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...

The paper deals with Bruggeman effective medium approximation (EMA) which is often used to model effective complex permittivity of a two-phase composite. We derive the Stieltjes integral representation of the 3D Bruggeman effective medium and use constrained Padé approximation method introduced in [39] to numerically reconstruct the spectral density function in this representation from the effe...

For a Bernstein function f the sequence sn = f(1)·. . .·f(n) is a Stieltjes moment sequence with the property that all powers sn, c > 0 are again Stieltjes moment sequences. We prove that sn is Stieltjes determinate for c ≤ 2, but it can be indeterminate for c > 2 as is shown by the moment sequence (n!)c, corresponding to the Bernstein function f(s) = s. Nevertheless there always exists a uniqu...

We consider certain quadrature rules of highest algebraic degree of precision that involve strong Stieltjes distributions (i.e., strong distributions on the positive real axis). The behavior of the parameters of these quadrature rules, when the distributions are strong c-inversive Stieltjes distributions, is given. A quadrature rule whose parameters have explicit expressions for their determina...

Using the notion of the truncated variation we obtain a new theorem on the existence and estimation of the Riemann-Stieltjes integral. As a special case of this theorem we obtain an improved version of the Loéve-Young inequality for the Riemann-Stieltjes integrals driven by irregular signals. Using this result we strengthen some results of Lyons on the existence of solutions of integral equatio...

In  this paper we define the $p$-analog of the restericted reperesentations and also the $p$-analog of the Fourier--Stieltjes algebras on the inverse semigroups . We improve some results about Herz algebras on Clifford semigroups. At the end of this paper we give the necessary and sufficient condition for amenability of these algebras on Clifford semigroups.