On applications of generalized splines and generalized inverses in regularization and projection methods

Abstract

A brief exposition on generalized splines is given. Generalized splines form a spectrum of functions Sα (that depend on a parameter 0<α<@@@@) which minimize a penalty-type functional (depending on α) associated with a variety of regularization and stabilization methods. Interpolating splines and least-squares splines are obtained as limiting cases of a generalized spline (as α→0 and α&rarr@@@@ respectively). Least-squares solutions (of minimal norm) of operator equations are considered in terms of generalized inverses of linear operators. Approximate minimization (of functionals that arise in these settings) using spline functions is indicated. Projection and least-squares methods (on subspaces of splines for example) are used to approximate least-squares solutions of minimal norm of linear operator equations.