On Approximation by Positive Sums of Powers

Abstract

The problem of approximating a sequence of real numbers y = ('0, V1t , VN) by the sequence obtained from the values of an expression of the form ,i= l ai exp (tt) at t = 0, 1, * * , N has been studied extensively [2], [4], [7], [9]. In some applications the sequence y obtained as experimental data is assumed to arise as a result of several decay processes so that it is appropriate to add the restriction that the Ai be >0. In [3] a standard test of gas mixing in the lungs is analyzed and under the assumptions made the restriction that the ai be > 0 can be added. For this application no restriction on the number of exponential components r is indicated. Thus we were led to study the approximation problem with the ai and Ai > 0 and with no restriction on r. The following results are offered because they may be more widely applicable. Put ui = exp (Ai); then the condition that Ai shall be _ 0 becomes 0 0. We note that the exponential polynomial which is 0 for each integer n has r = 0 when written in its reduced form and is a positive S-exponential polynomial for any choice of S. In what follows we shall frequently refer to a exponential polynomial; we shall always mean unique up to equivalence. We shall say that the exponential polynomial E represents the sequence V if E = V. In ? 1 we study the uniqueness of representation of sequences by positive S-exponential polynomials. In ? 2, we study the existence and the uniqueness of best approximations by positive S-exponential polynomials using various measures of best approximation. In ? 3 we give a numerical method for finding