Proceedings o the Interntionl Conerence on Theor nd Applictions o Mthemtics nd Inormtics ICTAMI 2003, Al Iuli CARACTERIZATIONS OF TE FUNCTIONS WIT BOUNDED VARIATION Dniel Lesnic Astrct. The present stud concerns the clss o the unctions with ounded vritions nd its reltions with other clsses o well-known chrcterized unctions. Deinition 1.1. A unction : [,] R is clled with ounded vrition on [,] i there is M > 0 such tht or n prtition = = o < 1 < < n = o the intervl [,] we hve: V = sup { V division o [,]} is clled the totl vrition o the unction on the intervl [,]. Remrks. i the concept o the unction with ounded vrition hs sense onl on compct intervls; ii the deinition cn esil e etended when the unction tkes vlues in metric spce. For etter understnding o the clss o unctions with ounded vrition one inds pproprite to list series o well-known results, see or instnce [1], the proo eing reserved onl or no clssicl results. 1.PROPERTIES OF TE FUNCTIONS WIT BOUNDED VARIATION Proposition 2.1. A unction : [,] R is constnt i nd onl i is unction with ounded vrition nd V = 0. Proposition 2.2. A unction : [,] R is monotonic i nd onl i is unction with ounded vrition nd V =. 47
Dniel Lesnic - Chrcteriztions o the unctions with ounded vrition Theorem 2.1. The set o the unctions with ounded vrition on given compct intervl orms n lger which is not closed. Furthermore, the set o unctions with ounded vrition nd with nonzero vlues on given compct intervl orms commuttive ield. Proposition 2.3. Let : [,] R e unction with ounded vrition on [,] nd let V: [,] R e unction deined s: V V = or n [,]. Then V nd V re incresing unctions nd stis the ollowing inequlit: V 2 1 d V d V d V d 2 1 d Proo. Let, [,] such tht >. Then V V = V 0 nd V V V + = + 0. ence, V nd V re incresing unctions. Appling the Ceîşev s inequlit or the unctions V nd V results in: 1 V V d V d V d d which is in ct the requested inequlit. Theorem 2.2. Jordn A unction is with ounded vrition i nd onl i it cn e represented s the dierence o two incresing decresing unctions. Remrks. i the two monotonic unctions cn e tken positive dding suicientl lrge constnt nd continuous i in ddition the initil unction is continuous; ii the decomposition is not unique. Corollr 2.1. I unction is with ounded vrition is not continuous it hs onl discontinuities o the irst kind. Corollr 2.2. Frod The set o the discontinuit points o unction with ounded vrition is t most countle. 48
Dniel Lesnic - Chrcteriztions o the unctions with ounded vrition Corollr 2.3. A unction with ounded vrition is Riemnn integrle. The reciprocl is lse nd contreemple is given the unction: sinπ/ i 0 < 2 : [0,2] R, = 0 i = 0 which is Riemnn integrle on [0,2] eing continuous, ut is not with ounded vrition on [0,2]. Corollr 2.4. Leesque A unction with ounded vrition is lmost everwhere derivle. Oservtion. Bsed on corollr 2.4., clss o continuous unctions which re not even locll with ounded vrition is the clss o continuous unctions nowhere derivle. Proposition 2.4. A unction with with ounded vrition is ounded. The reciprocl is lse nd contreemple is given the Dirichlet unction, nmel 0, i [, ] Q : [,] R, = 1, i [,] R - Q which is ounded, ut is not with ounded vrition rom corollr 2.2. Oservtion. The unction : [0,1] R 1/, i 0,1] = 0, i = 0 is not with ounded vrition eing unounded ut this emple shows tht monotonic unction on noncompct intervl, nmel g : 0,1] R, g=1/ could e structurll r w rom unction with ounded vrition. Proposition 2.5. Let,g : [,] R nd K > 0, c 1 e with the propert tht K g g c or n, [,]. I g is with ounded vrition then is with ounded vrition. Proo. From proposition 2.4. there is M > 0such tht g < M or n [,]. Then or n division = = o < 1 < < n = o the intervl [,] we hve: n 1 V = i= 0 n 1 i1 i i= 0 c i+ 1 g i K g 49
Dniel Lesnic - Chrcteriztions o the unctions with ounded vrition n 1 K 2M c-1 K g i+ 1 g i K 2M c-1 g. i= 0 ence, is with ounded vrition on [, ]. Corollr 2.5. A Lipschitz unction is with ounded vrition. Corollr 2.6. Let Φ : [,] R e Riemnn integrle unction. Then the unction : [,] R deined s: is with ounded vrition. V = Φ tdt or n [,] Corollr 2.7. A derivle unction with ounded derivtive on [, ] is with ounded vrition on [, ]. Theorem 2.3. A derivle unction : [,] R with integrle derivtive is with ounded vrition nd V = ' d. 2.RELATIONS WIT OTER CLASSES OF FUNCTIONS g Proposition 3.1. Let [,] [c,d] R e unctions with the properties tht is with ounded vrition on [c, d] nd g is monotonic on [, ]. Then the composition o g is with ounded vrition on [, ]. Proo. Suppose g is n incresing unction. Since is with ounded vrition, rom Jordn s theorem there re Φ nd Ψ incresing unctions such tht = Φ Ψ. Then o g = Φ o g - Ψ o g nd Φ o g nd Ψ o g re incresing unctions. Finll, rom Jordn s theorem, results tht o g is with ounded vrition on [, ]. Proposition 3.2. Let : [,] R e unction with Drou propert such tht is with ounded vrition. Then is continuous. Proo. It is es to oserve tht, since hs Drou propert, then hs this propert s well. First we shll prove tht is continuous. Assume, contrdiction, tht there is 0 discontinuit point or. 50
Dniel Lesnic - Chrcteriztions o the unctions with ounded vrition Since is with ounded vrition, rom corollr 2.1., results tht 0 is discontinuit point o the irst kind or. This conclusion is in contrdiction with the ct tht hs Drou propert since such unction cnnot hve discontinuit o the irst kind. ence, is continuous. Assume gin, contrdiction, tht there is 0 discontinuit point or. Since hs Drou propert results tht 0 is point o discontinuit o the second kind or nd thus 0 is et point o discontinuit or or which we hve proved tht is continuous. This is contrdiction nd hence, is continuous. Oservtion. We recll tht unction with ounded vrition nd with Drou propert is necessr continuous. This oservtion will e reerred s nlsis o discontinuities. The ollowing corollries cn esil e proved rom this nlsis. Corollr 3.1. Let, 0 nd e rel numers such tht < 0 < nd let : [,] R e unction with the ollowing properties: i is locl with ounded vrition t 0, i.e. there is compct intervl included in [, ] nd contining in interior the point 0 on which is with ounded vrition. ii possesses primitives on, 0 nd 0,. Then possesses primitives on, i nd onl i is continuous t 0. Corollr 3.2. Let : [,] I e unction, where I is n intervl included in [, ]. Then: ii o o... o is discontinuous nd with ounded vrition then hs not Drou propert. ii I hs Drou propert nd o is with ounded vrition then 2n = 1o 4243 o... o is continuous. 2ntimes Corollr 3.3. A unction with ounded vrition which cn e represented s rtio o two unctions possessing primitives is continuous. Lemm 3.1. Let : [,] R e continuous one to one unction nd let g : [,] R e unction possessing primitives. Then the product. g possesses primitives. Proo. Since is one-to-one there is n let-inverse -1, s, such tht 1 o = or n [, ]. In ddition, since is continuous results tht -1 is continuous. Let 1 e primitive o the unction g, so = g. then the unction o is continuous nd 1 hence possesses primitives. Let e such primitive, so = o. We prove now tht the unction. g possesses primitives showing tht the unction T : [,] 51
Dniel Lesnic - Chrcteriztions o the unctions with ounded vrition 52 R, deined s T =. o is derivle nd its derivtive is the unction. g. or this, let e n ritrr point in [, ]. Then, T T lim = + + lim = lim + lim. We ppl now Lgrnge s men vlue theorem to the unction on the intervl [, ]. Thus there is ξ [, ] such tht: = 1 ξ, whence it eists η [, ] such tht = η. Then the limit clculted ove ecomes: T T lim = g+ η lim = g+ + g η lim =. g. So, T is derivle nd T =. g, hence. g possesses primitives. Lemm 3.2. Let : [,] R e continuous monotonic unction nd let g : [,] R e unction possesses primitives. Proo. Suppose is n incresing unction on [, ]. Then the unction h : [,] R, deined s h = +, or n [, ] is strictl incresing unction nd hence one-to-one unction. From lemm 3.1. it results tht the unction p : [,] R deined s: p = hg = + g or n [, ] possesses primitive P, s, such tht P = p. Also, rom lemm 3.1.. it results tht the unction q : [,] R deined s: q =. g or n [, ] possesses primitive Q, s, such tht Q = q.
Dniel Lesnic - Chrcteriztions o the unctions with ounded vrition It is ovious now tht. g = p-q nd thus primitive o the product. g is the derivle unction P Q. Theorem 3.1. Let : [,] R e continuous unction with ounded vrition nd let g : [,] R e unction possesses primitives. Then the product. g possesses primitives. Proo. Since is continuous unction with ounded vrition, rom Jordn s theorem, there re Φ nd Ψ two continuous incresing unctions such tht = Φ Ψ. Then. g = Φ. g - Ψ. g. now using lemm 3.2 it results tht Φ. g nd Ψ. g possesses primitives nd hence, product. g possesses primitives. Proposition 3.3. Let : [,] R e unction with ounded vrition with the propert tht there is unction g : [,] R\{0} possesses primitives such tht the product. g possesses primitives. Then the product. h possesses primitives or n unction h : [,] R possessing primitives. Proo. Let Φ : [,] R e unction deined s Φ =. g or n [, ]. Then rom hpothesis the unctions g nd Φ possess primitives nd is with ounded vrition. Since g 0 or n [, ] nd = Φ /g, rom corollr 3.3 it results tht is continuous. Now the conclusion o the proposition is given the proposition 3.1. Deinition 3.1. Two sets re sid to e equipotent or crdinl equivlent i there is n univoc ppliction etween them. Proposition 3.4. For, R denote BV[, ] = { : [,] R is with ounded vrition on [, ]} B[, ] = { : [,] R is ounded on [, ]} DP[, ] = { : [,] R hs Drou propert on [, ]} Then BV[, ] nd B[, ] re not equipotent nd so there re BV[, ] nd DP[, ]. Proo. Assume, contrdiction, tht there is n univoc ppliction Φ : B[,] BV[, ] nd tke : [,] R deined s 0, i [, ] Q =. 1, i [, ] R - Q Oviousl, B[, ] nd Φ0, i [, ] Q Φ o =. Φ1, i [, ] R - Q 53
Dniel Lesnic - Chrcteriztions o the unctions with ounded vrition Recll now tht Φ is Dirichlet-tpe unction nd it is with ounded vrition i nd onl i is constnt, ut this in contrdiction with Φ is one-to-one unction. ence, BV[, ] nd B[, ] re not equipotent. For the second conclusion ssume gin, contrdiction, tht there is n univoc ppliction Ψ : BV[,] DP[, ] nd tke g : [,] R deined s g = 0, 1, i i [, + /2]. [ + /2, ] Oviousl, g BV[, ] nd Ψ0, Ψ o g =. Ψ1, i i [, + /2] [ + /2, ] Agin Ψ g is Dirichlet-tpe unction nd hs Drou propert i nd onl i is constnt, ut this in contrdiction with Ψ is one-to-one unction. ence, BV[, ] nd DP[, ] re not equipotent. REFERENCE 1. S. MĂRCUŞ, Mthemticl Anlsis in Romnin, Vol. 2, 1968. Author: Dniel Lesnic, Deprtment o Applied Mthemticl Studies, Leeds LS2 9JT, Englnd 54