An identity for the Chebychev functional is presented in which a Riemann-Stieltjes integral is involved. This allows bounds for the functional to be obtained for functions that are of bounded variation, Lipschitzian and monotone. Some applications are presented to produce bounds for moments of functions about a general point γ and for moment generating functions.

In the standard theory of delay equations, fundamental solution does not ‘live’ in state space. To eliminate this age-old anomaly, we enlarge As a consequence, lose strong continuity operators and this, turn, has as consequence that Riemann integral no longer suffices for giving meaning to variation-of-constants formula. compensate, develop Stieltjes-Pettis setting norming dual pair spaces. Par...

In order to approximate the Riemann–Stieltjes integral ∫ b a f (t) dg (t) by 2–point Gaussian quadrature rule, we introduce the quadrature rule ∫ 1 −1 f (t) dg (t) ≈ Af ( − √ 3 3 ) + Bf (√ 3 3 ) , for suitable choice of A and B. Error estimates for this approximation under various assumptions for the functions involved are provided as well.

We shall consider generated pseudo-operations of the following form: x⊕ y = g(−1) (g(x) + g(y)) , x ̄ y = g(−1) (g(x)g(y)) , where g is a positive strictly monotone generating function and g(−1) is its pseudo-inverse. Using this type of pseudo-operations, the Riemann-Stieltjes type integral will be introduced and investigated.

We study power series over the group ring CF of a free group F . We prove that the von Neumann trace maps rational power series over CF to algebraic power series. Using the Riemann-Stieltjes formula, we deduce the rationality and positivity of Novikov-Shubin invariants of matrices over CF .

New bounds are developed for the Čebyšev functional utilising an identity involving a Riemann-Stieltjes integral. A refinement of the classical Čebyšev inequality is produced for f monotonic non-decreasing, g continuous and M (g; t, b) −M (g; a, t) ≥ 0, for t ∈ [a, b] where M (g; c, d) is the integral mean over [c, d] .