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

Cerone, Pietro

Dragomir, Sever S

English

Victoria University Eprints Repository

APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL VIASOME MOMENTS OF THE INTEGRANDP. CERONE AND S.S. DRAGOMIRAbstract. Error bounds in approximating the Riemann-Stieltjes integral interms of some moments of the integrand are given. Applications for p convexfunctions and in approximating the Finite Foureir Transform are pointed outas well.1. IntroductionIn order to approximate the Riemann-Stieltjes integralR baf (t) du (t) with thearguably simpler expression(1.1)u (b)  u (a)b  a Z baf (t) dt;whereR baf (t) dt is the Riemann integral, Dragomir and Fedotov [8] considered in1998 the following Grüss type error functional:(1.2) D (f; u; a; b) :=Z baf (t) du (t)  1b  a [u (b)  u (a)]Z baf (t) dt:If the integrand f is Riemann integrable and  1 < m  f (t)  M < 1 for anyt 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 integralR baf (t) du (t) exists and the following boundholds:(1.4) jD (f; u; a; b)j  12L (M  m) (b  a) :In (1.4) the constant 12 is best possible in the sense that it cannot be replacedby a smaller quantity.A di¤erent bound for the Grüss error functional D (f; u; a; b) in the case that fis K Lipschitzian and u is of bounded variation has been obtained by the sameauthors in 2001, see [9], where they showed that(1.5) jD (f; u; a; b)j  12K (b  a)b_a(u) :HereWba (u) denotes the total variation of u on [a; b] : The constant12 is also bestpossible.Date : March 2, 2007.2000 Mathematics Subject Classication. 26D15, 41A55.Key words and phrases. Riemann-Stieltjes integral, p moments, p convex functions.12 P. CERONE AND S.S. DRAGOMIRFor other results concerning di¤erent bounds for the functional D (f; u; a; b) un-der various assumptions on f and u, see the recent papers [3], [4], [6] [7], [13] andthe references therein.The main aim of the present paper is to provide error bounds in approximatingthe Riemann-Stieltjes integralR baf (t) du (t) with the following expression contain-ing moments of the function f; namely the expressionp(b  a)p"u (b) Z ba(t  a)p 1 f (t) dt  u (a) Z ba(b  t)p 1 f (t) dt#;where p > 0 and the involved integrals exist.Some inequalities for monotonic integrands and p convex integrators as well aswhere u is an integral of a given weight are provided. An application for approxi-mating the Finite Fourier Transform is also given.The case p = 1 reduces to the Grüss error functional and in this way some earlierresults are recaptured as well.2. General ResultsThe following identity holds.Lemma 1. Let f; u : [a; b]! R such that the Riemann-Stieltjes integral R baf (t) du (t)and the Riemann integralsR ba(t  a)p 1 f (t) dt; R ba(b  t)p 1 f (t) dt for p > 0 ex-ist. Then(2.1)Z baf (t) du (t)=p(b  a)p"u (b) Z ba(t  a)p 1 f (t) dt  u (a) Z ba(b  t)p 1 f (t) dt#+Z ba(t  a)p u (b) + (b  t)p u (a)(b  a)p   u (t)df (t) :Proof. Integrating by parts of the Riemann-Stieltjes integral, we haveZ ba(t  a)p u (b) + (b  t)p u (a)(b  a)p   u (t)df (t)=(t  a)p u (b) + (b  t)p u (a)(b  a)p   u (t)f (t)ba Z baf (t) d(t  a)p u (b) + (b  t)p u (a)(b  a)p   u (t)= [u (b)  u (b)] f (b)  [u (a)  u (a)] f (a) "pu (b)(b  a)pZ ba(t  a)p 1 f (t) dt  pu (a)(b  a)pZ ba(b  t)p 1 f (t) dt Z baf (t) du (t)#APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL 3=   pu (b)(b  a)pZ ba(t  a)p 1 f (t) dt+ pu (a)(b  a)pZ ba(b  t)p 1 f (t) dt+Z baf (t) du (t) ;which is equivalent with the desired identity (2.1). Remark 1. For p = 1 we get the identity:(2.2)Z baf (t) du (t) =u (b)  u (a)b  a Z baf (t) dt+Z ba(t  a)u (b) + (b  t)u (a)b  a   u (t)df (t)that has been obtained in [5], see also [6].In order to approximate the Riemann-Stieltjes integralR baf (t) du (t) by the quad-rature(2.3)p(b  a)p"u (b) Z ba(t  a)p 1 f (t) dt  u (a) Z ba(b  t)p 1 f (t) dt#;we consider the error functional:(2.4) F (f; u; p; a; b) :=Z baf (t) du (t)  p(b  a)p"u (b) Z ba(t  a)p 1 f (t) dt  u (a) Z ba(b  t)p 1 f (t) dt#:The following result may be stated.Theorem 1. Let f; u : [a; b]! R be as in Lemma 1. For p > 0; dene(2.5) p (u; t; a; b) :=(t  a)p u (b) + (b  t)p u (a)(b  a)p   u (t) ;where t 2 [a; b] :If F (f; u; p; a; b) is the error functional dened by (2.4), then:(2.6) jF (f; u; p; a; b)j 8>>>>>>>>>><>>>>>>>>>>:supt2[a;b]jp (u; t; a; b)jWba (f)if f is of bounded variation;LR bajp (u; t; a; b)j dtif f is L  Lipschitzian;R bajp (u; t; a; b)j df (t)if f is monotonic nondecreasing.4 P. CERONE AND S.S. DRAGOMIRProof. It is well known that for the Riemann-Stieltjes integralR baw (t) dv (t) wehave the bounds(2.7)Z baw (t) dv (t) 8>>>>>>>>>><>>>>>>>>>>:supt2[a;b]jw (t)jWba (v)if v is of bounded variation;LR bajw (t)j dtif v is L  Lipschitzian;R bajw (t)j dv (t)if v is monotonic nondecreasing.Now, on utilising the representation (2.1) and applying (2.7) for w (t) := p (u; t; a; b) ;t 2 [a; b] and v = f; we deduce the desired result. Remark 2. For p = 1; by denoting(u; t; a; b) = 1 (u; t; a; b) =(t  a)u (b) + (b  t)u (a)b  a   u (t)andF (f; u; a; b) =Z baf (t) du (t)  u (b)  u (a)b  a Z baf (t) dt;we get from (2.6)(2.8) jF (f; u; a; b)j 8>>>>>>>>>><>>>>>>>>>>:supt2[a;b]j(u; t; a; b)jWba (f)if f is of bounded variation;LR baj(u; t; a; b)j dtif f is L  Lipschitzian;R baj(u; t; a; b)j df (t)if f is monotonic nondecreasing.The inequality (2.8) has been obtained in [5].Remark 3. If u (t) =R taw (s) ds; t 2 [a; b] ; then from (2.1) we get the representa-tion(2.9)Z baf (t)w (t) dt =p(b  a)pZ baw (s) dsZ ba(t  a)p 1 f (t) dt+1(b  a)p Z ba"(t  a)pZ baw (s) ds  (b  a)pZ taw (s) ds#df (t)for any p > 0, provided that the involved integrals exist.For p = 1; we obtain the identity due to Cerone in [2].3. Further Bounds for Monotonic IntegrandsIn this section some bounds for the error functional F (f; u; p; a; b) where theintegrator f is monotonic nondecreasing are given.APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL 5Theorem 2. Let f; u : [a; b] ! R be such that f is monotonic nondecreasing, usatises the bounds:(3.1)  1 < n  u (t)  N <1 for any t 2 [a; b]and the Riemann-Stieltjes integralR baf (t) du (t) exists. Thennp(b  a)p"Z ba(b  t)p 1 f (t) dt Z ba(t  a)p 1 f (t) dt#(3.2)  (N   n) [f (b)  f (a)] F (f; u; p; a; b) Np(b  a)p"Z ba(b  t)p 1 f (t) dt Z ba(t  a)p 1 f (t) dt#  (N   n) [f (b)  f (a)] ;where F (f; u; p; a; b) is given by (2.4).Proof. From (3.1) we obviously have:n (t  a)p  u (b)  N (t  a)p ;n (b  t)p  u (a)  N (b  t)p ; N (b  a)p   u (t) (b  a)p   n (b  a)pfor any t 2 [a; b] : Summing the above three inequalities, we have thatn  (t  a)p+ (b  t)p(b  a)p  N  p (u; t; a; b)  N (t  a)p + (b  t)p(b  a)p   nfor any t 2 [a; b] :Now, integrating over the monotonic nondecreasing function f we haven(b  a)p"Z ba(t  a)p df (t) +Z ba(b  t)p df (t)# N [f (b)  f (a)](3.3) F (f; u; p; a; b) N(b  a)p"Z ba(t  a)p df (t) +Z ba(b  t)p df (t)#  n [f (b)  f (a)] :Integrating by parts, we also haveZ ba(t  a)p df (t) = (b  a)p f (b)  pZ ba(t  a)p 1 f (t) dtand Z ba(b  t)p df (t) =   (b  a)p f (a) + pZ ba(b  t)p 1 f (t) dt:6 P. CERONE AND S.S. DRAGOMIRThenn(b  a)p"Z ba(t  a)p df (t) +Z ba(b  t)p df (t)# N [f (b)  f (a)](3.4)=n(b  a)p((b  a)p [f (b)  f (a)] + p"Z ba(b  t)p 1 f (t) dt Z ba(t  a)p 1 f (t) dt#) N [f (b)  f (a)]=np(b  a)p"Z ba(b  t)p 1 f (t) dt Z ba(t  a)p 1 f (t) dt#  (N   n) [f (b)  f (a)]andN(b  a)p"Z ba(t  a)p df (t) +Z ba(b  t)p df (t)#  n [f (b)  f (a)](3.5)=N(b  a)p((b  a)p [f (b)  f (a)] + p"Z ba(b  t)p 1 f (t) dt Z ba(t  a)p 1 f (t) dt#)  n [f (b)  f (a)]=Np(b  a)p"Z ba(b  t)p 1 f (t) dt Z ba(t  a)p 1 f (t) dt#+ (N   n) [f (b)  f (a)] :Now, on utilising (3.3) (3.5) we deduce the desired result (3.2). Remark 4. In the particular case when p = 1; the inequality (3.2) reduces to(3.6) jF (f; u; p; a; b)j  (N   n) [f (b)  f (a)] :4. An Inequality for Integrators that are s Convex in the SecondSenseFollowing Hudzik and Maligranda [12] (see also [11, p. 286]) we say that thefunction g : R+ ! R is p convex in the second sense, where p > 0 is xed, if:(4.1) g (tu+ (1  t) v)  tpg (u) + (1  t)p g (v)for any u; v  0 and t 2 [0; 1] :For di¤erent properties of this class of functions, see [12] and [11, pp. 286 293].The following inequality of Hermite-Hadamard type is due to Dragomir andFitzpatrick [10]:Theorem 3. Let g be a p convex function in the second sense on an intervalI  [0;1) with p 2 (0; 1] and let a; b 2 I with a < b: Then:(4.2) 2p 1ga+ b2 1b  aZ bag (t) dt  g (a) + g (b)p+ 1:APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL 7We can state and prove now the following result about the Riemann-Stieltjesintegral:Theorem 4. Let u be p convex with p > 0; f be monotonic nondecreasing on[a; b] and such that the Riemann-Stieltjes integralR baf (t) du (t) and the RiemannintegralsR ba(t  a)p 1 f (t) dt; R ba(b  t)p 1 f (t) dt exist. Then(4.3)Z baf (t) du (t) p(b  a)p"u (b)Z ba(t  a)p 1 f (t) dt  u (a)Z ba(b  t)p 1 f (t) dt#:Proof. Since u is p convex, then:u (t) = ut  ab  a  b+b  tb  a  at  ab  apu (b) +b  tb  apu (a) ;which shows, upon using the notations of (2.5), thatp (u; t; a; b)  0 for any t 2 [a; b] :Since f is monotonic nondecreasing on [a; b] ; we have thenZ bap (u; t; a; b) df (t)  0;which, via the representation (2.1), is equivalent with the desired inequality (4.3).Remark 5. The case p = 1; i.e., where the function u is convex in the usual sense,produces the following inequality(4.4)Z baf (t) du (t)  u (b)  u (a)b  aZ baf (t) dt;where f is monotonic nondecreasing, which has been obtained in [5], see also [6].5. Approximating the Finite Fourier TransformThe Fourier Transform is one of the most important mathematical tools in awide variety of elds in science and engineering [1, p. xi].Throughout this section f : [a; b] ! R will be a Riemann integrable functiondened on the nite interval [a; b] and F (g) will be its Finite Fourier Transform.That is,F (f) (t) :=Z baf (s) e 2itsds:Consider also the exponential mean of two complex numbers z; w dened byE (z; w) :=8<:ez ewz w if z 6= w;exp (w) if z = w;z; w 2 C:8 P. CERONE AND S.S. DRAGOMIRNow, for w (s) = exp( 2its); on applying the identity (2.9), which holds forcomplex-valued functions as well, we get the following representation of the FiniteFourier Transform(5.1) F (f) (t) = p(b  a)p 1 E ( 2itb; 2ita)Z ba(s  a)p 1 f (s) ds+R (f) (t) ;where the remainder R (f) has the representationR (f) (t) = 1(b  a)p 1Z ba(s  a)h(s  a)p 1E ( 2itb; 2ita)  (b  a)p 1E ( 2its; 2ita)idf (s) :In order to provide a composite rule in approximating the Finite Fourier Transformin terms of moments for the function f; we consider a division In : a = x0 < x1 <::: < xn 1 < xn = b and the quadrature ruleA (In; t) : =n 1Xi=0p(xi+1   xi)p 1 E ( 2itxi+1; 2itxi)(5.2)Z xi+1xi(s  xi)p 1 f (s) dsthat has been obtained from (5.1) applied on each subinterval [xi; xi+1] and theresults were summed over i from 0 to n   1: It is an open question as to whetheror not A (In; t) is uniformly convergent to F (f) (t) on [a; b] and what the order ofconvergence is ?The following numerical experiment obtained by implementing the quadrature(5.2) for the function f(s) = s + 1 and p = 2 shows the behavior of the absoluteerror value(5.3) En (t) := jF (f) (t) A (In; t)j ; t 2 [a; b] ;for a division with 10 points (Figure 1) respectively 100 points (Figure 2).References[1] P.L. BUTZER and R.J. NESSEL, Fourier Analysis and Approximation Theory, I, AcademicPress, New York and London, 1971.[2] P. CERONE, On an identity for the Chebyshev functional and some ramications, J. Ineq.Pure & Appl. Math., 3(1) (2002), Art. 2.[3] P. CERONE and S.S. DRAGOMIR, New bounds for the three-point rule involving theRiemann-Stieltjes integral, in Advances in Statistics, Combinatorics and Related Areas,Chandra Gulati, Yan-Xia Lin, Satya Mishra and John Rayner (Eds.), World Scientic Pub-lishers, New Jersey - London - Singapore - Hong Kong, 2002, 53-62.[4] P. CERONE and S.S. DRAGOMIR, Approximation of the Stieltjes integral and applicationsin numerical integration, Appl. Math., 51(1) (2006), 37-47.[5] S.S. DRAGOMIR, Inequalities of Grüss type for the Stieltjes integral and applications, Kragu-jevac J. Math., 26 (2004), 89-122.[6] S.S. DRAGOMIR, Inequalities for Stieltjes integrals with convex integrators and applications,Applied Math. Lett., 20(1) (2007), 123-130.[7] S.S. DRAGOMIR, A generalisation of Cerones identity and applications, Tamsui Oxford J.Math. (Taiwan), (in press).[8] S.S. DRAGOMIR and I. FEDOTOV, An inequality of Grüss type for the Riemann-Stieltjesintegral and applications for special means, Tamkang J. Math., 29(4) (1998), 287-292.APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL 9Figure 1. The behaviour of En (t) from (5.3) for n = 10:Figure 2. The behaviour of En (t) from (5.3) for n = 100:[9] S.S. DRAGOMIR and I. FEDOTOV, A Grüss type inequality for mappings of boundedvariation and applications for numerical analysis, Nonlinear Funct. Anal. Appl., 6(3) (2001),425-433.[10] S.S. DRAGOMIR and S. FITZPATRICK, The Hadamard inequalities for s-convex functionsin the second sense. Demonstratio Math. 32 (1999), no. 4, 687696[11] S.S. DRAGOMIR and C.E.M. PEARCE, Selected Topics on Hermite-Hadamard In-equalities and Applications, RGMIA Monographs, Victoria University, 2000. (ONLINE:http://rgmia.vu.edu.au/monographs/).10 P. CERONE AND S.S. DRAGOMIR[12] H. HUDZIK and M. MALIGRANDA, Some remarks on s-convex functions. AequationesMath. 48 (1994), no. 1, 100111.[13] Z. LIU, Renement of an inequality of Grüss type for Riemann-Stieltjes integral, Soochow J.Math., 30(4) (2004), pp. 483-489.School of Computer Science and Mathematics, Victoria University, PO Box 14428,Melbourne City, VIC 8001, Australia.E-mail address : pietro.cerone@vu.edu.auURL: http://rgmia.vu.edu.au/ceroneE-mail address : sever.dragomir@vu.edu.auURL: http://rgmia.vu.edu.au/dragomir

Approximating the Riemann-Stieltjes Integral via Some Moments of the Integrand

Error bounds in approximating the Riemann-Stieltjes integral in\ud
terms of some moments of the integrand are given. Applications for p-convex\ud
functions and in approximating the Finite Foureir Transform are pointed out\ud
as well

Approximating the Riemann-Stieltjes Integral via Some Moments of the Integrand

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