On the Reduced Moment Problem

Abstract

For a special class of cumulative distribution functions which are solutions of a given reduced moment problem (cf. paragraph 3, pages 27 and 28, of [4]), the well known expression for the least upper bound of the absolute difference between any two solutions of the same reduced moment problem is improved upon by the introduction of a constant nonnegative multiplier which is smaller than unity in the case of the special class of solutions. Useful properties of the determinantal form of the classical expression for the least upper bound are derived. The numerical value of the constant multiplier is computed in the case of a well known class of cumulative distribution functions. In addition, a simple method is given for constructing, over a finite range, an infinite set of continuous and differentiable cumulative distribution functions which are solutions of the same reduced moment problem when one such solution is known. The new expression for the least upper bound, when applied to members of the constructed class of continuous solutions, may be helpful in deriving general, but crude, inequalities among orthogonal polynomials over a finite interval.