Approximating the Riemann-Stieltjes integral via some moments of the integrand

Error bounds in approximating the Riemann-Stieltjes integral in terms of some moments of the integrand are given. Applications for p convex functions and in approximating the Finite Foureir Transform are pointed out as well. 1. Introduction In order to approximate the Riemann-Stieltjes integral R b a f (t) du (t) with the arguably simpler expression (1.1) u (b) u (a) b a Z b a f (t) dt; where R b a f (t) dt is the Riemann integral, Dragomir and Fedotov [8] considered in 1998 the following Grüss type error functional: (1.2) D (f; u; a; b) := Z b a f (t) du (t) 1 b a [u (b) u (a)] Z b a f (t) dt: If the integrand f is Riemann integrable and 1 < m f (t) M < 1 for any t 2 [a; b] while the integrator u is L Lipschitzian, namely, (1.3) ju (t) u (s)j L jt sj for each t; s 2 [a; b] ; then the Riemann-Stieltjes integral R b a f (t) du (t) exists and the following bound holds: (1.4) jD (f; u; a; b)j 1 2 L (M m) (b a) : In (1.4) the constant 1 2 is best possible in the sense that it cannot be replaced by a smaller quantity. A di¤erent bound for the Grüss error functional D (f; u; a; b) in the case that f is K Lipschitzian and u is of bounded variation has been obtained by the same authors in 2001, see [9], where they showed that (1.5) jD (f; u; a; b)j 1 2 K (b a) b _ a (u) : Here Wb a (u) denotes the total variation of u on [a; b] : The constant 1 2 is also best possible. Date : March 2, 2007. 2000 Mathematics Subject Classi…cation. 26D15, 41A55.