Sparse discretization matrices for Volterra integral operators with applications to numerical differentiation

Abstract

We consider Volterra integral equations having a finite dimensional feature space. This provides us flexibility to construct an orthonormal basis with small support that can be preserved by the Volterra integral operator. Under the projection method, such a basis yields a sparse discretization matrix for the Volterra integral operator. When the feature space is refinable, we introduce a construction of such an orthonormal basis from existing references. Finally, we present applications to numerical differentiation for which we obtain a quasi-linear lossless compression of the discretization matrix.