On the condition number of matrices arising in the Tikhonov regularization method (Q1338483)

An integral equation of the first kind (1) \((Kf)(s) = \int^ b_ a k(s,t) f(t)dt = g(s)\), \(K : L^ 2 [a,b] \to L^ 2 [a,b]\), \(k \in L^ 2 ([a,b] \times [a,b])\) is known to be an ill-posed problem. As an expression of this ill-posedness, for many numerical methods (e.g., Galerkin, collocation) the approximate solution of (1) fails in general to the exact solution \(f\). Even if the numerical methods converge, it happens that the rate of decrease of the singular values of \(K\) is directly related to the rate of increase of the condition numbers of the matrices representing discrete versions of \(K\). The main result of the paper is bounding the condition number from both sides, in terms of the singular values of the operator \(K\), the numerical method being a variant of finite-dimensional Tikhonov regularization. In order to obtain such estimates, a compatibility condition is needed between the operator \(K\) and the approximation spaces. Classes of operators satisfying the compatibility condition are presented and studied. The approximation spaces are spaces of spline functions, the latter being chosen because of their special approximation properties.