A One Parameter Family of Spline-Type Schemes for Approximation of Delay Systems

This paper presents a one parameter family of approximation schemes for systems of linear autonomous retarded functional differential equations. The schemes are based on transformation of the given system into an abstract Cauchy problem on the state space Rn × L2 and on approximation of the corresponding solution semigroup on sequences of finite dimensional subspaces of the state space. The parameter ranges over a certain real interval, whose extreme points are determined by the values generating an orthogonal basis consisting of piecewise linear functions respectively a basis of linear splines for the finite dimensional subspaces. This construction of the one parameter family is used to investigate the eigenvalue behaviour of linear spline approximating systems and to compare it with that of other approximating systems of the family. Moreover, a new approach for proving consistency is presented, showing and utilizing a close relationship between approximating schemes for delay systems and ODE-methods.