further the connections between convolution equations and realization triples. Here the emphasis is on equations on a finite interval. For each system of equations an operator (called indicator) is introduced which can be used to describe the inversion properties of the system. This indicator may be of simpler form than the convolution operator defined by the equations even for the case when the corresponding symbol is non-rational     

On convolution equations with semi-almost periodic symbols on a finite interval

This note concerns a class of Wiener-Hopf operators on a finite interval, acting between Sobolev multi-index spaces. Necessary and sufficient conditions for such an operator to be Fredholm are given, as well as a formula for the index. The argument is based on a reduction procedure of convolution operators on a finite interval to operators of the same type on the half-line.supported by the Netherlands organization for scientific research (NWO)supported in part by NSF Grant 9101143     

Integro-differential equations of the convolution on a finite interval with Kernel having a logarithmic singularity

The integro-differential equations $$\frac{{d^{2n} }}{{dx^{2n} }}\int\limits_{ - 1}^1 {(a[(x - t)^2 ]1n|x - t| + b[(x - t)^2 )\varphi (t)dt = f(x)} $$ of the convolution on an interval with infinitely differentiable functions a(s) and b(s) decreasing at infinity are considered. The Fourier symbol is assumed to be sectorial, that is, it has positive projection on some direction in the complex plane. The existence and uniqueness of solutions in the classes of functions representable in the form $$\varphi (t) = (1 - t^2 )^{\delta n} \psi (t),{\text{ }}\delta _n = n - 1 + \varepsilon ,{\text{ }}\varepsilon > 0,{\text{ }}\psi \in C^1 [ - 1,1]$$ are proved. Properties concerning the smoothness of solutions are described. Bibliography: 4 titles.     

A new approach to the convolution operator on a finite interval

The convolution operator on a finite interval defined on a space ofL ₂ functions is studied by relating it to a singular integral operator acting on a space of functions defined on a system of two parallel straight lines in the complex plane . The approach followed in the paper applies both to the case where the Fourier transform of the kernel functions is anL function and to the case where the kernel function is periodic, thus yielding a unified treatment of these two classes of kernel functions. In the non-periodic case it is possible, for a special class of kernel functions, to study the invertibility property of the operator giving an explicit formula for the inverse. An example is presented and generalizations are suggested.     

Remarks on the solvability of a convolution integral equation on a finite interval

We present some results on the solvability of an integral equation of the second kind with a difference kernel on a finite interval, construct a counterexample to an assertion, earlier believed to have been proved, on the solvability of this equation, and pose an open problem.     

A finite element method for some integral equations of the first kind

This paper discusses a finite element approximation for a class of singular integral equations of the first kind. These integral equations are deduced from Dirichlet problems for strongly elliptic differential equations in two independent variables. By a variation of technique due to Aubin, it is shown that the Galerkin method with finite elements as trial functions leads to an optimal rate of convergence.     

New results on the invertibility of the finite interval convolution operator

Conditions for the invertibility and explicit formulas for the inverse of the convolution operator on a finite interval are obtained making use of solutions of corona problems. Using these results, a family of classes of functions is defined for which the study of invertibility can be carried through. An example of one class of this family is presented and a smaller class, for which the calculations are simpler, is more thoroughly studied.Work sponsored by F.C.T. (Portugal) under Project Praxis XXi/2/2.1/MAT/441/94     

On the theory of convolution equations of the third kind

The autoconvolution equation of the third kind with coefficient of general power type is dealt with by the method of weighted norms developed for equations with coefficients of linear and integer power type in recent joint work of the author with L. Berg, J. Janno, and B. Hofmann. For this equation two existence theorems and a uniqueness theorem are proved. Further, as an auxiliary equation a linear singular integral equation of Abel is treated anew and the existence of solutions to a related class of linear Volterra equations of the third kind is derived.     

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.     

Lattice-Nyström method for Fredholm integral equations of the second kind with convolution type kernels

We consider Fredholm integral equations of the second kind of the form , where g and k are given functions from weighted Korobov spaces. These spaces are characterized by a smoothness parameter α>1 and weights γ₁≥γ₂≥. The weight γ_j moderates the behavior of the functions with respect to the jth variable. We approximate f  by the Nyström method using n rank-1 lattice points. The combination of convolution and lattice group structure means that the resulting linear system can be solved in O(nlogn) operations. We analyze the worst case error measured in sup norm for functions g in the unit ball and a class of functions k in weighted Korobov spaces. We show that the generating vector of the lattice rule can be constructed component-by-component to achieve the optimal rate of convergence O(n^-α/2+δ), δ>0, with the implied constant independent of the dimension d under an appropriate condition on the weights. This construction makes use of an error criterion similar to the worst case integration error in weighted Korobov spaces, and the computational cost is only O(nlognd) operations. We also study the notion of QMC-Nyström tractability: tractability means that the smallest n needed to reduce the worst case error (or normalized error) to is bounded polynomially in ^-1 and d; strong tractability means that the bound is independent of d. We prove that strong QMC-Nyström tractability in the absolute sense holds iff , and QMC-Nyström tractability holds in the absolute sense iff .