Two results on optimal modeling with differential-delay systems

Let
G(i\omega)
be a fixed causal filter. The first result of this paper shows that for any positive integer
k
and any delay time
T > 0
there is an "optimal model" for
G(i\omega)
out of the class of differential-delay systems of order
\leq k
(and utilizing delay
T
). The second and main result shows that unless
G(i\omega)
corresponds exactly to a purely differential system of order
\leq k
then there always exists some delay time
T > 0
and corresponding differential-delay system of order
\leq k
that is a better model for
G(i\omega)
than even the optimal purely differential system of order
\leq k
.