A note on the regularization of discrete approximation problems

Abstract

Discrete approximation problems may not have a solution and, in case a sequence of discrete approximation problems is solved, the discretization errors may not converge, even if the uniform approximation problem possesses a unique solution. In this note we show how the discrete problems can be “regularized,” i.e., we exhibit how the existence of solutions and the requested convergence can be enforced. Our result here supplements the results in [ J. Approx. Theory 49 1987, 256–273]. It is particularly useful for the method studied in [R. Reemtsen, “Defect Minimization in Operator Equations: Theory and Applications,” Pitman Research Notes in Mathematics, Vol. 163, Longman Scientific and Technical, Harlow, Essex/New York 1987].