Abstract: We consider a method for constructing uniformly stable spline approximations for scalar delay equations.

From the Introduction:
In this paper we discuss a method for constructing `stability preserving' finite dimensional spline-based approximation schemes for the scalar delay equation $$ \dot x(t)=ax(t)+bx(t-r)\eqno{(1.1)} $$ where $a,b,r\in\RR$ and $r>0$. The approximation problem for linear functional differential equations, of which (1.1) is a simple scalar case, has been the subject of much attention in recent years....... we consider a method for defining the spline based approximation schemes for (1.1) in terms of a suitable equivalent inner product so that the uniform stability condition is satisfied. In section 2 we discuss a state space formulation for (1.1) and define an equivalent inner product, which is then used in section 3 to define a new spline based approximation scheme. Numerical results are given which compare and contrast the stability properties of the various schemes.