Approximation with sums of exponentials in Lp[0, ∞)

AbstractWe consider the problem of approximating a given f from Lp [0, ∞) by means of the family Vn(S) of exponential sums; Vn(S) denotes the set of all possible solutions of all possible nth order linear homogeneous differential equations with constant coefficients for which the roots of the corresponding characteristic polynomials all lie in the set S. We establish the existence of best approximations, show that the distance from a given f to Vn(S) decreases to zero as n becomes infinite, and characterize such best approximations with a first-order necessary condition. In so doing we extend previously known results that apply in Lp[0, 1]