A MEAN VALUE THEOREM FOR GENERALIZED RIEMANN DERIVATIVES. 1. Introduction Throughout this article, will denote the following functional difference:

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volme 136, Nmber 2, Febrry 200, Pges S (07) Article electroniclly pblished on November 6, 2007 A MEAN VALUE THEOREM FOR GENERALIZED RIEMANN DERIVATIVES H. FEJZIĆ, C. FREILING, AND D. RINNE (Commnicted by Dvid Preiss) Abstrct. Fnctionl differences tht led to generlized Riemnn derivtives were stdied by Ash nd Jones in (197). They gve prtil nswer s to when these differences stisfy n nlog of the Men Vle Theorem. Here we give complete clssifiction. 1. Introdction Throghot this rticle, will denote the following fnctionl difference: f(x) = i f(x + b i h), where 1,..., n nd b 1 <b 2 < <b n re constnts. We will lso sppose tht there is some positive integer d, clled the order of, sch tht (1.1) j = i b j i =0for0 j<d, (1.2) d = i b d i = d! (normlized). A generlized Riemnn derivtive is creted by f(x) lim h 0 + h d when this limit exists. It is well-known fct tht if the ordinry dth derivtive, f (d) (x), exists, then so does the generlized Riemnn derivtive nd the two mst gree. In the cse where f (d) is continos t x this cn be esily seen by d- pplictions of l Hôpitl s Rle. More generlly, it follows from vrint of Tylor s Theorem first proved by Peno (see pp of [2] for sttement nd history). Definition 1.1. We will sy tht possesses the men vle property if for ny x, nyh>0, nd ny fnction f(x) sch tht f (d 1) (x) is continos on Received by the editors Mrch 2, 2006 nd, in revised form, October 10, Mthemtics Sbject Clssifiction. Primry 26A06, 26A24. Key words nd phrses. Men vle theorem, generlized derivtives. The second thor ws spported in prt by NSF. 569 c 2007 Americn Mthemticl Society License or copyright restrictions my pply to redistribtion; see

2 570 H. FEJZIĆ, C. FREILING, AND D. RINNE [x + b 1 h, x + b n h] nd differentible on (x + b 1 h, x + b n h), we hve f(x) h d = f (d) (z) for some z (x + b 1 h, x + b n h). In [1] Ash nd Jones gve prtil clssifiction for the men vle property. First of ll, it mst be tht d n 1 since if (1) holds for ll 0 j n 1, then it holds for ll j, rendering (2) impossible. The difference, (n 1) d is clled the excess of. Inthecsewhered = n 1, i.e. the excess is zero, the thors of [1] show tht the vles of i re niqely determined from the vles of b i. The men vle property holds in this cse, nd this is well-known nmericl nlysis fct (see e.g. [4]). If d<n 1, then cn be expressed ntrlly s liner combintion of differences whose excess is zero. These re the niqe excess zero differences tht come from the sets {b 1,...,b d+1 }, {b 2,...,b d+2 },...,{b n d,...,b n }. If this trns ot to be convex combintion, then stisfies the men vle property, nd this is esy to show from the well-known Drbox property (or Intermedite Vle Property) of derivtives. The thors then showed tht the converse of this fct holds in the cses d =1, 2,n 2, nd sked whether the converse holds in generl. They lso gve specific order-three difference for which the men vle property ws n open qestion: 5 f(x)+ 17 f(x + h) 26 f(x +2h)+26 f(x +3h) 17 f(x +4h)+5 f(x +5h). This difference cold not be decided sing their method since the corresponding liner combintion hd negtive coefficient. In this note we will give condition tht is both necessry nd sfficient for ll d>0. Using this clssifiction, we will qickly determine tht the prticlr difference given bove does not stisfy the men vle property. We lso exmine slight vrition of this difference tht retins its negtive coefficient nd yet it does stisfy the men vle property. This shows tht the condition in [1] is not necessry for the cse d =3,n =6. As with the proof in [1], or pproch will be bsed on the Drbox property of derivtives. As for the nmericl nlysis fct stted bove, rther thn sing it s lemm, we will derive it s corollry. Or bsic techniqe is to first represent the difference s n integrl nd then pply severl integrtions by prts. All of or integrls will be Denjoy integrls, nd we smmrize some of their relevnt properties in the next section. 2. Denjoy integrls Althogh it wold be possible to crry ot ll of the rgments in this pper sing the Riemnn integrl, the proofs re simplified by mking se of the Denjoy integrl. The tility of the Denjoy integrl comes from the fct tht it is powerfl enogh to integrte ny derivtive. More generlly, it gives s the following integrtion by prts forml. Theorem 2.1 (see e.g. [3], Theorem 12.6). Let F be continos fnction on [, b] tht is differentible on (, b) with F (x) =f(x), ndletg(x) be Lebesge integrble License or copyright restrictions my pply to redistribtion; see

3 A MEAN VALUE THEOREM FOR GENERALIZED RIEMANN DERIVATIVES 571 on [, b] with G(x) = x g(t)dt + c. ThenFg is Lebesge integrble, fg is Denjoy integrble nd Fgdx = F (x)g(x)] b fgdx. Setting g(x) =0ndG(x) = 1 in the bove theorem reslts in the Fndmentl Theorem of Clcls, f(x)dx = F (x)] b for ny differentible fnction F (x). Other expected properties of integrtion tht hold for the Lebesge integrl contine to hold for the Denjoy integrl. For exmple, the Denjoy integrl is both liner nd monotone: Theorem 2.2 (see e.g. [3], Theorem 7.4). If f(x) nd g(x) re two Denjoy integrble fnctions on [, b] nd k is constnt, then kf(x) +g(x) is lso Denjoy integrble on [, b] nd kf(x)+g(x)dx = k Frthermore, if f(x) g(x), then f(x)dx f(x)dx + g(x)dx. g(x)dx. Also, if f is Denjoy integrble on n intervl, then it is Denjoy integrble on every sbintervl (see e.g. [3], Theorem 7.4). Finlly, for mny fnctions, the two notions of integrtion gree: Theorem 2.3 (see e.g. [3], Theorem 7.7). If f(x) is Lebesge integrble on [, b], then it is lso Denjoy integrble on [, b] nd the two integrls gree. If f(x) is Denjoy integrble nd f is bonded from below, then f(x) is Lebesge integrble. 3. Differences s integrls To represent the difference s n integrl, we will se the Dirc delt fnction δ(x). Althogh techniclly not fnction, δ(x) cts jst like fnction when sed inside n integrl. There re slight vritions of δ(x) in the litertre, depending on the exct property needed. For s, the relevnt property is tht for ny fnction f(t) continos from bove t 0, we hve (3.1) 0 δ(t)f(t)dt = f(0). The ntiderivtive of δ(x) is the Heviside nit step fnction, which is n ctl fnction, x { 0 if x 0, H(x) = δ(t)dt = 1 if x>0. Using H(x) we my write (3.1) more generlly s (3.2) δ(t)f(t)dt = f(0)[h(v) H()]. License or copyright restrictions my pply to redistribtion; see

4 572 H. FEJZIĆ, C. FREILING, AND D. RINNE One of the fnction-like properties tht δ(x) enjoys is integrtion by prts. Ths, if f(x) is differentible on (, v) nd continos on [, v], then δ(t)f(t)dt = f(t)h(t)] v f (t)h(t)dt, fct tht follows esily from (3.2). Given fnctionl difference we define n ssocited distribtion n 1 (3.3) D(t) = i δ(t b i h)+ n δ(b n h t). The reson for choosing different δ-fnction for b n h is becse we wnt to work with fnction f tht is continos on the closed intervl [x+b 1 h, x+b n h], so tht it is continos from bove t x + b i h s long s i<n, bt it is continos from below t x + b n h. Then, ssming f is sch fnction, we my se (3.2) to write n h (3.4) f(x) = i f(x + b i h)= D(t)f(t + x)dt, which gives s the desired integrl for. We wish to integrte (3.4) severl times, so we let D [j] (x)denotethejth-order ntiderivtive of D(x) defined indctively by D [0] = D nd D [k+1] (x) = x D[k] (t)dt. The next lemm tells s how to qickly compte the vles of D [j] t the endpoints b 1 h nd b n h. Lemm 3.1. If is normlized difference of order d>0, thend [j] (b 1 h)=0 for ll j, D [j] (b n h)=0for j =0,...,d nd D [d+1] (b n h)=( 1) d h d. Proof. Let = b 1 h nd v = b n h.thevlestre immedite from the definition, since D is identiclly zero on (,). At the other endpoint, for j =0thissys D(v) = 0, which follows by the definition of D nd the fct tht d > 0. We proceed by indction. Assme the lemm holds for j =0, 1,...,k d. Then sing integrtion by prts, D [k+1] (v) = D [k] (t)dt ] v ] v = D [k] (t)t] v D [k 1] (t) t2 + +( 1) k 1 D [1] (t) tk 2 k! + ( 1) k D(t) tk k! dt. By the indction hypothesis, ll of the terms on the right re zero except possibly the lst one, which is ( 1) k D(t) tk (b i h) k dt = ( 1)k i k! k! = ( 1) k k h k. k! Since hs order d, thisiszerofork<d. Since is normlized, this is ( 1) d h d when k = d. b 1 h License or copyright restrictions my pply to redistribtion; see

5 A MEAN VALUE THEOREM FOR GENERALIZED RIEMANN DERIVATIVES 573 Lemm 3.1 llows s to do mny integrtion by prts very esily, s in the next lemm. Lemm 3.2. Let be normlized difference of order d>0 nd j d. Iff (j 1) is continos on [x+, x+v] =[x+b 1 h, x+b n h] nd differentible on (x+, x+v), then f(x) =( 1) j D [j] (t)f (j) (t + x)dt. Proof. Using integrtion by prts, we compte f(x) = D(t)f(t + x)dt = D [1] (t)f(t + x)] v D [2] (t)f (t + x)] v + +( 1) j 1 D [j] (t)f (j 1) (t + x)] v +( 1)j D [j] (t)f (j) (t + x)dt. Bt by Lemm 3.1, ll of the terms on the right except the lst one re zero. We now smmrize or reslts so fr. Lemm 3.3. If is normlized difference of order d>0nd f (d 1) is continos on [x +, x + v] =[x + b 1 h, x + b n h] nd differentible on (x +, x + v), then h d = ( 1) d D [d] (t)dt, f(x) = ( 1) d D [d] (t)f (d) (t + x)dt. 4. Min reslts Theorem 4.1. If is normlized difference of order d>0. Then the generlized Riemnn derivtive f(x) lim h 0 + h d possesses the men vle property if nd only if D [d] does not chnge sign. Proof. Sppose first tht D [d] does not chnge sign nd let f(x) be fnction sch tht f (d 1) is continos on [x +, x + v] =[x + b 1 h, x + b n h] nd differentible on (x +, x + v). From Lemm 3.3 we hve f(x) =( 1) d D [d] (t)f (d) (t + x)dt. Let m nd M be the infimm nd spremm, respectively, of f (d) on (x +, x + v). Since D [d] does not chnge sign, ssme ( 1) d D [d] (t) 0; in the other cse, the following ineqlities will be reversed. Then by monotonicity, m( 1) d D [d] (t)dt f(x) M( 1) d D [d] (t)dt License or copyright restrictions my pply to redistribtion; see

6 574 H. FEJZIĆ, C. FREILING, AND D. RINNE or, sing Lemm 3.3, we my write m f(x) h d M. If both of these ineqlities re strict, then f (d) ttins vles on ech side of f(x). The theorem follows since f (d) possesses the Drbox property. If one of h d the ineqlities is not strict, sy if m = f(x),thenmis finite. Using Theorem 2.3 h d lmost everywhere on (x +, x + v) wehveeitherf (d) = m or D [d] (t) =0. By Lemm 3.3, D [d] (t) cnnot be zero lmost everywhere, so the theorem follows. Now sppose tht D [d] hs t lest one sign chnge. We set x =0ndh =1. We will find fnction f(x) thtisd times differentible, bt for which the men vle property fils. We my ssme tht D [d] (t) is positive on n open intervl I (b 1,b n ), nd negtive on n open intervl J (b 1,b n ). If d is odd, we pick g to be continos fnction tht is negtive on I nd 0 elsewhere nd set f (d) = g. Using Lemm 3.3, we get f(0) = n b 1 D [d] (t)f (d) (t)dt > 0 while f (d) 0. If d is even, we pick g to be continos fnction tht is negtive on J nd 0 elsewhere nd set f (d) = g. Ths f(0) = n b 1 D [d] (t)f (d) (t)dt > 0 while f (d) 0. In either cse the men vle property fils. Althogh the nmber of sign chnges in D [d] is certinly comptble, it my be tedios to determine by hnd. The next theorem gives qick bt rogh estimte. Theorem 4.2. Let be normlized difference of degree d. Letc k be the nmber of sign chnges in D [k], 1 k d. Then the seqence c 1,c 2,...,c d is strictly decresing, c 1 is the nmber of sign chnges in the prtil sms of n i =0, c k n k 1. Proof. For ech 1 k d we hve from Lemm 3.1 tht D [k] (b 1 h)=d [k] (b n h)=0. Since D is zero otside the intervl (b 1 h, b n h) ll sign chnges mst occr in this intervl. Between ny two sign chnges in D [k] (x) = x D[k 1] (t)dt there mst be sign chnge in D [k 1].TostisfythereqirementsD [k] (b 1 h)=d [k] (b n h)=0, there mst lso be sign chnge in D [k 1] before the first nd fter the lst sign chnge in D [k]. Therefore, c k 1 c k + 1. The second prt follows directly from the definition D [1] = n 1 ih(t b i h) n H(b n h t). The third prt follows directly from the first two prts, observing tht the prtil sms of n i =0cnhve t most n 2 sign chnges, nd so c 1 n 2. If d = n 1, then Theorem 4.2 sys tht c d = 0. Combining this with Theorem 4.1 we get the following well-known reslt. Corollry 4.3. If d = n 1, then possesses the men vle property. 5. Exmples Exmple 5.1. In [1] the thors sked whether the following degree-difference three possesses the men vle property. For i =0, 1, 2, let i f(x) = f(x + ih)+3f(x +(i +1)h) 3f(x +(i +2)h)+f(x +(i +3)h) License or copyright restrictions my pply to redistribtion; see

7 A MEAN VALUE THEOREM FOR GENERALIZED RIEMANN DERIVATIVES 575 Figre 1. The first three ntiderivtives of D for Exmples 5.1 nd 5.2. The men vle property fils in the first cse nd scceeds in the second. nd define f(x) = 5 0f(x) 1 4 1f(x)+ 5 2f(x) = 5 f(x)+17 f(x + h) 26 f(x +2h) f(x +3h) f(x +4h)+5 f(x +5h). This qestion ws posed in [1] since writing f(x) s liner combintion of i involves negtive coefficients. In this exmple D [1] tkes on the vles 5, 12, 14, 12, 5 on the intervls (i, i +1), 0 i 4. It is then firly esy to determine tht License or copyright restrictions my pply to redistribtion; see

8 576 H. FEJZIĆ, C. FREILING, AND D. RINNE D [3] hs two sign chnges, so Theorem 4.1 nswers the qestion in the negtive. Figre 1 shows the fnctions D [1], D [2],ndD [3] with h =1. Noticehowthegrph of D [3] climbs slightly bove the x-xis. Exmple 5.2. Chnging the coefficients of the i slightly cn chnge the nswer to yes. Consider f(x) = 5 0f(x) 1 1f(x)+ 4 2f(x) = 5 f(x)+16 f(x + h) 22 f(x +2h) f(x +3h) f(x +4h)+4 f(x +5h). Now D [1] tkes on the vles 5, 11, 11, 9, 4 on the intervls (i, i+1), 0 i 4. It is then esy to determine tht D [3] does not chnge sign, so Theorem 4.1 nswers the qestion in the positive. Agin, we grph the fnctions D [1], D [2],ndD [3] with h = 1 in Figre 1, bt this time the grph of D [3] stys below the x-xis. Exmple 5.3. This exmple illstrtes the se of Theorem 4.2. It is esy to check tht the difference f(x) = 4 15 ( f(x)+f(x + h)+f(x h) f(x + 7 h) f(x +4h)+f(x +5h)) 2 hs degree three nd is normlized. Also, the seqence of prtil sms of i is 4 15, 0, 4 15, 0, 4 15, 0, which hs two sign chnges, so c 1 =2. Btc 1 >c 2 >c 3,so c 3 = 0, nd possesses the men vle property by Theorem 4.1. References 1. J.M. Ash nd R.L. Jones, Men Vle Theorems for Generlized Riemnn Derivtives, Proc. Amer. Mth. Soc., vol. 101, no. 2, October, 197. MR (i:26011) 2. J.M. Ash, A. E. Gtto, nd S. Vági, A mltidimensionl Tylor s Theorem with miniml hypotheses, Colloq. Mth., (1990), MR (92b:26001) 3. Rssell A. Gordon, The Integrls of Lebesge, Denjoy, Perron, nd Henstock, Grdte Stdies in Mthemtics, vol. 4, Americn Mthemticl Society, MR12751 (95m:26010) 4. E. Iscson nd H. B. Keller, Anlysis of Nmericl Methods, Wiley, New York, MR (34:924) Deprtment of Mthemtics, Cliforni Stte University, Sn Bernrdino, Cliforni E-mil ddress: hfejzic@cssb.ed Deprtment of Mthemtics, Cliforni Stte University, Sn Bernrdino, Cliforni E-mil ddress: cfreilin@cssb.ed Deprtment of Mthemtics, Cliforni Stte University, Sn Bernrdino, Cliforni E-mil ddress: drinne@cssb.ed License or copyright restrictions my pply to redistribtion; see