Approximating the Riemann-Stieltjes integral by a trapezoidal quadrature rule with applications

In this paper we provide sharp bounds for the error in approximating the Riemann-Stieltjes integral R b a f (t) du (t) by the trapezoidal rule f (a) + f (b) 2 [u (b) u (a)] under various assumptions for the integrand f and the integrator u for which the above integral exists. Applications for continuous functions of selfadjoint operators in Hilbert spaces are provided as well. 1. Introduction In Classical Analysis, a trapezoidal type inequality is an inequality that provides upper and/or lower bounds for the quantity f (a) + f (b) 2 (b a) Z b a f (t) dt; that is the error in approximating the integral by a trapezoidal rule, for various classes of integrable functions f de…ned on the compact interval [a; b] : In the following we recall some trapezoidal inequalities for various classes of scalar functions of interest, such as: functions of bounded variation, monotonic, Lipschitzian, absolutely continuous or convex functions. The case of functions of bounded variation was obtained in [6] (see also [5, p. 68]): Theorem 1. Let f : [a; b] ! C be a function of bounded variation. We have the inequality (1.1) Z b a f (t) dt f (a) + f (b) 2 (b a) 12 (b a) b _ a (f) ; where Wb a (f) denotes the total variation of f on the interval [a; b]. The constant 1 2 is the best possible one. This result may be improved if one assumes the monotonicity of f as follows (see [5, p. 76]): 1991 Mathematics Subject Classi…cation. 26D15, 41A55, 47A63. Key words and phrases. Riemann-Stieltjes integral, Trapezoidal Quadrature Rule, Selfadjoint operators, Functions of Selfadjoint operators, Spectral representation, Inequalities for selfadjoint operators. 1 2 S.S. DRAGOMIR Theorem 2. Let f : [a; b] ! R be a monotonic nondecreasing function on [a; b]. Then we have the inequalities: Z b a f (t) dt f (a) + f (b) 2 (b a) (1.2) 1 2 (b a) [f (b) f (a)] Z b a sgn t a+ b 2 f (t) dt 1 2 (b a) [f (b) f (a)] : The above inequalities are sharp. If the mapping is Lipschitzian, then the following result holds as well [9] (see also [5, p. 82]). Theorem 3. Let f : [a; b] ! C be an L Lipschitzian function on [a; b] ; i.e., f satis…es the condition: (L) jf (s) f (t)j L js tj for any s; t 2 [a; b] (L > 0 is given). Then we have the inequality: (1.3) Z b a f (t) dt f (a) + f (b) 2 (b a) 14 (b a) L: The constant 14 is best in (1.3). If we would assume absolute continuity for the function f , then the following estimates in terms of the Lebesgue norms of the derivative f 0 hold [5, p. 93]: Theorem 4. Let f : [a; b]! C be an absolutely continuous function on [a; b]. Then we have Z b a f (t) dt f (a) + f (b) 2 (b a) (1.4) 8>>><>>>>>>>>>: 1 4 (b a) kf k1 if f 0 2 L1 [a; b] ; 1 2 (q + 1) 1 q (b a) kf kp if f 0 2 Lp [a; b] ; p > 1; 1 p + 1 q = 1; 1 2 (b a) kf k1 ; where k kp (p 2 [1;1]) are the Lebesgue norms, i.e., kf k1 = ess sup s2[a;b] jf 0 (s)j and kf kp := Z b a jf 0 (s)j ds ! 1 p ; p 1: The case of convex functions is as follows [13]: APPROXIMATING THE RIEMANN-STIELTJES INTEGRALS 3 Theorem 5. Let f : [a; b] ! R be a convex function on [a; b] : Then we have the inequalities 1 8 (b a) f 0 + a+ b 2 f 0 a+ b 2 (1.5) f (a) + f (b) 2 (b a) Z b a f (t) dt 1 8 (b a) f 0 (b) f 0 + (a) : The constant 1 8 is sharp in both sides of (1.5). For other scalar trapezoidal type inequalities, see [5]. Motivated by the above results, we endeavour in the following to provide sharp bounds for the error in approximating the Riemann-Stieltjes integral R b a f (t) du (t) by the trapezoidal rule (1.6) f (a) + f (b) 2 [u (b) u (a)] under various assumptions for the integrand f and the integrator u for which the above integral exists. Applications for continuous functions of selfadjoint operators in Hilbert spaces are provided as well. The above quadrature (1.6) is di¤erent from the one considered in the papers [2], [4], [12] and [14] where error bounds in approximating the Riemann-Stieltjes integral R b a f (t) du (t) by the generalized trapezoidal formula u (b) u a+ b 2 f (b) + u a+ b 2 u (a) f (a) were provided. In [21], P.R. Mercer has obtained some Hadamard’s type inequalities for the Riemann-Stieltjes integral when the integrand is convex while in [24] M. Munteanu has provided error bounds in approximating the Riemann-Stieltjes integral by the use of Weyl derivatives and the method of approximation. For other results and techniques that are di¤erent from the ones outlined below, we recommend the papers [1], [3], [7], [8] and the classical paper on the closed Newton–Cotes quadrature rules for the Riemann-Stieltjes integrals [26]. 2. The Case of Hölder-Continuous Integrands 2.1. The Case of Bounded Variation Integrators. The following theorem generalizing the classical trapezoid inequality for integrators of bounded variation and Hölder-continuous integrands was obtained by the author in 2001, see [11]. For the sake of completeness and since parts of it will be used in the proofs of other results, we will present it here as well. Theorem 6 (Dragomir, 2001, [11]). Let f : [a; b] ! C be a p H-Hölder type function, that is, it satis…es the condition (2.1) jf (x) f (y)j H jx yj for all x; y 2 [a; b] ; 4 S.S. DRAGOMIR where H > 0 and p 2 (0; 1] are given, and u : [a; b] ! C is a function of bounded variation on [a; b] : Then we have the inequality: (2.2) f (a) + f (b) 2 [u (b) u (a)] Z b a f (t) du (t) 1 2p (b a) b _ a (u) : The constant C = 1 on the right hand side of (2:2) cannot be replaced by a smaller quantity. Proof. Using the inequality for the Riemann-Stieltjes integral of continuous integrands and bounded variation integrators, we have f (a) + f (b) 2 [u (b) u (a)] Z b a f (t) du (t) (2.3)