Collocation method for integral equations of the first kind in a certain class of distributions (Q1902724)

The article deals with the Fredholm integral equation of the first kind (1) \(\int^1_0 K(t,s) x(s)ds = y(t)\) \((0 \leq t \leq 1)\), where \(K(t,s) = L(t,s)\) \((s \leq t)\) and \(K(t,s) = M(t,s)\) \((s > t)\). Here \(L,M\) and \(y\) are given sufficiently smooth functions, and \(x\) is the desired element. We search a solution to equation (1) in a certain space of distributions. Equations under consideration are explicitly solvable only in some specific cases. Therefore, the development of approximative technique in order to solve these is of importance both for theory and for its applications. Here we offer and prove a direct projection method for solving (1), which maximally takes into account the characteristics of functions from the main space. We prove that the offered method is optimal with respect to order on the class \(H^r_\omega\) among all the polynomial projection methods for solving (1).