We present a new approach for expressing and solving boundary problems for linear ordinary differential equations in the language of differential algebras. Starting from an algebra with a derivation and integration operator, we construct a ring of linear integro-differential operators that is expressive enough for specifying regular boundary problems with arbitrary Stieltjes boundary conditions...

We obtain a characterization of generalized Stieltjes functions of any order λ > 0 in terms of inequalities for their derivatives on (0,∞). When λ = 1, this provides a new and simple proof of a characterization of Stieltjes functions first obtained by Widder in 1938.

Some inequalities for the Stieltjes integral and applications in numerical integration are given. The Stieltjes integral is approximated by the product of the divided difference of the integrator and the Lebesgue integral of the integrand. Bounds on the approximation error are provided. Applications for the Fourier Sine and Cosine transforms on finite intervals are mentioned as well.

For a certain constant δ > 0 (a little less than 1/4), every function f : N0 → ]0,∞[ satisfying f(n)2 ≤ δf(n − 1)f(n + 1), n ∈ N, is a Stieltjes indeterminate Stieltjes moment sequence. For every indeterminate moment sequence f : N0 → R there is a positive definite matrix sequence (an) which is not of positive type and which satisfies tr(an+2) = f(n), n ∈ N0. For a certain constant ε > 0 (a lit...

For a real-valued, measurable function f defined on [0,∞), its nth moment is defined as sn( f ) = ∫∞ 0 x n f (x)dx, n ∈N = {0,1, . . .}. Let (sn)n≥0 be a sequence of real numbers. If f is a real-valued, measurable function defined on [0,∞) with moment sequence (sn)n≥0, we say that f is a solution to the Stieltjes moment problem (related to (sn)n≥0). If the solution is unique, the moment problem...