Non-uniform UE-spline quasi-interpolants and their application to the numerical solution of integral equations (Q6169249)

Splines and kernel-based methods are central to the numerical solution of differential and integral equations. In this article, they are used for solving integral equations by computer algorithms. Among the most important results for formulating spline approximations is the so-called Marsden identity. This is particular relevant when Schoenberg-type quasi-interpolants are used for the numerical methods. In the paper, the authors use this -- it is one of the most successful approximation schemes -- quasi-interpolation with splines in order to perform the mentioned numerical solutions of integral equations. Basically, this requires quadrature formulae so that the given integral equations can be put into a suitable discrete form for the numerical algorithms. The said quasi-interpolation, which is a very useful method, is used here even for nonequally spaced (scattered data). In order to get the desired approximation orders of the quasi-interpolants, one wants the quasi-interpolant to reproduce certain finite dimensional subspaces of approximants exactly (classically polynomials, but now trigonometric polynomials and hyperbolic ones). The Marsden identity that is required is not the standard one known for piecewise polynomial splines only, but a new one that allows trigonometric and hyperbolic splines too. Its proof applies for ``UE-splines'', i.e., uniform generalized Bs-Splines, is not for the classical B-splines only and is one of the many important contributions of the authors.