Cordial Volterra integral equations and collocation methods for solving them

We introduce the class of so-called cordial Volterra integral operators $V$ (they are non-compact!) and cordial integral equations $u = V u + f$. For example, Diogo's, Lighthill's and Abel's integral equations are cordial under certain conditions. We discuss the relations between cordial equations, Mellin convolution equations and Wiener-Hopf integral equations. We determine the spectra of a cordial operator in different spaces and, in this way, we obtain results about the smoothness of a solution to a cordial integral equation. Our final purpose is to examine the convergence and convergence speed of polynomial and piecewise polynomial (spline) collocation methods for cordial integral equations.