ON PERTURBED TRAPEZOIDAL AND MIDPOINT RULES. f (t) dt

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1 ON PERTURBED TRAPEZOIDAL AND MIDPOINT RULES P. CERONE Abstrct. Explicit bounds re obtined for the perturbed or corrected trpezoidl nd midpoint rules in terms of the Lebesque norms of the second derivtive of the function. It is demonstrted tht the bounds obtined re the sme for both rules lthough the perturbtion or the correction term is different.. Introduction Let f : b] R nd define the functionls.) I f) : f t) dt.) I T ) f) : b f ) + f b)] nd ) + b.) I M) f) : b ) f. Here I T ) f) nd I M) f) re the well known trpezoidl nd midpoint rules used to pproximte the functionl I f). Atkinson ] defined the corrected or perturbed trpezoidl nd midpoint rules by.4) P I T ) f) : I T ) f) c f b) f )] nd.5) P I M) f) : I M) f) + c 6 f b) f )] respectively where c b. Atkinson ] uses n symptotic error estimte which does not redily produce estimtes of the bounds in using.4) nd.5) to pproximte.) by.4) or.5). In recent rticle Brnett nd Drgomir ] obtined explicit bounds for I f) P I T ) f) in terms of the Lebesque norms of f t) f ; b] where f ; b] : f b) f ) b is the divided difference. If the Lebesque norms re defined in the usul wy such tht by h L p b] we men ) b p.6) h p : h t) p dt p < Dte: April Mthemtics Subject Clssifiction. Primry 05C8 5A5; Secondry 05A5 5A8. Key words nd phrses. Keyword one keyword two keyword three.

2 P. CERONE nd.7) h : ess sup h t). t b] Brnett nd Drgomir ] obtined the following theorem. Theorem. Let f : b] R be differentible nd f bsolutely continuous on b] then.8) b ) f f ; b] 4 if f L b] ; I f) P I T ) f) b ) + q 8 q + ) q f f ; b] p if f L p b] b ) f f ; b] 8 p > p + q ; where h; b] : hb) h) b is the divided difference. Further the Grüss integrl inequlity for function h : b] R with < m h x) m < for lmost every x b] then see for exmple 0 p. 96]).9) 0 ) M m b h M h) where M h) b b h t) dt is the integrl men. Brnett nd Drgomir ] lso obtined the following results. Let f : b] R then.0) nd.) I f) P I T ) f) I f) P I T ) f) b ) 8 5 ] b f f ; b] f L b] b ) 6 5 Γ γ) γ f t) Γ.e. t b]. Result.0) is obtined from the second inequlity in.8) nd.) from.9) nd.0). It is the intention of the current rticle to demonstrte tht bounds for I f) P I T ) f) my be obtined involving the trditionl Lebesque norms of f p p where p re s defined by.6) nd.7) rther thn f f ; b] p s obtined in.8). Further bounds will be obtined for I f) P I M) f). These will be shown to be the sme s those obtined for the perturbed trpezoidl rule lthough the correction term is different.. Identities nd Inequlities for Trpezoidl Like Rules Let the trpezoidl functionl T f; b) be defined by.) T f; b) : I f) I T ) f) b f t) dt b f ) + f b)]

3 TRAPEZOIDAL AND MIDPOINT RULES then it is well known tht the identity.) T f; b) t ) b t) f t) dt holds. The following theorem ws obtined in 9] using identity.). Theorem. Let f : b] R be twice differentible function on b). Then we hve the estimte.) T f; b) f b ) if f L b] ; f q B q + q + )] p b ) + p if f L p b] p + q p > f 8 b ) if f L b] where B is the Bet function tht is Let B r s) : 0 t r t) s dt r s > 0..4) P T f; b) : I f) P I T ) f) T f; b) + c f b) f )] where c b then the following lemm holds. Lemm. Let f : b] R be such tht f is bsolutely continuous on b] then.5) P T f; b) is vlid with.6) κ t) Proof. From.4) nd.6) we hve κ t) f t) dt t + b ) c c b P T f; b) t ) b t) f t) dt + c f b) f )] c ] t ) b t) f t) dt ] t + b) t + b + c f t) dt b t + b ) ] c f t) dt nd so.5) holds with κ t) s given by.6).

4 4 P. CERONE Theorem. Let f : b] R be such tht f is bsolutely continuous on b] then.7) P T f; b) 4c 9 f if f L b] ; { c c 6 B q + ) + u ) } q q du f p if f L p b] p + q p > c 6 f if f L b] where B is the bet function nd c b. Proof. From identity.5) nd using.6) we hve.8) P T f; b) κ t) f t) dt f ) b p κ t) q dt q p >. Now we need to exmine the behviour of κ t) in order to proceed further. We notice from.6) tht κ ) κ b) c nd κ t) 0 where t +b ± c. Further < 0 t < +b ; κ t) t + b ) 0 t +b ; > 0 t > +b. Also κ t) is symmetric function bout +b since κ +b + x ) κ +b x ) so tht from.8).9) κ q q +b + c b κ t)] q dt + κ q t) dt +b +b + c : I q) + I q)]. Further from.6) I q) +b + c +b c t + b ) ] q dt c Let u t +b then.0) I q) c ) c q u ) q c q+ du 0 q+ since ) 0 u q du B q + ). Also b I q) t + b +b + c ) ] q c dt ) B q +

5 TRAPEZOIDAL AND MIDPOINT RULES 5 nd substituting c v t +b gives.) I q) c ) c q v ] q dv. Combining.0) nd.) into.9) gives from.8) the second inequlity in.7). The first inequlity is obtined by tking q in the second inequlity of.7) s my be noticed from.8). Thus κ t) dt c 6 c 6 B 0 ) + ] 4c Now for the finl inequlity from.8) we obtin 5 P T f; b) sup κ t) f t b] nd so from the behviour of κ t) discussed erlier { sup κ t) mx t b] c } c c.. u ) du ] The following corollry involving the Eucliden norm is of prticulr importnce. Corollry. Let f; b] R be such tht f L b]. Then we hve the inequlity.) 5 5 c P T f; b) 5 f b ) 6 f 5 where c b. Proof. Tking p q in.7) gives P T f; b) c 6 { c B ) + which upon using the fcts tht ) B Γ ) Γ ) Γ ) nd u ) ) du 4 5 gives the stted result.) fter some simplifiction. u ) du ]} f

6 6 P. CERONE. Identities nd Inequlities for Midpoint Like Rules Let from.) nd.) the midpoint functionl M f; b) be defined by.) M f; b) : I f) I M) f) ) + b f t) dt b ) f b then the identity.) M f; b) is well known where t ) t.) φ t) b t) t φ t) f t) dt + b ] ] + b b. The following theorem concerning the clssicl midpoint functionl.) with bounds involving the L p b] norms of the second derivtive is known see 8] nd 6]). Theorem 4. Let f : b] R be such tht f is bsolutely continuous on b]. Then.4) M f; b) b ) 4 f if f L b] ; b ) + q 8q+) q f p if f L p b] p + q p > b ) 8 f if f L b] The first inequlity in.4) is the one tht is trditionlly most well known. Further from.5) nd.) define the perturbed or corrected midpoint functionl s.5) P M f; b) : I f) P I M) f) M f; b) c 6 f b) f )] where c b. The following lemm concerning P M f; b) holds. Lemm. Let f : b] R be such tht f is bsolutely continuous on b]. Then.6) P M f; b) where t ) c t.7) χ t) with c b. b t) c t χ t) f t) dt + b ] ] + b b.

7 TRAPEZOIDAL AND MIDPOINT RULES 7 Proof. From.5) nd.7) we hve on utilising.) P M f; b) M f; b) c 6 f b) f )] ] φ t) c f t) dt 6 Now from the definition of φ t) from.) gives t ) c φ t) c b t) c ] φ t) c f t) dt. t + b ] ] + b t b. nd the result s stted in.6) redily follows nd the lemm is thus proved. Theorem 5. The Lebesgue norms for the perturbed midpoint functionl P M f; b) s given by.6) re the sme s those for the perturbed trpezoid function P T f; b) given by.7). Proof. To prove the theorem it suffices to demonstrte tht.8) χ p κ p p. The properties of κ were investigted in the proof of Theorem. Now for χ t). χ ) χ b) c c nd χ t) 0 when t + b c for t b]. Further χ t) is continuous t t +b nd χ ) +b c. Also t ) > 0 t + b ] χ t) ] + b b t) < 0 t b. As mtter of fct for t ] +b χ t) κ + +b t ) nd for t +b b] χ t) κ b + +b t ). Tht is χ t) nd κ t) re symmetric bout +b 4 the midpoint of ] +b nd +b 4 the midpoint of +b b]. Thus.8) holds nd the theorem is vlid s stted. Remrk. The bound given in.) lso holds for P M f; b) given the results of Theorem Perturbed Rules from the Chebychev Functionl For g h : b] R the following T g h) is well known s the Chebychev functionl. Nmely 4.) T g h) M gh) M g) M h) where M g) b g t) dt is the integrl men.

8 8 P. CERONE The Chebychev functionl 4.) is known to stisfy number of identities including 4.) T g h) b h t) g t) M g)] dt. Further number of shrp bounds for T g h) exist under vrious ssumptions bout g nd h including see 5] for exmple) 4.) T g g)] T h h)] g h L b] T g h) A u A l T h h)] A l g t) A u t b] ) ) Au A l Bu B l B l h t) B u t b] Grüss). It will be demonstrted how 4.) my be used to obtin perturbed results which 4.) will provide n identity with which to obtin bounds. The following theorem holds. Theorem 6. Let f : b] R be such tht f is bsolutely continuous then 4.4) P T f; b) b ) 5 ] b f f ; b] f L b] b ) 4 5 B u B l ) B l f t) B u t b] where P T f; b) is the perturbed trpezoidl rule defined by.4). Proof. Let g t) t ) b t) the trpezoidl kernel nd h t) f t) then from 4.) 4.5) b ) T g t) f t)) g t) f t) dt M g) f t) dt T f; b) + c f b) f )] where M g) c. Now from 4.) 4.6) b ) T g t) f t)) nd so 4.4) nd 4.5) produce identities.5).6). ] f t) g t) + c dt κ t) f t) dt

9 TRAPEZOIDAL AND MIDPOINT RULES 9 Thus from 4.) nd 4.4) we get 4.7) b ) T g g)] b ) T f f )] g h L b] ) Au A l P T f; b) b ) T f f )] A l g t) A u Here from 4.) Specificlly ) ) Au A l Bu B l b ) B l f t) B u. b ) T h h)] h t) dt b ) M h) b ) b ) T f f )] b ) 4.8) ] ] b h M h). b f f b) f ) b ] ] nd 4.9) b ) b ) T g t) g t))] ) g g t) dt t) dt b b ) b b f f; b] t ) b t) dt ] t ) b t) dt b ) b ) 5 b ) 5 B ) B ) ]!) 5! ) ] b ) 5! 5. Utilising 4.8) nd 4.9) into the first result in 4.7) gives the first result 4.4). For the second result in 4.4) we utilise.9) giving the stted corser bound. Remrk. Even though A l c g t) 0 A u it is not worthwhile using this in the second nd third inequlity of 4.5) s this would produce corser bound thn those stted in Theorem 6. Remrk. The results of Theorem 6 s represented by 4.4) re tighter thn.0) nd.). For different proof of the shrpness of 4.4) see ]. The following bounds for the perturbed midpoint rule holds.

10 0 P. CERONE Corollry. Let f : b] R be such tht f is bsolutely continuous then ] b ) P M f; b) 5 b f f ; b] f L b] b ) 4 5 B u B l ) B l f t) B u t b]. Proof. The proof follows redily from Theorem 5 since T φ φ) χ where φ t) nd χ t) re s given by.) nd.7) respectively. Acknowledgement. The work for this pper ws done while the uthor ws on sbbticl t L Trobe University Bendigo. References ] K.E. ATKINSON An Introduction to Numericl Anlysis Wiley nd Sons Second Edition 989. ] N.S. BARNETT nd S.S. DRAGOMIR On the perturbed trpezoid formul Preprint: RGMIA Res. Rep. Coll. 4) Article ONLINE] ] N.S. BARNETT P. CERONE nd S.S. DRAGOMIR A shrp bound for the error in the corrected trpezoid rule nd pplictions submitted. 4] P. CERONE On reltionships between Ostrowski trpezoidl nd Chebychev identities nd inequlities Preprint: RGMIA Res. Rep. Coll. 4) Article ONLINE] 5] P. CERONE Perturbed rules in numericl integrtion from product brnched Peno kernels Theory of Inequlities nd Applictions C.J. Cho S.S. Drgomir nd J.-K. Kim Ed.)). 6] P. CERONE nd S. S. DRAGOMIR Midpoint type rules from n inequlities point of view Accepted for publiction in Anlytic-Computtionl Methods in Applied Mthemtics G.A. Anstssiou Ed) CRC Press New York ] P. CERONE nd S. S. DRAGOMIR Trpezoidl type rules from n inequlities point of view Accepted for publiction in Anlytic-Computtionl Methods in Applied Mthemtics G.A. Anstssiou Ed) CRC Press New York ] S.S. DRAGOMIR P. CERONE nd A. SOFO Some remrks on the midpoint rule in numericl integrtion submitted 999. ONLINE] 9] S.S. DRAGOMIR P. CERONE nd A. SOFO Some remrks on the trpezoid rule in numericl integrtion Indin J. of Pure nd Appl. Mth. in press) 999. Preprint: RGMIA Res. Rep. Coll. 5) Article 999. ONLINE] 0] D.S. MITRINOVIĆ J.E. PEČARIĆ nd A.M. FINK Clssicl nd New Inequlities in Anlysis Kluwer Acdemic Publishers 99. School of Communictions nd Informtics Victori University of Technology PO Box 448 Melbourne City MC Victori 800 Austrli E-mil ddress: pc@mtild.vu.edu.u URL: