APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL BY A TRAPEZOIDAL QUADRATURE RULE WITH APPLICATIONS

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1 APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL BY A TRAPEZOIDAL QUADRATURE RULE WITH APPLICATIONS S.S. DRAGOMIR Astrct. In this pper we provide shrp ounds for the error in pproximting the Riemnn-Stieltjes integrl R du (t) y the trpezoidl rule f () + f () [u () u ()] under vrious ssumptions for the integrnd f nd the integrtor u for which the ove integrl exists. Applictions for continuous functions of selfdjoint opertors in Hilert spces re provided s well.. Introduction In Clssicl Anlysis, trpezoidl type ineulity is n ineulity tht provides upper nd/or lower ounds for the untity f () + f () ( ) dt; ths the error in pproximting the integrl y trpezoidl rule, for vrious clsses of integrle functions f de ned on the compcntervl [; ] : In the following we recll some trpezoidl ineulities for vrious clsses of sclr functions of interest, such s: functions of ounded vrition, monotonic, Lipschitzin, solutely continuous or convex functions. The cse of functions of ounded vrition ws otined in [6] (see lso [5, p. 68]): Theorem. Let f : [; ]! C e function of ounded vrition. We hve the ineulity f () + f () (.) dt ( ) _ ( ) (f) ; where W (f) denotes the totl vrition of f on the intervl [; ]. The constnt is the est possile one. This result my e improved if one ssumes the monotonicity of f s follows (see [5, p. 76]): 99 Mthemtics Suject Clssi ction. 6D5, 4A55, 47A63. Key words nd phrses. Riemnn-Stieltjes integrl, Trpezoidl Qudrture Rule, Selfdjoint opertors, Functions of Selfdjoint opertors, Spectrl representtion, Ineulities for selfdjoint opertors.

2 S.S. DRAGOMIR Theorem. Let f : [; ]! R e monotonic nondecresing function on [; ]. Then we hve the ineulities: f () + f () (.) dt ( ) ( ) [f () f ()] + sgn t dt ( ) [f () f ()] : The ove ineulities re shrp. If the mpping is Lipschitzin, then the following result holds s well [9] (see lso [5, p. 8]). Theorem 3. Let f : [; ]! C e n L stis es the condition: Lipschitzin function on [; ] ; i.e., f (L) jf (s) j L js tj for ny s; t [; ] (L > 0 is given). Then we hve the ineulity: (.3) dt The constnt 4 is esn (.3). f () + f () ( ) 4 ( ) L: If we would ssume solute continuity for the function f, then the following estimtes in terms of the Leesgue norms of the derivtive f 0 hold [5, p. 93]: Theorem 4. Let f : [; ]! C e n solutely continuous function on [; ]. Then we hve f () + f () (.4) dt ( ) 8 4 ( ) kf 0 k if f 0 L [; ] ; >< ( ) += kf 0 k ( + ) p if f 0 L p [; ] ; p > ; p + = ; >: ( ) kf 0 k ; where kk p (p [; ]) re the Leesgue norms, i.e., nd kf 0 k p := kf 0 k = ess sup jf 0 (s)j s[;] jf 0 (s)j ds! p The cse of convex functions is s follows [3]: ; p :

3 APPROXIMATING THE RIEMANN-STIELTJES INTEGRALS 3 Theorem 5. Let f : [; ]! R e convex function on [; ] : Then we hve the ineulities + + (.5) ( ) f+ 0 f 0 8 f () + f () ( ) 8 ( ) f 0 () f 0 + () : dt The constnt 8 is shrp in oth sides of (.5). For other sclr trpezoidl type ineulities, see [5]. Motivted y the ove results, we endevour in the following to provide shrp ounds for the error in pproximting the Riemnn-Stieltjes integrl R du (t) y the trpezoidl rule (.6) f () + f () [u () u ()] under vrious ssumptions for the integrnd f nd the integrtor u for which the ove integrl exists. Applictions for continuous functions of selfdjoint opertors in Hilert spces re provided s well. The ove udrture (.6) is di erent from the one considered in the ppers [], [4], [] nd [4] where error ounds in pproximting the Riemnn-Stieltjes integrl R du (t) y the generlized trpezoidl formul + + u () u f () + u u () f () were provided. In [], P.R. Mercer hs otined some Hdmrd s type ineulities for the Riemnn-Stieltjes integrl when the integrnd is convex while in [4] M. Muntenu hs provided error ounds in pproximting the Riemnn-Stieltjes integrl y the use of Weyl derivtives nd the method of pproximtion. For other results nd techniues tht re di erent from the ones outlined elow, we recommend the ppers [], [3], [7], [8] nd the clssicl pper on the closed Newton Cotes udrture rules for the Riemnn-Stieltjes integrls [6].. The Cse of Hölder-Continuous Integrnds.. The Cse of Bounded Vrition Integrtors. The following theorem generlizing the clssicl trpezoid ineulity for integrtors of ounded vrition nd Hölder-continuous integrnds ws otined y the uthor in 00, see []. For the ske of completeness nd since prts of it will e used in the proofs of other results, we will present here s well. Theorem 6 (Drgomir, 00, []). Let f : [; ]! C e p function, ths, it stis es the condition H-Hölder type (.) jf (x) f (y)j H jx yj p for ll x; y [; ] ;

4 4 S.S. DRAGOMIR where H > 0 nd p (0; ] re given, nd u : [; ]! C is function of ounded vrition on [; ] : Then we hve the ineulity: Z f () + f () (.) [u () u ()] du (t) _ p H ( )p (u) : The constnt C = on the right hnd side of (:) cnnot e replced y smller untity. Proof. Using the ineulity for the Riemnn-Stieltjes integrl of continuous integrnds nd ounded vrition integrtors, we hve Z f () + f () (.3) [u () u ()] du (t) f () + f () = du (t) sup f () + f () _ u (t) : t[;] As f is of p H-Hölder type, then we hve f () + f () = f () + f () (.4) jf () j + jf () j H [(t )p + ( t) p ] ; for ny t [; ] : Now, consider the mpping (t) = (t ) p +( t) p ; t [; ] ; p (0; ] : Then 0 (t) = p (t ) p p ( t) p = 0 i t = + nd 0 (t) > 0 on ; + ; 0 (t) < 0 on + ; ; which shows thts mximum is relized t t = + nd + mx (t) = = p ( ) p : t[;] Conseuently, y (.4), we hve sup f () + f () t[;] p H : Using (.3) we otin the desired ineulity (.). To prove the shrpness of the constnt ; ssume tht (.) holds with constnt C > 0: Ths (.5) f () + f () [u () u ()] du (t) C p H ( )p _ (u) : Choose f : [0; ]! R; = t p ; p (0; ] nd u (t) = t; t [0; ] : We oserve tht f is of p H-Hölder type with H = nd u is of ounded vrition, then, y (.5) we otin p + C ; for ny p (0; ] : p

5 APPROXIMATING THE RIEMANN-STIELTJES INTEGRALS 5 Ths, C p p + p ; for ny p (0; ] : Letting p! 0+; we get C nd the theorem is completely proved. Remrk. We notice tht, if f is H-Lipschitzin, then (.) ecomes Z f () + f () (.6) [u () u ()] du (t) _ H ( ) (u) : The constnt is est possile in (.6). Remrk. If we ssume tht g : [; ]! C is Leesgue integrle on [; ] ; then u (x) = R x g (t) ds di erentile lmost everywhere, u () = R g (t) dt; u () = 0 nd W (u) = R jg (t)j dt: Conseuently, y (.) we otin Z f () + f () (.7) g (t) dt g (t) dt p H ( )p jg (t)j dt: From (.7) we get weighted version of the trpezoid ineulity, Z f () + f () (.8) R g (t) dt g (t) dt p H ( )p ; provided tht g (t) 0; for lmost every t [; ] nd R g (t) dt 6= 0: Exmple. By pplying the ineulity (.8), we give now some exmples of weighted trpezoid ineulities for some of the most populr weights. ) (Legendre) If g (t) = nd t [; ] ; then we get the following trpezoid ineulity for Hölder type mppings f: Z f () + f () dt p H ( )p : ) (Logrithm) If g (t) = ln t ; t (0; ] ; f is of p-hölder type on [0; ] nd the integrl R 0 ln t ds nite, then we hve Z f (0) + f () ln dt t p H: 0 c) (Jcoi) If g (t) = p t ; t (0; ] ; f is s ove nd the integrl R 0 nite, then we otin f (0) + f () Z 0 p dt t p H: f(t) p t ds d) (Cheychev) If g (t) = p t ; t ( ; ) ; f is of p-hölder type on ( ; ) nd the integrl R f(t) p t ds nite, then f ( ) + f () Z p t dt H:

6 6 S.S. DRAGOMIR.. The Cse of Lipschitzin Integrtors. The cse when the integrtor is Lipschitzin is s follows: Theorem 7. Let f : [; ]! C e p H-Hölder type mpping where H > 0 nd p (0; ] re given, nd u : [; ]! C is Lipschitzin function on [; ] ; this mens tht (.9) ju (x) u (y)j L jx yj for ll x; y [; ] ; where L > 0 is given. Then we hve the ineulity: Z f () + f () (.0) [u () u ()] du (t) p + HL ( )p+ : Proof. Using the ineulity for the Riemnn-Stieltjes integrl of Riemnn integrle integrnds nd Lipschitzin integrtors, we hve Z f () + f () (.) [u () u ()] du (t) f () + f () = du (t) L f () + f () dt: On utilizing the ineulity (.4) we hve (.) f () + f () dt H [(t ) p + ( t) p ] dt = p + H ( )p+ ; which, y (.), provides the desired result (.0). Remrk 3. If we ssume tht g : [; ]! C is Leesgue mesurle on [; ] with kgk := ess sup t[;] jg (t)j <, then on choosing u (x) = R x g (t) dt we get tht u is Lipschitzin with the constnt L = kgk : Conseuently, y (.0) we otin Z f () + f () (.3) g (t) dt g (t) dt p + H kgk ( )p+ : Remrk 4. We oserve thf the function u : [; ]! R is Lipschitzin function on [; ] with the constnt L then is of ounded vrition, nd, oviously, W (u) ( ) L: On utilizing the ineulity (.) we deduce Z f () + f () (.4) [u () u ()] du (t) p HL ( )p+ : Now, in order to compre which one of the ineulities (.0) nd (.4) is etter, we consider the uxiliry function (p) := p p nd p 0: We oserve tht 0 (p) = p ln nd the eution 0 (p) = 0 hs uniue solution p 0 = log (ln ) (0; ) : Since 0 (p) < 0 on (0; p 0 ) nd 0 (p) > 0 on (p 0 ; ) it follows tht min p[0;) (p) = (p 0 ) = ln ln : Also, we oserve tht (p) < 0 on (0; ) ; (p) > 0 on (; ) nd (0) = () = 0: In conclusion p + < for ll p (0; ) ; p

7 APPROXIMATING THE RIEMANN-STIELTJES INTEGRALS 7 showing tht the ineulity (.0) is lwys etter thn the ineulity (.4). Remrk 5. We notice tht, if f is H-Lipschitzin, then (.0) ecomes Z f () + f () [u () u ()] du (t) HL ( ) :.3. The Cse of Monotonic Nondecresing Integrtors. In the cse when u is monotonic nondecresing, we hve the following result s well: Theorem 8. Let f : [; ]! C e p H-Hölder type mpping where H > 0 nd p (0; ] re given, nd u : [; ]! R monotonic nondecresing function on [; ] : Then we hve the ineulity: Z f () + f () (.5) [u () u ()] du (t) Z " # ) (( H ) p ( t) p (t ) p [u () u ()] p ( t) p u (t) dt (t ) p p H ( )p [u () u ()] : The ineulities in (.5) re shrp. Proof. Using the ineulity for the Riemnn-Stieltjes integrl of continuous integrnds nd monotonic integrtors, we hve Z f () + f () (.6) [u () u ()] du (t) f () + f () du (t) : Utilising (.4) we then hve (.7) f () + f () du (t) H [(t ) p + ( t) p ] du (t) : Integrting y prts in the Riemnn-Stieltjes integrl we get (.8) [(t ) p + ( t) p ] du (t) h = [(t ) p + ( t) p ] u (t)j p (t ) p ( t) p i u (t) dt Z " # = ( ) p ( t) p (t ) p [u () u ()] p ( t) p u (t) dt; (t ) p which together with (.7) produces the rst prt of the ineulity (.5). Now, since [(t ) p + ( t) p ] p ( ) p for ny t [; ], then [(t ) p + ( t) p ] u (t) dt p ( ) p [u () u ()]

8 8 S.S. DRAGOMIR nd the lst prt of (.5) is lso proved. Choose f : [0; ]! R; = t; nd 8 < 0 for t [0; ); u (t) = : for t = : We oserve tht f is of H-Hölder type with H = nd u is monotonic nondecresing on [0; ], then we otin in ll sides of the ineulity (.5) the sme untity : Remrk 6. We notice thf f is H-Lipschitzin, then (.5) reduces to Z f () + f () (.9) [u () u ()] du (t) H ( ) [u () u ()] : The constnt is est possile in (.9). 3. The Cse of Hölder-Continuous Integrtors 3.. The Cse of Bounded Vrition Integrnds. Theorem 9 (Drgomir, 00, []). Let f : [; ]! C e function of ounded vrition on [; ] nd u : [; ]! C p K-Hölder continuous function on tht intervl. Then we hve the ineulity: Z f () + f () (3.) [u () u ()] du (t) _ p K ( )p (f) : The constnt C = on the right hnd side of (3:) cnnot e replced y smller untity. Proof. Using the integrtion y prts formul for the Riemnn-Stieltjes integrls, we hve the following eulity of interest u () + u () (3.) u (t) d = = u (t) f () + f () u () + u () [u () u ()] du (t) du (t) : Utilising similr pproch to the one in the proof of Theorem 6 we deduce the desired ineulity (3.). To prove the shrpness of the constnt ; ssume tht (3.) holds with constnt C > 0; i.e., (3.3) f () + f () [u () u ()] du (t) C p K ( )p _ (f) : Choose u (t) = t p ; p (0; ] ; t [0; ] which is of p-hölder type with the constnt K = nd f : [0; ]! R given y: 0 if t [0; ) ; = if t = ; which is of ounded vrition on [0; ] :

9 APPROXIMATING THE RIEMANN-STIELTJES INTEGRALS 9 Sustituting in (3.3) we otin Z pt p dt C p ( However, 0 Z t p 0 dt = 0 nd ) _ (f) : 0 _ (f) = ; nd then C p for ll p (0; ] : Choosing p = ; we deduce C nd the theorem is completely proved. Remrk 7. Let f : [; ]! C e s in Theorem 9 nd u e K Lipschitzin mpping on [; ] ; where K > 0 is given. Then we hve the ineulity Z f () + f () (3.4) [u () u ()] du (t) _ K ( ) (f) : The constnt in (3.4) is est possile. We now point out some results in estimting the integrl of product. Corollry. Let f : [; ]! C e mpping of ounded vrition on [; ] nd g e Leesgue mesurle on [; ] : Put kgk := ess sup t[;] jj ; nd if kgk < ; then we hve the ineulity: (3.5) f () + f () g (s) ds 0 g (t) dt kgk ( _ ) (f) : Proof. De ne the mpping u : [; ]! C; u (t) := R t g (s) ds: Then u is L Lipschitzin with the constnt L = kgk : Therefore, y the properties of Riemnn-Stieltjes integrls, we hve du (t) = g (t) dt; nd then, y (3.4) we deduce the desired result (3.5). The following corollry is lso nturl conseuence of Theorem 9. Corollry. Let f : [; ]! C e mpping of ounded vrition on [; ] nd g e Leesgue mesurle on [; ] : Put kgk := jg (s)j ds! ; > : If kgk < ; then we hve the ineulity Z f () + f () (3.6) g (s) ds g (t) dt kgk ( ) _ (f) : Proof. Consider the mpping u s in the proof of Corollry. Then, y Hölder s integrl ineulity, we cn stte tht Z t Z ju (t) u (s)j = g (z) dz jt sj t jg (z)j dz jt sj kgk ; s s

10 0 S.S. DRAGOMIR for ll t; s [; ] ; which shows tht the mpping u is of r K-Hölder type with r := (0; ) nd K = kgk < : Applying Theorem 9 we deduce the desired result (3.6). Exmple. We give now some exmples of weighted trpezoid ineulities for some of the most populr weights. (3.7) (3.8) ) (Legendre). If g(t) = ; t [; ] then y (3:5) nd (3:6) we get the trpezoid ineulities Z f () + f () ( ) dt _ ( ) (f) nd f () + f () ( ) dt _ ( ) (f) ; > : = We remrk tht the rsneulity is etter thn the second one. ) (Jcoi). If g(t) = p t ; t (0; ]; then oviously kgk = +; so we cnnot pply the ineulity (3:5) : If we ssume tht (; ); then we hve Z = = p kgk = dt = t 0 nd pplying the ineulity (3:6) we deduce Z f(0) + f() p f(t)dt 0 t 4 ( )= ( ) = _ (f) for ll (; ): c) (Cheychev). If g(t) = p ; t ( ; ); then oviously kgk t = +; so we cnnot pply the ineulity (3:5) : If we ssume tht (; ) then we hve Z = kgk = p dt t Z = (t + ) ( t) = = B dt ; = : Applying the ineulity (3:6) we deduce Z f( ) + f() f(t) p dt t B ; = _ (f) for ll (; ): 0 = 0

11 APPROXIMATING THE RIEMANN-STIELTJES INTEGRALS 3.. The Cse of Lipschitzin Integrnds. Theorem 0. Let f : [; ]! C e Lipschitzin function with the constnt S > 0 on [; ] nd u : [; ]! C p K-Hölder continuous function on thntervl. Then we hve the ineulity: Z f () + f () (3.9) [u () u ()] du (t) p + KS ( )p+ : Proof. Is similr to the one from the proof of Theorem 7 nd we omit the detils. Corollry 3. Let f : [; ]! C e Lipschitzin function with the constnt S > 0 on [; ] nd g e Leesgue mesurle on [; ] : If kgk < ; where ; then we hve the ineulity (3.0) f () + f () g (s) ds g (t) dt kgk S ( ) : Some pplictions for vrious clsses of weights my e provided, however the detils re left to the interested reder The Cse of Monotonic Nondecresing Integrnds. Theorem. Let f : [; ]! R e monotonic nondecresing function on [; ] nd u : [; ]! K p K-Hölder continuous function on thntervl. Then we hve the ineulities: Z f () + f () (3.) [u () u ()] du (t) Z " # # "( K ) p ( t) p (t ) p [f () f ()] p ( t) p dt (t ) p p K ( )p [f () f ()] : The ineulities in (3.) re shrp. Proof. Is similr to the one from the proof of Theorem 8 nd we omit the detils. Remrk 8. We oserve thf u is K-Lipschitzin, then (3.) reduces to Z f () + f () (3.) [u () u ()] du (t) K ( ) [f () f ()] nd the constnt is est possile in (3.). 4. Other Trpezoidl Ineulities 4.. The Cse of Bounded Vrition Integrnds nd Integrtors. The cse when oth the integrnd nd the integrtor re of ounded vrition is s follows: Theorem. Let f, u : [; ]! C e of ounded vrition on [; ] : If one of them is continuous on [; ] ; then the Riemnn-Stieltjes integrl R du (t) exists nd we hve the ineulity Z f () + f () (4.) [u () u ()] du (t) (f) (u) :

12 S.S. DRAGOMIR The constnt is est possile in (4.). Proof. Since f is of ounded vrition, then we hve f () + f () [jf () j + j f ()j] _ (f) for ll t [; ] : If we ssume tht f is continuous on [; ] ; then it follows tht the Riemnn- Stieltjes integrl exists nd Z f () + f () (4.) [u () u ()] du (t) f () + f () = du (t) mx f () + f () _ t[;] u (t) (f) u (t) ; which proves the desired result (4.). Now, if we choose in (4.) = t; then we get the ineulity Z + (4.3) [u () u ()] tdu (t) _ ( ) (u) ; ths of interesn itself s well. We show tht the constnt is est possile in this ineulity. Assume tht (4.3) with constnt E > 0; i.e. Z + _ (4.4) [u () u ()] tdu (t) E ( ) (u) ; for ny function of ounded vrition u : [; ]! C. If we choose the function u : [; ]! R, with 8 < ; t = ; u (t) = 0; t (; ) ; : ; t = ; then we oserve tht u is of ounded vrition nd prts in the Riemnn-Stieltjes integrl we hve tdu (t) = tu (t)j _ (u) = : Also, integrting y u (t) dt = nd y (4.4) we deduce tht E ( ) ; which implies tht E ; nd the shrpness of the constnn (4.) is proven.

13 APPROXIMATING THE RIEMANN-STIELTJES INTEGRALS 3 If u is continuous, then on utilizing the integrtion y prts formul for the Riemnn-Stieltjes integrl, we hve the following eulity of interest (4.5) f () + f () [u () u () + u () = which, s ove, gives tht f () + f () [u () mx u () + u () t[;] u ()] u (t) d ; du (t) u ()] du (t) _ u (t) : On utilizing the sme rgument s in the rst prt, we deduce the desired result (4.). If in (4.) we tke u (t) = t; then we get the ineulity of interest Z f () + f () ( ) dt _ ( ) (f) ; tht holds for ny function of ounded vrition f : [; ]! C for which est possile constnt. Note tht this results ws otined for the rst time in [0]. is the Remrk 9. If we ssume tht g : [; ]! C is Leesgue integrle on [; ] ; then u (x) = R x g (t) ds di erentile lmost everywhere, u () = R g (t) dt; u () = 0 nd W (u) = R jg (t)j dt: Conseuently, y (4.) we otin Z f () + f () (4.6) g (t) dt g (t) dt _ (f) jg (t)j dt: From (4.6) we get weighted version of the trpezoid ineulity, Z f () + f () (4.7) R g (t) dt g (t) dt _ (f) ; provided tht g (t) 0; for lmost every t [; ] nd R g (t) dt 6= 0: 4.. The Cse of End-point Lipschitzin Functions. In this susection we consider the cse when the function f : [; ]! C stis es the end-point Lipschitzin conditions (4.8) j f ()j L (t ) nd jf () j L ( t) for ny t (; ) where the constnts L ; L > 0 nd ; > re given. Theorem 3. Assume tht the function f stis es the condition (4.8).

14 4 S.S. DRAGOMIR ) If u : [; ]! C is Lipschitzin with the constnt K > 0; then we hve the ineulity (4.9) f () + f () [u () u ()] K L + ( )+ + L + ( du (t) )+ : ) If ; > 0 nd u : [; ]! R is monotonic nondecresing on [; ] ; then (4.0) f () + f () [u () u ()] " L + L " ( ) u () du (t) (t ) u (t) dt ( t) u (t) dt ( ) u () h L ( ) + L ( ) i [u () u ()] : # # Proof. ). Since u : [; ]! C is Lipschitzin with the constnt K > 0; then we hve f () + f () [u () u ()] du (t) K [jf () j + j f ()j] dt K h L ( t) + L (t ) i dt = K L + ( )+ + L + ( )+ nd the ineulity (4.9) is otined. ) Since u : [; ]! R is monotonic nondecresing on [; ] ; then (4.) f () + f () [u () u ()] du (t) [jf () j + j f ()j] du (t) h L ( t) + L (t ) i du (t) :

15 APPROXIMATING THE RIEMANN-STIELTJES INTEGRALS 5 Utilising the integrtion y prts formul for the Riemnn-Stieltjes integrl nd the fct tht u is monotonic nondecresing on [; ] we hve tht (4.) nd (4.3) (t ) du (t) = ( ) u () ( ) u () u () = ( ) [u () u ()] ( t) du (t) = ( ) u () + u () (t ) u (t) dt (t ) dt ( t) u (t) dt ( t) dt ( ) u () = ( ) [u () u ()] : On mking use of (4.)-(4.3) we deduce the desired ineulity (4.0). Remrk 0. We notice tht the dul cse, i.e., when the integrtor stis es the condition (4.8) nd the integrnd f is either Lipschitzin or monotonic nondecresing produces similr ineulities. However they will not e stted here. 5. A Qudrture Rule for the Riemnn-Stieltjes Integrl Consider the prtition I n : = t 0 < t < ::: < t n < t n = of the intervl [; ] ; nd de ne h i := + (i = 0; :::; n ), (h) := mx fh i ji = 0; :::; n g nd the generlized trpezoidl udrture rule (5.) T n (f; u; I n ) := nx f ( ) + f (+ ) [u (+ ) u ( )] : The following result for the numericl pproximtion of the Riemnn-Stieltjes integrl holds. Theorem 4. Let f : [; ]! C e p H-Hölder continuous function on [; ] (p (0; ]) nd u : [; ]! C e function of ounded vrition on [; ]. Then (5.) du (t) = T n (f; u; I n ) + R n (f; u; I n ) ; where T n (f; u; I n ) is the generlized trpezoidl formul given y (5:) ; nd the reminder R (f; u; I n ) stis es the estimte (5.3) jr n (f; u; I n )j _ p H [ (h)]p (u) : If the integrtor u : [; ]! C is Lipschitzin function with the constnt L > 0; then the reminder R (f; u; I n ) stis es the ound (5.4) jr n (f; u; I n )j p + HL ( ) [ (h)]p :

16 6 S.S. DRAGOMIR Proof. We pply Theorem 6 on every suintervl [ ; + ] (i = 0; :::; n Z f ( ) + f (+ ) ti+ (5.5) [u (+ ) u ( )] du (t) p Hhp i t_ i+ (u) : ) to otin Summing the ineulities (5:5) over i from 0 to n nd using the generlized tringle ineulity, we otin n X Z jr (f; u; I n )j f ( ) + f (+ ) ti+ [u (+ ) u ( )] du (t) n p H X h p i t_ i+ = _ p H [ (h)]p (u) ; (u) H [ (h)]p p nx nd the ound (5:3) is proved. The cse of Lipschitzin integrtors follows from Theorem 7 nd the detils re omitted. Let us ssume tht g : [; ]! C is Leesgue integrle nd f : [; ]! C is of r H-Hölder type on [; ] : For given prtition I n of the intervl [; ] ; consider the udrture nx Z f ( ) + f (+ ) ti+ (5.6) W T n (f; g; I n ) := g (s) ds: We cn stte the following corollry: Corollry 4. Let f : [; ]! C e of r H-Hölder type nd g : [; ]! C e Leesgue integrle on [; ] : Then we hve the formul (5.7) t_ i+ (u) g (t) dt = W T n (f; g; I n ) + W R n (f; g; I n ) ; where the reminder term W R n (f; g; I n ) stis es the estimte in terms of the integrl of jgj (5.8) jw R n (f; g; I n )j p H [ (h)]p jg (s)j ds nd the estimte in terms of the essentil supremum of jgj (5.9) jw R n (f; g; I n )j p + H ( ) [ (h)]p kgk ; provided tht, in this cse, kgk < : The previous corollry llows us to otin dptive udrture formule for di erent weighted integrls. Exmple 3. We point out only few exmples for the most common weights.

17 APPROXIMATING THE RIEMANN-STIELTJES INTEGRALS 7 ) (Legendre) If g (t) = ; nd t [; ] ; then we get for the mpping f : [; ]! C of p H-Hölder type: dt = T (f; I n ) + R (f; I n ) ; where T (f; I n ) is the usul trpezoidl udrture rule T (f; I n ) = nx f ( ) + f (+ ) nd the reminder stis es the estimte h i jr (f; I n )j p H ( ) [ (h)]p : ) (Logrithm) If g (t) = ln t ; t (; ] (0; ]; f is of p H-Hölder type nd the integrl R ln t dt <, then we hve the generlized trpezoid formul: ln dt = T L (f; I n ) + R L (f; I n ) ; t where T L (f; I n ) is the Logrithm-Trpezoid udrture rule nx f ( ) + f (+ ) ti ti+ T L (f; I n ) = ln + ln e e nd the reminder term R L (f; I n ) stis es the estimte jr L (f; I n )j p H [ (h)]p ln ln : e e c) (Jcoi) If g (t) = p t ; t (; ] (0; ) ; f is of p H-Hölder type nd R f(t) p t dt < ; then we hve the generlized trpezoid formul p t dt = T J (f; I n ) + R J (f; I n ) ; where T J (f; I n ) is the Jcoi-Trpezoid udrture rule T J (f; I n ) = nx f ( ) + f (+ ) pti+ nd the reminder term R J (f; I n ) stis es the estimte jr J (f; I n )j p+ H [ p p (h)]p : p ti d) If g (t) = p t ; t (; ) ( ; ) ; f is of p H-Hölder type nd R f(t) p t dt < ; then we hve the generlized trpezoid formul p t dt = T C (f; I n ) + R C (f; I n )

18 8 S.S. DRAGOMIR where T C (f; I n ) is the Cheychev-Trpezoid udrture rule T C (f; I n ) := nx f ( ) + f (+ ) [rcsin (+ ) rcsin ( )] nd the reminder term R C (f; I n ) stis es the estimte 5.. More Error Bounds. jr C (f; I n )j p H [ (h)]p [rcsin () rcsin ()] : Theorem 5. Let u : [; ]! C e p K-Hölder continuous function on [; ] (p (0; ]) nd f : [; ]! C e function of ounded vrition on [; ]. Then (5.0) du (t) = T n (f; u; I n ) + R n (f; u; I n ) ; where T n (f; u; I n ) is the generlized trpezoidl formul given y (5:) ; nd the reminder R (f; u; I n ) stis es the estimte (5.) jr n (f; u; I n )j _ p K [ (h)]p (f) : If the integrnd f : [; ]! C is Lipschitzin function with the constnt S > 0; then the reminder R (f; u; I n ) stis es the ound (5.) jr n (f; u; I n )j p + KS ( ) [ (h)]p : Proof. Follows from Theorems 9 nd 0 nd the detils re omitted. The cse of weighted integrls is s follows: Corollry 5. Let f : [; ]! C e mpping of ounded vrition on [; ] nd g e Leesgue mesurle on [; ] : Then we hve the formul (5.3) g (t) dt = W T n (f; g; I n ) + W R n (f; g; I n ) ; where the reminder term W R n (f; g; I n ) stis es the estimte (5.4) jw R n (f; g; I n )j kgk [ (h)] _ (f) ; provided tht kgk < ; : Let f : [; ]! C e Lipschitzin function with the constnt S > 0 on [; ] nd g e Leesgue mesurle on [; ] : If kgk < ; where ; then we hve the ineulity (5.5) jw R n (f; g; I n )j kgk S [ (h)] : Remrk. Vrious prticulr udrtures nd their errors ounds for the usul weights cn e provided, however the detils re left to the interested reder.

19 APPROXIMATING THE RIEMANN-STIELTJES INTEGRALS Other Error Bounds. Theorem 6. Let f : [; ]! C e continuous nd of ounded vrition on [; ] nd u : [; ]! C e function of ounded vrition on [; ]. Then (5.6) du (t) = T n (f; u; I n ) + R n (f; u; I n ) ; where T n (f; u; I n ) is the generlized trpezoidl formul nd the reminder R (f; u; I n ) stis es the estimte ( (5.7) jr n (f; u; I n )j ti+ ) mx (f) (u) : if0;:::;n g In prticulr, if f is Lipschitzin with the constnt L > 0; then (5.8) jr n (f; u; I n )j _ L (h) (u) : ) to o- Proof. We pply Theorem on every suintervl [ ; + ] (i = 0; :::; n tin Z f ( ) + f (+ ) ti+ (5.9) [u (+ ) u ( )] du (t) t_ i+ t_ i+ (f) (u) : t_ i+ (f) (u) ( ti+ ) (f) (u) ; mx if0;:::;n g Summing the ineulities (5:9) over i from 0 to n nd using the generlized tringle ineulity, we otin nx Z jr (f; u; I n )j f ( ) + f (+ ) ti+ [u (+ ) u ( )] du (t) nx t_ i+ ( ti+ ) _ nx t_ i+ = mx if0;:::;n g nd the ound (5:7) is proved. The second prt follows from the fct thf f is L-Lipschitzin, then W + (f) Lh i for ll i f0; :::; n g : In prticulr we hve: Corollry 6. Let f : [; ]! C e continuous nd of ounded vrition on [; ] nd g : [; ]! C e Leesgue integrle on [; ] : Then we hve the formul (5.0) (f) g (t) dt = W T n (f; g; I n ) + W R n (f; g; I n ) ; where the reminder term W R n (f; g; I n ) stis es the estimte ( (5.) jw R n (f; g; I n )j ti+ ) _ mx (f) jg (s)j ds: if0;:::;n g (u)

20 0 S.S. DRAGOMIR In prticulr, if f is Lipschitzin with the constnt L > 0; then (5.) jr n (f; u; I n )j _ L (h) (u) : 6. Applictions for Functions of Selfdjoint Opertors Let A e selfdjoint liner opertor on complex Hilert spce (H; h:; :i) : The Gelfnd mp estlishes -isometriclly isomorphism etween the set C (Sp (A)) of ll continuous functions de ned on the spectrum of A; denoted Sp (A) ; nd the C -lger C (A) generted y A nd the identity opertor H on H s follows (see for instnce [9, p. 3]): For ny f; g C (Sp (A)) nd ny ; C we hve (i) (f + g) = (f) + (g) ; (ii) (fg) = (f) (g) nd f = (f) ; (iii) k (f)k = kfk := sup tsp(a) jj ; (iv) (f 0 ) = H nd (f ) = A; where f 0 (t) = nd f (t) = t; for t Sp (A) : With this nottion we de ne f (A) := (f) for ll f C (Sp (A)) nd we cll it the continuous functionl clculus for selfdjoint opertor A: If A is selfdjoint opertor nd f is rel vlued continuous function on Sp (A), then 0 for ny t Sp (A) implies tht f (A) 0; i:e: f (A) is positive opertor on H: Moreover, if oth f nd g re rel vlued functions on Sp (A) then the following importnt property holds: (P) g (t) for ny t Sp (A) implies tht f (A) g (A) in the opertor order of B (H) : For recent monogrph devoted to vrious ineulities for continuous functions of selfdjoint opertors, see [9] nd the references therein. For other recent results see [5], [6], [7], [0], [], [3] nd [5]. Let U e selfdjoint opertor on the complex Hilert spce (H; h:; :i) with the spectrum Sp (U) included in the intervl [m; M] for some rel numers m < M nd let fe g e its spectrl fmily. Then for ny continuous function f : [m; M]! R, is well known tht we hve the following spectrl representtion in terms of the Riemnn-Stieltjes integrl: (6.) hf (U) x; yi = Z M m 0 f () d (he x; yi) ; for ny x; y H: The function g x;y () := he x; yi is of ounded vrition on the intervl [m; M] nd g x;y (m 0) = 0 nd g x;y (M) = hx; yi for ny x; y H: Is lso well known tht g x () := he x; xi is monotonic nondecresing nd right continuous on [m; M]. Let A e selfdjoint opertor in the Hilert spce H with the spectrum Sp (A) [m; M] for some rel numers m < M nd let fe g e its spectrl fmily. Consider the prtition I n : m = t 0 < t < ::: < t n < t n = M of the intervl [m; M] ; nd de ne h i := + (i = 0; :::; n ), (h) := mx fh i ji = 0; :::; n g

21 APPROXIMATING THE RIEMANN-STIELTJES INTEGRALS nd the generlized trpezoidl udrture rule ssocited to the continuous function f : [m; M]! C, selfdjoint opertor A nd the vectors x; y H (6.) T n (f; A; I n ; x; y) := nx f ( ) + f (+ ) Eti+ E ti x; y : Theorem 7. Let A e selfdjoint opertor in the Hilert spce H with the spectrum Sp (A) [m; M] for some rel numers m < M nd let fe g e its spectrl fmily. ) If f : [m; M]! C is continuous nd with ounded vrition on [m; M] ; then for ny x; y H (6.3) hf (A) x; yi = T n (f; A; I n ; x; y) + R n (f; A; I n ; x; y) (6.4) (6.5) nd the reminder R n (f; A; I n ; x; y) stis es the error ounds ( jr n (f; A; I n ; x; y)j ti+ ) M mx (f) E() x; y if0;:::;n g m ( ti+ ) _ mx if0;:::;n g (f) kxk kyk : ) Let f : [m; M]! C e p H-Hölder continuous function on [m; M] ; then for ny x; y H we hve the eulity (6.3) nd the reminder R n (f; A; I n ; x; y) stis es the error ounds jr n (f; A; I n ; x; y)j p H [ (h)]p M _ m p H [ (h)]p kxk kyk : E() x; y Proof. The rsneulities in (6.4) nd (6.5) follow from Theorem 4 nd Theorem 6 pplied for the integrtor of ounded vrition u (t) = he t x; yi with t [m; M] : If P is nonnegtive opertor on H; i.e., hp x; xi 0 for ny x H; then the following ineulity is generliztion of the Schwrz ineulity in H (6.6) jhp x; yij hp x; xi hp y; yi for ny x; y H: Further, if d : m = t 0 < t < ::: < t n < t n = M is n ritrry prtition of the intervl [m; M] ; then we hve y Schwrz s ineulity for nonnegtive opertors tht M_ E() x; y m = sup d sup d ( n X ) E ti+ E ti x; y ( n X h = i ) = Eti+ E ti x; x Eti+ E ti y; y := I:

22 S.S. DRAGOMIR By the Cuchy-Bunikovski-Schwrz ineulity for seuences of rel numers we lso hve tht 8" < nx # = " nx # 9 = = I sup Eti+ E ti x; x Eti+ E ti y; y d : ; 8" < nx # = " nx # 9 = = sup Eti+ E d : ti x; x sup Eti+ E ti y; y d ; = " M _ m E() x; x # = " M _ m E() y; y # = = kxk kyk for ny x; y H: These prove the lst prts of (6.4) nd (6.5). Remrk. In the prticulr cse when the prtition reduces to the whole intervl [m; M] we hve the following trpezoidl ineulities for ny x; y H f (M) + f (m) (6.7) hx; yi hf (A) x; yi M_ M_ (f) E() x; y M_ (f) kxk kyk ; m m when f : [m; M]! C is continuous nd with ounded vrition on [m; M] ; nd f (M) + f (m) (6.8) hx; yi hf (A) x; yi p H (M m)p M _ m m E() x; y p H (M m)p kxk kyk ; when f : [m; M]! C is p H-Hölder continuous function on [m; M] : Moreover, if we use the ineulity (.4) for the monotonic nondecresing function u (t) = he t x; xi with x H; then we get f (M) + f (m) kxk (6.9) hf (A) x; xi Z " # ) M ((M H m) p kxk (M t) p (t m) p p (M t) p he t x; xi dt (t m) p p H (M m)p kxk ; m 0 for ny x H where f : [m; M]! C is p H-Hölder continuous function on [m; M] : Finlly, if the function f : [m; M]! C stis es the end-point Lipschitzin conditions j f (m)j L m (t m) nd jf (M) j L M (M t)

23 APPROXIMATING THE RIEMANN-STIELTJES INTEGRALS 3 for ny t (m; M) where the constnts L m ; L M > 0 nd ; > 0 re given, then on pplying the ineulity (4.0) we hve f (M) + f (m) kxk (6.0) hf (A) x; xi " # L m + Z M L M (M m) kxk m 0 Z M m 0 (M t) he t x; xi dt h L m (M m) + L M (M m) i kxk : (t m) he t x; xi dt Acknowledgement. The uthor would like to thnk the nonymous referees for their vlule suggestions tht hve een implemented in the nl version of this pper. References [] C.T.H. Bker, A perspective on the numericl tretment of Volterr eutions, J. Comput. Appl. Mth. 5 (000) [] N.S. Brnett, W.-S. Cheung, S.S. Drgomir, A. Sofo, Ostrowski nd trpezoid type ineulities for the Stieltjes integrl with Lipschitzin integrnds or integrtors, Computers & Mthemtics with Applictions, 57(009), Issue, [3] T.K. Boehme, W. Preuss, W. vn der Wll, On simple numericl method for computing Stieltjes integrls in reliility theory, Pro. Engrg. Inform. Sci. 5 (99) 3 8. [4] P. Cerone, W.S. Cheung, S.S. Drgomir, On Ostrowski type ineulities for Stieltjes integrls with solutely continuous integrnds nd integrtors of ounded vrition, Computers & Mthemtics with Applictions, 54(007), Issue, [5] P. Cerone nd S.S. Drgomir, Trpezoidl-type rules from n ineulities point of view, in Hndook of Anlytic-Computtionl Methods in Applied Mthemtics, G.A. Anstssiou (Ed), Chpmn & Hll/CRC Press, New York, 000, [6] P. Cerone, S.S. Drgomir nd C.E.M. Perce, A generlised trpezoid ineulity for functions of ounded vrition, Turkish J. of Mth., 4() (000), [7] D. Cruz-Urie, C.J. Neugeuer, An elementry proof of error estimtes for the trpezoidl rule, Mth. Mg. 76 (003) [8] K. Diethelm, A note on the midpoint rectngle formul for Riemnn Stieltjes integrls, J. Stt. Comput. Simul. 73 (004) [9] S.S. Drgomir, On the trpezoid udrture formul for Lipschitzin mppings nd pplictions, Tmkng J. of Mth., 30() (999), [0] S.S. Drgomir, The Ostrowski s integrl ineulity for mppings of ounded vrition, Bull. Austrl. Mths. Soc., 60(999), [] S.S. Drgomir, Some ineulities for Riemnn-Stieltjes integrl nd pplictions, in: A. Ruinov nd B. Glover (eds.), Optimiztion nd Relted Topics, 97-35, Kluwer Acdemic Pulishers, 00. [] S.S. Drgomir, Some ineulities of midpoint nd trpezoid type for the Riemnn Stieltjes integrl, Nonliner Anlysis, 47(00), Issue 4, [3] S.S. Drgomir, An ineulity improving the second Hermite-Hdmrd ineulity for convex functions de ned on liner spces nd pplictions for semi-inner products, J. Ineul. Pure & Appl. Mth., 3(00), No. 3, Article 00. [4] S.S. Drgomir, Approximting the Riemnn Stieltjes integrl in terms of generlised trpezoidl rules, Nonliner Anlysis: Theory, Methods & Applictions, 7(009), e6-e7. [5] S.S. Drgomir, µceyšev s type ineulities for functions of selfdjoint opertors in Hilert spces, Preprint RGMIA Res. Rep. Coll., (e) (008), Art. 9. [6] S.S. Drgomir, Grüss type ineulities for functions of selfdjoint opertors in Hilert spces, Preprint RGMIA Res. Rep. Coll., (e) (008), Art..

24 4 S.S. DRAGOMIR [7] S.S. Drgomir, Ineulities for the µceyšev functionl of two functions of selfdjoint opertors in Hilert spces, Prerpint RGMIA Res. Rep. Coll., (e) (008), Art. 7. [8] S.S. Drgomir, Some trpezoidl vector ineulities for continuous functions of selfdjoint opertors in Hilert spces, Preprint RGMIA Res. Rep. Coll. 3(00), Issue, Article 4. [9] T. Furut, J. Mićić Hot, J. Peµcrić nd Y. Seo, Mond-Peµcrić Method in Opertor Ineulities. Ineulities for Bounded Selfdjoint Opertors on Hilert Spce, Element, Zgre, 005. [0] A. Mtković, J. Peµcrić nd I. Perić, A vrint of Jensen s ineulity of Mercer s type for opertors with pplictions. Liner Alger Appl. 48 (006), No. -3, [] P. R. Mercer, Hdmrd s ineulity nd trpezoid rules for the Riemnn Stieltjes integrl, J. Mth. Anl. Appl. 344 (008) [] B. Mond nd J. Peµcrić, Convex ineulities in Hilert spces, Houston J. Mth., 9(993), [3] B. Mond nd J. Peµcrić, Clssicl ineulities for mtrix functions, Utilits Mth., 46(994), [4] M. Muntenu, Qudrture formuls for the generlized Riemnn-Stieltjes integrl. Bull. Brz. Mth. Soc. (N.S.) 38 (007), no., [5] J. Peµcrić, J. Mićić nd Y. Seo, Ineulities etween opertor mens sed on the Mond- Peµcrić method. Houston J. Mth. 30 (004), no., [6] M. Tortorell, Closed Newton Cotes udrture rules for Stieltjes integrls nd numericl convolution of life distriutions, SIAM J. Stt.Comput. (990) Mthemtics, School of Engineering & Science, Victori University, PO Box 448, Melourne City, MC 800, Austrli. E-mil ddress: sever.drgomir@vu.edu.u URL: School of Computtionl nd Applied Mthemtics, University of the Witwtersrnd, Privte Bg-3, Wits-050, Johnnesurg, South Afric