A trigonometrical approach for some projection methods (Q1300265)

In order to obtain an approximate solution of a Fredholm integral equation of the first kind \(Kf=g,\;(Kf)(s) = \int_a^b k(s, t)f(t) dt,\;k\in L_2([a, b] \times [a, b]),\;K: L_2[a, b]\to L_2[a, b],\) an abstract projection method is proposed. Its convergence is characterized in terms of the minimal angle between the range and nullspace of the projections, and expressed as a compatibility condition between the approximation spaces and the operator \(K.\) It is optimal for approximation spaces generated by singular functions of \(K.\) Convergence of some known projection methods is analyzed as particular cases of the general method. Numerical methods for calculating the angles, based on eigenvalues of pencils of matrices, are given in each special case. As an example, the ``\(p\)-iterated integration operator'' \((Kf)(s) = \int_0^s (s- t)^pf(t) dt,\;K: L_2[0, 1]\to L_2[0, 1],\) is considered.