Asymptotic optimum Laguerre series

If a functionh(t)is approximated by the firstNterms of the set of Laguerre functionsIₙ(pt), then the minimum integral-square error isIN(p) = ∫₀∞ h²(t)dt - ∑ₙ₌₀N-1 cₙ²(p)in whichcₙ(p)are the coefficients of the Laguerre expansion ofh(t)andpis a scale factor by which the Laguerre functions can be stretched or compressed. The errorIN(p)can be minimized further by an optimum choice ofp. Generally, it is not simple to determine the optimum scale factorpNby 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 factorp∞ = n {lim}{→} ∞ pₙ. The method is illustrated by determiningp∞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 factorp∞can be expected to be a good approximation to the optimum scale factorpN.