Solvability of singular integral equations with rotations and degenerate kernels in the vanishing coefficient case

By means of Riemann boundary value problems and of certain convenient systems of linear algebraic equations, this paper deals with the solvability of a class of singular integral equations with rotations and degenerate kernel within the case of a coefficient vanishing on the unit circle. All the possibilities about the index of the coefficients in the corresponding equations are considered and described in detail, and explicit formulas for their solutions are obtained. An example of application of the method is shown at the end of the last section

On The Approximation Of Singular Integral Equations By Equations With Smooth Kernels

Singular integral equations with Cauchy kernel and piecewise--continuous matrix coefficients on open and closed smooth curves are replaced by integral equations with smooth kernels of the form (t \Gamma ø )[(t \Gamma ø ) 2 \Gamma n 2 (t)" 2 ] \Gamma1 , " ! 0, where n(t), t 2 \Gamma, is a continuous field of unit vectors non--tangential to \Gamma. We give necessary and sufficient conditions under which the approximating equations have unique solutions and these solutions converge to the solution of the original equation. For the scalar case and the space L 2 (\Gamma) these conditions coincide with the strong ellipticity of the given equation

Exploration of toeplitz-like matrices with unbounded symbols is not a purely academic journey

It is often asked why Toeplitz-like matrices with unbounded symbols are worth studying. This paper gives an answer by presenting several concrete problems that motivate such studies. It surveys the central results of the theory of Generalized Locally Toeplitz (GLT) sequences in a self-contained tool-kit fashion, and gives a new extension from bounded Riemann integrable functions to unbounded almost everywhere continuous functions. The emergence of unbounded symbols is illustrated by local grid refinements in finite di erence and finite element discretizations and also by preconditioning strategies