A projection method for solving Fredholm integral equations of the second kind (Q1122351)

The Fredholm integral equation of the second kind \(x(t)-\int k(t,s)x(s)ds=y(t),\) \(a\leq t\leq b\), is considered. The degenerated kernel methods of approximating the solution of this equation consists of replacing the kernel k(t,s) by a finite rank approximation using bivariate functions. For that purpose a projection of a space of bivariate functions onto its subspace, consisting of functions, each of which is a finite sum of products of univariate functions, is established. These ``basis'' functions can be selected. One can chose them in such a way that the integral can be easy to compute, for example if it refers to the B-spline functions, then the values of such integrals can be used from the tabulograms. Corresponding theorems are presented, proved and used as an example, reducing the problem of constructing projections of univariate functions' spaces.