On the optimal choice of nodes in the collocation-projection method for solving linear operator equations

AbstractH R is a Hilbert space with norm ∥ ∥R and K is a linear operator mapping H R into l2(T), where T is a closed, bounded interval of the real line. K and H R jointly have the property that there exist M such that ¦ (Kf)(t)¦ ⩽ M ∥f∥R, t ϵ T. In the collocation-projection method for approximately solving the equation Kf = g, the approximate solution fn is taken as that element of H R that minimizes ∥f;∥R subject to (Kf)(t) = g(t) for t ϵ Tn, where Tn + 1 = {ti}i = 0n and ti ϵ T. We show how results obtained elsewhere are applicable to the problem of choosing the node set Tn to minimize ∥f − fn∥R. The results are applicable to the approximate solution of 2-point boundary value problems using a spline basis. We make some remarks concerning the problem of choosing the nodes to minimize sups ¦f(s) − fn(s)¦, when K is a differential operator