Asymptotic optimum Laguerre series

If a function
h(t)
is approximated by the first
N
terms of the set of Laguerre functions
I_{n}(pt)
, then the minimum integral-square error is
I_{N}(p) = \int_{0}^{\infty} h^{2}(t)dt - \sum_{n=0}^{N-1} c_{n}^{2}(p)
in which
c_{n}(p)
are the coefficients of the Laguerre expansion of
h(t)
and
p
is a scale factor by which the Laguerre functions can be stretched or compressed. The error
I_{N}(p)
can be minimized further by an optimum choice of
p
. Generally, it is not simple to determine the optimum scale factor
p_{N}
by analytical methods. In this paper an analytical method based on the power series equivalence of the Laguerre series is presented for determining the asymptotic optimum scale factor
p_{\infty} = n \stackrel{\lim}{\rightarrow} \infty p_{n}
. The method is illustrated by determining
p_{\infty}
for some classes of functions of importance in system and signal theory. In engineering applications the number of terms used often is sufficiently large so that the asymptotic optimum scale factor
p_{\infty}
can be expected to be a good approximation to the optimum scale factor
p_{N}
.