An Approximate Convolution Equation of a Given Response

Abstract

In constructing models of certain linear control processes, one encounters the problem of approximation by exponential sums. A function f_N(t) is called an exponential sum of order N+1 if it satisfies a linear differential equation of the form

$${u^{(N + 1)}}(t)\; + \;{a_0}{u^{(N)}}(t)\; + \ldots + \;{a_N}\;u(t)\,\; = \;\,0$$

((1.1))

for some constant, real coefficients a₀,a₁,a₂,... a_N. The problem of approximation by exponential sums, then, is to find a differential equation of the form (1.1) such that the solution is close, in some sense, to a given function f(t).

The work of this author was supported by the U.S. Army Research Office — Durham under Grant DA-ARO-D-31-124-73-G51.