Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsuku Tsuku Irki 5-857 Jp tski@mth.tsuku.c.jp Keywords : covergece rte; Riem sum; Riem itegrl; trpezoidl sum Astrct We represet the covergece rtes of the Riem sums d the trpezoidl sums with respect to regulr divisios d optiml divisios of ouded closed itervl to the Riem itegrls s some limits of their expded error terms. Itroductio The Riem sums d the trpezoidl sums of fuctios defied o ouded closed itervl re well kow s pproximte sums of the Riem itegrls of the fuctios. I this pper the uthor represets the covergece rtes of the Riem sums d the trpezoidl sums s some limits of their expded error terms. Let [, ] e ouded closed itervl. We tke -divisio of [, ] defied y : = s s s s =. We deote y D the divisio of [, ] defied y s i = + i( / d cll it the regulr -divisio. For fuctio f defied o [, ] d s i ξ i s i we defie the Riem sum R(f;, ξ i y R(f;, ξ i = (s i s i f(ξ i.
The width d( of is defied s d( = mx{s i s i i }. The Riem itegrl of f is defied s f(xdx = lim R(f;, ξ i d( d textooks o clculus usully show tht this limit exists for cotiuous fuctio f. I this pper we cosider some limits of expded error terms like f(xdx R(f;, ξ i, f(xdx R(f;, ξ i s. Chui [] otied such limit of expded error term. Theorem. (Chui If f is twice differetile d f is oud d lmost everywhere cotiuous o [, ], the { ( f(xdx R f; D, } (s i + s i = lim ( 4 f (xdx = ( (f ( f (. 4 I [] the ove theorem is formulted for the itervl [, ]. We cosider ot oly regulr divisios D ut lso optiml divisios for lower Riem sums d trpezoidl sums, so we expli out optiml divisios. We tke cotiuous fuctio f defied o [, ]. For y divisio of [, ] we tke s i ξ i s i which stisfies f(ξ i = mi f d defie [s i,s i ] the lower Riem sum R(f;, mi = R(f;, ξ i. The set of ll -divisios of [, ] is compct d R(f;, mi is cotiuous, so there exists -divisio # t which the ove fuctio ttis its mximum. This -divisio # is optiml for the lower Riem sum R(f;, mi. It my ot e uique, ut the sum R(f; #, mi is uique. Thus we c cosider R(f; #, mi. Oe of the mi theorems of this pper is s follows:
Theorem. If f is fuctio of clss C defied o [, ], the { } lim f(xdx R(f; #, mi = ( f (x dx /. The trpezoidl sum T (f; of f is defied s T (f; = (s i s i (f(s i + f(s i. We c oti the limit of the expded error term of the trpezoidl sum s follows: Theorem. If f is twice differetile d f is oud d lmost everywhere cotiuous o [, ], the { } f(xdx T (f; D lim ( = f ( (xdx = (f ( f (. We cosider optiml divisio for the trpezoidl sum. f(xdx T (f; is cotiuous, so there exists -divisio t# t which the ove fuctio ttis its miimum. This -divisio t# is optiml for the trpezoidl sum T (f;. It my ot e uique, ut f(xdx T (f; t# is uique. Thus we c cosider it. Theorem.4 If f is fuctio of clss C defied o [, ] which stisfies f or f, the lim f(xdx T (f; t# = ( f (x dx /. I the cse where f(x = x o [, ], if we tke y divisio of [, ] ll of whose poits re symmetric t, the stisfies however its trpezoidl sum o [, ] my ot e close to we oly cosider the cse where f or f. f(xdx T (f; =, f(xdx. So
Trpezoidl sums for regulr divisios I this sectio we tret regulr divisios of itervls d prove Theorem.. We c prove it i wy similr to tht of Chui []. First we ssume tht [, ] = [, ]. For ech positive iteger we defie fuctio v defied y the followig grph. This is fuctio of ouded vritio which stisfies < v (t, v ( = v ( =. The Riem-Stieltjes itegrl of f with respect to v is give y ( k ( f(tdv (t = f f(tdt. k= Thus we get f(tdv (t = ( k f f(tdt k= = T (f; D + (f( f( f(tdt. Sice f is Riem itegrle d v is of ouded vritio, we hve f(tdv (t = [f(tv (t] v (tdf(t = (f( f( v (tdf(t 4
d We defie T (f; D which stisfies f(tdt = u (x = u ( = u ( k v (tdf(t = x v (tdt, = (k =,..., f (tv (tdt. d is periodic with period /. From the ove results we hve {T (f; D = = = = = k= / k= k= For x we hve u (tf (tdt = / u ( x } f(tdt ( u t + k ( u (tf u ( x k/ k= (k / ( f t + k ( x + k f k= ( x x/ u = f ( x + k u (tf (tdt t + k dt dx dx. ( t dt = x x. I prticulr u (/ = /8. We defie fuctio w y w(x = (x x ( x. dt The we get u ( x = w(x d 5 w(xdx =.
Usig w we oti {T (f; D } f(tdt = = = ( x ( x + k u f dx k= w(x ( x + k f dx k= ( w(xr f ; D, x + k dx. Therefore we oti { } lim T (f; D f(tdt ( = lim w(xr f ; D, x + k ( = = = w(x lim R w(x f (tdtdx f ; D, x + k f (tdt = (f ( f (, which completes the proof of Theorem. i the cse where [, ] = [, ]. We c get the geerl sttemet of the theorem y the vrile chge x = + ( t. Lower Riem sums for optiml divisios We prove Theorem. i this sectio. We eed the followig lemm otied y Gleso [] d Lemm. i order to cosider the lower Riem sums for optiml divisios. Lemm. (Gleso Let ϕ(t e oegtive cotiuous fuctio defied o [, ]. For y positive iteger there exists divisio of [, ]: such tht ll of = s < s < < s < s = (s i s i mx ϕ(t ( i [s i,s i ] 6 dx dx
re equl to ech other. We deote y J the equl vlue. The we oti lim J = ϕ(tdt. Lemm. For y fuctio of clss C defied o [, ] we hve f(xdx ( mi f(x [,] ( mx f (x. [,] The estimte i this lemm is well kow, so we omit its proof. Proof of Theorem. We first prove the followig iequlity. lim sup f(xdx R(f; #, mi ( f (x dx /. We pply Lemm. to the fuctio f (x / d oti divisio e : s, s,..., s of [, ] such tht ll of (s i s i mx [s i,s i ] f (x / ( i re equl to ech other. We deote y J the equl vlue. The we oti lim J = f (x / dx. By the estimte of Lemm. we hve f(xdx R(f; e, mi ( si = f(xdx (s i s i mi f(x s i [s i,s i ] Thus we oti lim sup lim sup lim sup (s i s i mx f (x = [s i,s i ] J. f(xdx R(f; #, mi f(xdx R(f; e, mi J = lim (J = ( f (x dx /. 7
I order to complete the proof of Theorem. we hve to estimte ( f(xdx R(f; #, mi from elow. We prepre the followig lemms for this purpose. Lemm. Uder the ssumptio of Theorem., we defie ω y ω (r = sup{ f (x f (y x, y [, ], x y r}. The ω is cotiuous fuctio defied o [, ] which is mootoe icresig d stisfies lim ω (r =. If f (x i suitervl [p, q] of r [, ], the for y ξ i [p, q] we hve the followig iequlity. q f(xdx (q p mi f(x p [p,q] (q p f (ξ ω (q p(q p. By the use of the me vlue theorem we c prove this lemm. Lemm.4 Uder the ssumptio of Theorem., for y ɛ > there exists positive iteger N such tht for y N d y -divisio of [, ] we hve the followig iequlity. f(xdx R(f;, mi ( f (x dx / ɛ. Proof For the proof of the lemm, we show the followig sttemet: For y δ > there exists positive iteger r such tht for y -divisio of [, ] we hve the followig iequlity. / ( + r / f(xdx R(f;, mi / f (x / dx δ(. Sice the fuctio x x / is uiformly cotiuous o [,, there exists δ > such tht for y x d y i [, if x y < δ the x / y / < δ. 8
We tke suitervl [p, q] of [, ] d suppose tht f (x i [p, q]. We c use Lemm. d get q f(xdx (q p mi f(x p [p,q] (q p f (ξ ω (q p for y ξ i [p, q]. Becuse of cotiuity of ω d ω ( =, there exists η > such tht z η implies ω (z/ δ. Thus q p η implies ω (q p/ δ. Therefore we hve tht is, ( q p ( q p f(xdx (q p mi [p,q] f(x q p / / f(xdx (q p mi f(x [p,q] f (ξ / / δ, f (ξ / (q p / δ(q p. Sice f is uiformly cotiuous o [, ], for the ove δ > there exists β > such tht x y < β implies f (x f (y δ. We deote y Z(f the zero set of f : Z(f = {x [, ] f (x = } d defie the β-eighorhood Z(f β of Z(f y Z(f β = {y [, ] x Z(f x y < β}. The for y y i Z(f β we hve f (y δ d f is ot equl to o the complemet of Z(f β. By the defiitio of Z(f β we c see tht Z(f β is disjoit uio of fiitely my itervls. We deote y r the umer of ll edpoits of the itervls costructig Z(f β. For η > otied ove we tke positive iteger r stisfyig ( /r η d set r = r + r. For y -divisio of [, ] we c dd t most r poits to such tht the width of ech suitervl is less th or equl to η. Moreover we dd ll the edpoits of the itervls costructig Z(f β d deote the ew divisio y : s =, s,..., s t =. 9
By the defiitio of we hve t + r d s i s i η. Ech itervl [s i, s i ] stisfies [s i, s i ] Z(f β or [s i, s i ] [, ] Z(f β. I oth cses, ccordig to the first me vlue theorem for itegrtio we c tke s i i [s i, s i ] stisfyig si s i f (x / dx = f (s i / (s i s i. I the cse where [s i, s i ] Z(f β f (s i / (s / i s i δ(s / i s i si / f(xdx (s i s i mi f(x + δ(s i s s i [s i,s i ] i holds. I the cse where [s i, s i ] [, ] Z(f β, f is ot equl to i [s i, s i ], thus f (s i / (s / i s i si / f(xdx (s i s i mi f(x + δ(s i s s i [s i,s i ] i. Filly the sme iequlity holds i oth cses. We dd the ove iequlities for i =,..., t d get f (x / dx = / ( t si t / f (s i / (s i s i s i f(xdx (s i s i mi [s i,s i ] f(x / + δ(. ( We pply the Cuchy-Schwrz iequlity to the first term of ( d get ( t si / f(xdx (s i s i mi f(x s i [s i,s i ] ( t ( t / si f(xdx (s i s i mi f(x s i [s i,s i ] ( / = t / f(xdx R(f;, mi. /
From these we hve / The iequlity ( / f (x / dx t / f(xdx R(f;, mi + δ(. f(xdx R(f;, mi the estimte otied ove d t + r imply / f(xdx R(f;, mi, ( / f (x / dx δ( ( + r / f(xdx R(f;, mi. Usig the result otied ove, we prove Lemm.4. Sice the fuctio x x is cotiuous, for y ɛ > there exists ξ > such tht if the we hve f (x / dx x ξ / ( f (x / dx x ɛ. So we tke δ > which stisfies δ( ξ. We c pply the result otied ove d get positive iteger r such tht for y -divisio of [, ] ξ δ( / which implies ( / f (x / dx ( + r / f(xdx R(f;, mi, ɛ ( f (x dx / ( + r f(xdx R(f;, mi. We c sustitute the optiml divisio # for i the ove iequlity d get ( + r f(xdx R(f; #, mi f (x / dx ɛ.
Sice ( lim f(xdx R(f; #, mi we c choose positive iteger N such tht for N holds. Thus for N we hve =, r f(xdx R(f; #, mi ɛ f(xdx R(f; #, mi ( + r f(xdx R(f; #, mi ɛ ( f (x dx / ɛ. Therefore for y -divisio of [, ] we hve f(xdx R(f;, mi which completes the proof of Lemm.4. Proof of Theorem. ( lim sup d y Lemm.4 we c see tht f(xdx R(f; #, mi ( f (x dx / ɛ, We hve lredy proved f(xdx R(f; #, mi f (x / dx ( f (x dx / lim if ( f(xdx R(f; #, mi. Therefore the limit of the left-hd side stted i the theorem exists d the equtio holds.
4 Trpezoidl sums for optiml divisios We prove Theorem.4 i this sectio. We eed Lemms. d 4. i order to cosider the trpezoidl sums for optiml divisios. Lemm 4. For y fuctio f of clss C defied o [, ] we hve f(xdx (f( + f(( ( mx f (x. [,] By the use of the me vlue theorem we c prove this lemm. Proof of Theorem.4 We first prove the followig iequlity for y fuctio f of clss C defied o [, ]. lim sup f(xdx T (f; t# ( f (x dx /. We pply Lemm. to the fuctio f (x / d oti divisio te : s, s,..., s of [, ] such tht ll of (s i s i mx [s i,s i ] f (x / ( i re equl to ech other. We deote y J the equl vlue. The we oti lim J = f (x / dx. By the estimte of Lemm 4. we hve f(xdx T (f; te si f(xdx s i (f(s i + f(s i (s i s i (s i s i mx{ f (x s i x s i } = J.
Thus we oti lim sup lim sup lim sup f(xdx T (f; t# f(xdx T (f; te J = lim (J = ( f (x dx /. I order to complete the proof of Theorem.4 we hve to estimte f(xdx T (f; t# from elow. We prepre the followig lemms for this purpose. Lemm 4. Let f e fuctio of clss C defied o [, ]. We defie ω y ω (r = sup{ f (x f (y x, y [, ], x y r}. The ω is cotiuous fuctio defied o [, ] which is mootoe icresig d stisfies lim ω (r =. If f (x or f (x i suitervl [p, q] of [, ], the for y ξ i [p, q] we hve the followig r iequlity. q f(xdx p (f(p + f(q(q p (q p f (ξ ω (q p(q p. By the use of the me vlue theorem we c prove this lemm. Lemm 4. Uder the ssumptio of Theorem.4, for y ɛ > there exists positive iteger N such tht for y N d y -divisio of [, ] we hve the followig iequlity. f(xdx T (f; ( f (x dx / ɛ. 4
Proof For the proof of the lemm, we show the followig sttemet: For y δ > there exists positive iteger r such tht for y -divisio of [, ] we hve the followig iequlity. ( + r / f(xdx T (f; / f (x / dx δ(. / Sice the fuctio x x / is uiformly cotiuous o [,, there exists δ > such tht for y x d y i [, if x y < δ the x / y / δ. We tke suitervl [p, q] of [, ]. We c use Lemm 4. d get q f(xdx p (f(p + f(q(q p (q p f (ξ ω (q p for y ξ i [p, q]. Becuse of cotiuity of ω d ω ( =, there exists η > such tht z η implies ω (z/ δ. Thus q p η implies ω (q p/ δ. Therefore we hve tht is, q f(xdx / p (f(p + f(q(q p q p f (ξ / / δ, q f(xdx / p (f(p + f(q(q p f (ξ / (q p / δ(q p. For η > otied ove we tke positive iteger r stisfyig ( /r η. For y -divisio of [, ] we c dd t most r poits to such tht the width of ech suitervl is less th or equl to η. We deote the ew divisio y : s =, s,..., s t =. we hve t + r. Accordig to the first me vlue theorem for itegrtio we c tke s i i [s i, s i ] stisfyig si s i f (x / dx = f (s i / (s i s i. 5
By the estimte otied ove we get f (s i / (s / i s i si f(xdx s i (f(s i + f(s i (s i s i / + δ(s i s i. We dd the ove iequlities for i =,..., t d get / t si t f (x / dx = f (s i / (s / i s i f(xdx / s i (f(s i + f(s i (s i s i + δ(. ( We pply the Hölder iequlity to the first term of ( d get t si f(xdx / s i (f(s i + f(s i (s i s i ( t t / si f(xdx s i (f(s i + f(s i (s i s i / = t / f(xdx T (f;. From these we hve / The iequlity f (x / dx t / f(xdx T (f; / f(xdx T (f; f(xdx T (f;, the estimte otied ove d t + r imply / / + δ(. f (x / dx δ( ( + r / f(xdx T (f; /. 6
Usig the result otied ove, we prove Lemm 4.. Sice the fuctio x x is cotiuous, for y ɛ > there exists ξ > such tht if the we hve ξ f (x / dx x / ɛ ( f (x dx / x. So we tke δ > which stisfies δ( ξ. We c pply the result otied ove d get positive iteger r such tht for y -divisio of [, ] ξ δ( / which implies ɛ ( f (x / dx ( + r / f(xdx T (f; f (x / dx ( + r We c sustitute the optiml divisio t# d get ( + r f(xdx T (f; t# Sice lim sup (r + r f(xdx T (f;. / for i the ove iequlity f (x / dx ɛ. f(xdx T (f; t# =, we c choose positive iteger N such tht for N (r + r f(xdx T (f; t# ɛ holds. Thus for N we hve f(xdx T (f; t# ( + r f(xdx T (f; t# ɛ ( f (x dx / ɛ 7,
Therefore for y -divisio of [, ] we hve f(xdx T (f; which completes the proof of Lemm 4.. Proof of Theorem.4 lim sup f(xdx T (f; t# ( f (x dx / ɛ, We c comie the iequlity f(xdx T (f; t# ( f (x dx /. d Lemm 4. d see the ssertio of the theorem. Refereces [] Chrles K. Chui, Cocerig rtes of covergece of Riem sums, Jourl of Approximtio Theory 4 (97 79 87. [] Adrew M. Gleso, A curvture formul, Amer. J. Mth. (979 86 9. 8