7088:
6344:
6001:
5676:
2841:
716:
43:
6339:{\displaystyle {\begin{aligned}&\lim _{N\to \infty }\int _{\Omega }f\operatorname {div} \theta _{N}^{*}\\&=\lim _{N\to \infty }\int _{\{\nabla f\neq 0\}}\mathbb {I} _{\left}\nabla f\cdot {\frac {\nabla f}{\left|\nabla f\right|}}\\&=\lim _{N\to \infty }\int _{\left\cap {\{\nabla f\neq 0\}}}\nabla f\cdot {\frac {\nabla f}{\left|\nabla f\right|}}\\&=\int _{\Omega }\left|\nabla f\right|\end{aligned}}}
5499:
1011:
4444:
5671:{\displaystyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } =-\int _{\Omega }\mathbf {\varphi } \cdot \nabla f\leq \left|\int _{\Omega }\mathbf {\varphi } \cdot \nabla f\right|\leq \int _{\Omega }\left|\mathbf {\varphi } \right|\cdot \left|\nabla f\right|\leq \int _{\Omega }\left|\nabla f\right|}
5021:
1640:
1528:
821:
1898:
5411:
4116:
5823:
1739:
5483:
4889:
5303:
3134:
6501:
608:
4807:
2790:
483:
2509:
5202:
4933:
5126:
3394:
4716:
2132:
3261:
2058:
1534:
1422:
6567:
6006:
1006:{\displaystyle V(f,\Omega ):=\sup \left\{\int _{\Omega }f(x)\operatorname {div} \phi (x)\,\mathrm {d} x\colon \phi \in C_{c}^{1}(\Omega ,\mathbb {R} ^{n}),\ \Vert \phi \Vert _{L^{\infty }(\Omega )}\leq 1\right\},}
7453:
6388:
3697:
1781:
4036:
6690:
4121:
4439:{\displaystyle {\begin{aligned}V_{a}^{b}(f)&=V_{a}^{a_{1}}(f)+V_{a_{1}}^{a_{2}}(f)+\,\cdots \,+V_{a_{N}}^{b}(f)\\&=|f(a)-f(a_{1})|+|f(a_{1})-f(a_{2})|+\,\cdots \,+|f(a_{N})-f(b)|\end{aligned}}}
5309:
1406:
6878:
1362:
696:
1157:
5709:
1073:
7548:
4925:
1651:
6431:
4559:
1236:
6758:
5417:
4827:
4498:
3450:
5208:
2348:, its upper and lower variation cannot be defined and the Hahn–Jordan decomposition theorem can only be applied to its real and imaginary parts. However, it is possible to follow
3808:
2618:
2299:
3006:
2242:
3575:
1186:
330:
6436:
5701:
5046:
502:
1959:
4582:
2998:
5855:
2951:
5875:
6917:
5902:
1315:
4745:
5966:
5934:
3965:
6785:
2189:
2162:
5066:
3600:
1108:
5993:
4609:
2696:
2684:
2660:
2536:
2400:
2370:
2338:
1773:
1280:
338:
2411:
6593:
5016:{\displaystyle \int _{\Omega }\operatorname {div} \left(f\mathbf {\varphi } \right)=\int _{\partial \Omega }\left(f\mathbf {\varphi } \right)\cdot \mathbf {n} }
4631:
4518:
4076:
4056:
3831:
3720:
3532:
3470:
3157:
2821:
2559:
287:
6949:(for the case of functions of several variables). As a functional, total variation finds applications in several branches of mathematics and engineering, like
4108:
3863:
3296:
5132:
5074:
3304:
4640:
64:
57:
7492:
7061:
3160:
2917:
2902:
2064:
1635:{\displaystyle {\underline {\mathrm {W} }}(\mu ,E)=\inf \left\{\mu (A)\mid A\in \Sigma {\text{ and }}A\subset E\right\}\qquad \forall E\in \Sigma }
31:
1523:{\displaystyle {\overline {\mathrm {W} }}(\mu ,E)=\sup \left\{\mu (A)\mid A\in \Sigma {\text{ and }}A\subset E\right\}\qquad \forall E\in \Sigma }
3177:
1993:
7876:
7778:
6515:
7887:(with in-depth coverage and extensive applications of Total Variations in modern image processing, as started by Rudin, Osher, and Fatemi).
6352:
1893:{\displaystyle |\mu |(E)={\overline {\mathrm {W} }}(\mu ,E)+\left|{\underline {\mathrm {W} }}(\mu ,E)\right|\qquad \forall E\in \Sigma }
2911:
is exactly one, therefore it is not interesting as a means of investigating the properties of such measures. However, when μ and ν are
7870:
3608:
5406:{\displaystyle \int _{\Omega }f\partial _{x_{i}}\mathbf {\varphi } _{i}=-\int _{\Omega }\mathbf {\varphi } _{i}\partial _{x_{i}}f}
3970:
6659:
7014:" is the name for the application of total variation to image noise reduction; further details can be found in the papers of (
1367:
107:
7754:
7600:
6797:
1323:
5818:{\displaystyle \theta _{N}:=-\mathbb {I} _{\left}\mathbb {I} _{\{\nabla f\neq 0\}}{\frac {\nabla f}{\left|\nabla f\right|}}}
617:
79:
1734:{\displaystyle {\overline {\mathrm {W} }}(\mu ,E)\geq 0\geq {\underline {\mathrm {W} }}(\mu ,E)\qquad \forall E\in \Sigma }
1115:
17:
5478:{\displaystyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } =-\int _{\Omega }\mathbf {\varphi } \cdot \nabla f}
1022:
4897:
4529:
1206:
86:
6698:
6393:
4884:{\displaystyle \int _{\Omega }\operatorname {div} \mathbf {R} =\int _{\partial \Omega }\mathbf {R} \cdot \mathbf {n} }
7884:
7772:
7176:
7131:
7109:
2888:
763:
126:
7828:
Blomgren, Peter; Chan, Tony F. (1998), "Color TV: total variation methods for restoration of vector-valued images",
7490:
Adams, C. Raymond; Clarkson, James A. (1933), "On definitions of bounded variation for functions of two variables",
7102:
5298:{\displaystyle \int _{\Omega }\mathbf {\varphi } _{i}\partial _{x_{i}}f+f\partial _{x_{i}}\mathbf {\varphi } _{i}=0}
2870:
745:
7797:
Rudin, Leonid I.; Osher, Stanley; Fatemi, Emad (1992), "Nonlinear total variation based noise removal algorithms",
2862:
737:
4460:
3412:
238:. The extension of the concept to functions of more than one variable however is not simple for various reasons.
93:
6965:
its value. As an example, use of the total variation functional is common in the following two kind of problems
7900:
7619:. Monografie Matematyczne. Vol. 7 (2nd ed.). Warszawa–Lwów: G.E. Stechert & Co. pp. VI+347.
7449:"Sui gruppi di punti e sulle funzioni di variabili reali (On groups of points and functions of real variables)"
4818:
4728:
3129:{\displaystyle \|\mu -\nu \|=|\mu -\nu |(X)=2\sup \left\{\,\left|\mu (A)-\nu (A)\right|:A\in \Sigma \,\right\}}
2866:
741:
3732:
2567:
2248:
7383:
7373:
7369:
7360:
7350:
7341:
7322:
7303:
7284:
7265:
6496:{\textstyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } \leq \int _{\Omega }\left|\nabla f\right|}
1087:
7710:
7226:"Sulle funzioni di due variabili a variazione limitata (On functions of two variables of bounded variation)"
2201:
603:{\displaystyle {\mathcal {P}}=\left\{P=\{x_{0},\dots ,x_{n_{P}}\}\mid P{\text{ is a partition of }}\right\}}
75:
7225:
6611:
defined on the space of measures of bounded variation. The space of measures on a σ-algebra of sets is a
3540:
1171:
295:
7378:
7355:
7336:
7317:
7298:
7279:
7260:
7066:
7046:
6978:
5684:
5029:
7312:
1913:
7274:
6788:
1973:
7448:
7331:
7255:
4564:
2956:
7422:
7293:
7051:
7010:
6625:
5828:
4449:
7660:, McGraw-Hill Series in Higher Mathematics (1st ed.), New York: McGraw-Hill, pp. xi+412,
7614:
7096:
6935:
4802:{\displaystyle \int _{\Omega }f\operatorname {div} \varphi =-\int _{\Omega }\nabla f\cdot \varphi }
3481:
3168:
3164:
2924:
2851:
726:
497:
5860:
7642:
7476:
6890:
5880:
3512:
2855:
1288:
730:
53:
7113:
6958:
6942:
6573:
5939:
5907:
3868:
3535:
3499:
2795:
where the supremum is as above. This definition is slightly more general than the one given by
1083:
611:
290:
267:
228:
174:
156:
6770:
3163:). Informally, this is the largest possible difference between the probabilities that the two
2167:
2140:
7071:
6973:
5051:
2785:{\displaystyle |\mu |(E)=\sup _{\pi }\sum _{A\in \pi }\|\mu (A)\|\qquad \forall E\in \Sigma }
2192:
1984:
1976:: according to his version of this theorem, the upper and lower variation are respectively a
1093:
160:
100:
478:{\displaystyle V_{a}^{b}(f)=\sup _{\mathcal {P}}\sum _{i=0}^{n_{P}-1}|f(x_{i+1})-f(x_{i})|,}
7837:
7806:
7665:
7569:
7523:
7022:). A sensible extension of this model to colour images, called Colour TV, can be found in (
5971:
4587:
2504:{\displaystyle |\mu |(E)=\sup _{\pi }\sum _{A\in \pi }|\mu (A)|\qquad \forall E\in \Sigma }
1201:
7693:
7673:
7632:
7624:
7577:
7561:
7531:
7515:
7466:
7436:
7418:). This is, according to Boris Golubov, the first paper on functions of bounded variation.
7410:
7241:
2669:
2645:
2521:
2385:
2355:
2323:
1758:
1265:
215:
The concept of total variation for functions of one real variable was first introduced by
8:
7543:
7426:
7056:
6946:
2912:
2908:
167:
7841:
7810:
7638:
7245:
3580:
2953:
where the norm is the total variation norm of signed measures. Using the property that
181:, its total variation on the interval of definition is a measure of the one-dimensional
6954:
6633:
associated to the norm gives rise to the total variation distance between two measures
6608:
6578:
5197:{\displaystyle \int _{\Omega }\partial _{x_{i}}\left(f\mathbf {\varphi } _{i}\right)=0}
4616:
4503:
4061:
4041:
3816:
3723:
3705:
3517:
3455:
3142:
2806:
2544:
1163:
1160:
272:
7506:
7230:
Rendiconto delle
Sessioni della Reale Accademia delle Scienze dell'Istituto di Bologna
7193:
7166:
5121:{\displaystyle \int _{\Omega }\operatorname {div} \left(f\mathbf {\varphi } \right)=0}
4081:
3836:
3389:{\displaystyle \delta (\mu ,\nu )={\frac {1}{2}}\sum _{x}\left|\mu (x)-\nu (x)\right|}
3269:
7880:
7853:
7818:
7596:
7595:, Graduate Studies in Mathematics, American Mathematical Society, pp. xxii+734,
7470:
7172:
7036:
7001:
6962:
6630:
6596:
2305:
1076:
493:
235:
231:
203:
7729:
7610:
7845:
7814:
7788:
7669:
7628:
7620:
7573:
7557:
7527:
7511:
7501:
7462:
7432:
7406:
7394:
7237:
6993:
6989:
2824:
2629:
7774:
The discontinuity set of solutions of the TV denoising problem and some extensions
7221:
6652:
and the total variation of a function, as described above, goes as follows. Given
7872:
Image
Processing and Analysis - Variational, PDE, Wavelet, and Stochastic Methods
7661:
7565:
7539:
7519:
7444:
6976:. Applications of total variation to these problems are detailed in the article "
6950:
6619:, relative to this norm. It is contained in the larger Banach space, called the
4727:
The first step in the proof is to first prove an equality which follows from the
4711:{\displaystyle V(f,\Omega )=\int _{\Omega }\left|\nabla f(x)\right|\mathrm {d} x}
4450:
The form of the total variation of a differentiable function of several variables
2352:, pp. 137–139) and define the total variation of the complex-valued measure
2345:
1080:
7390:
5703:
could be omitted, because by definition its essential supremum is at most one.
2687:
2663:
2539:
2341:
1907:
is defined as the value of this measure on the whole space of definition, i.e.
1260:
263:
224:
216:
148:
7544:"Sulle funzioni a variazione limitata (On the functions of bounded variation)"
7894:
7744:
7646:
2127:{\displaystyle \mu ^{-}(\cdot )=-{\underline {\mathrm {W} }}(\mu ,\cdot )\,,}
1283:
3500:
The form of the total variation of a differentiable function of one variable
7866:
7857:
7787:(a work dealing with total variation application in denoising problems for
7653:
6928:
6787:. In general, the total variation of a signed measure can be defined using
6612:
3256:{\displaystyle \delta (\mu ,\nu )=\sum _{x}\left|\mu (x)-\nu (x)\right|\;.}
2403:
2053:{\displaystyle \mu ^{+}(\cdot )={\overline {\mathrm {W} }}(\mu ,\cdot )\,,}
1981:
1977:
1318:
701:
7836:(3), Image Processing, IEEE Transactions on, vol. 7, no. 3: 304-309: 304,
7782:
6629:(as opposed to countably additive) measures, also with the same norm. The
7041:
6939:
6931:
4521:
2823:: this implies that it can be used also to define the total variation on
2666:: its total variation is defined as above. This definition works also if
1239:
784:
259:
164:
140:
7748:
7697:
6562:{\displaystyle \varphi \to {\frac {-\nabla f}{\left|\nabla f\right|}}.}
1189:
223:). He used the new concept in order to prove a convergence theorem for
7849:
7758:
7734:
7714:
6767:
is equal to the total variation, in the above sense, of the function
4110:
can be written as the sum of local variations on those subintervals:
246:
182:
6383:{\textstyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } }
2840:
715:
42:
7399:
Comptes rendus hebdomadaires des séances de l'Académie des sciences
6620:
6616:
4739:
Under the conditions of the theorem, the following equality holds:
4524:
3477:
3473:
2515:
489:
152:
7000:
reconstructed from data obtained by electronic means, for example
7781:, Multiscale Modeling and Simulation, vol. 6 n. 3, archived from
7415:
7158:
6509:
It can be seen from the proof that the supremum is attained when
2623:
3171:
it is possible to write the total variation distance as follows
147:
identifies several slightly different concepts, related to the (
7005:
6945:(for the case of functions of one variable) or on the space of
6504:
7805:(1–4), Physica D: Nonlinear Phenomena 60.1: 259-268: 259–268,
7168:
Functions of
Bounded Variation and Free Discontinuity Problems
3692:{\displaystyle V_{a}^{b}(f)=\int _{a}^{b}|f'(x)|\mathrm {d} x}
3404:
3159:
above is usually dropped (as is the convention in the article
7771:
Caselles, Vicent; Chambolle, Antonin; Novaga, Matteo (2007),
6997:
5493:
Under the conditions of the theorem, from the lemma we have:
4031:{\displaystyle a<a_{1}<a_{2}<\cdots <a_{N}<b}
1964:
6685:{\displaystyle \varphi \colon \mathbb {R} \to \mathbb {R} }
2315:
1972:, p. 11) uses upper and lower variations to prove the
6992:, denoising is a collection of methods used to reduce the
2830:
2690:: the variation is then defined by the following formula
201:∈ . Functions whose total variation is finite are called
7726:
7582:
7431:(in German), Berlin: Springer Verlag, pp. VII+600,
6972:: it is the science of finding approximate solutions to
2561:
into a countable number of disjoint measurable subsets.
1401:{\displaystyle {\underline {\mathrm {W} }}(\mu ,\cdot )}
7770:
7165:
Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego (2000).
7019:
6873:{\displaystyle \|\mu \|_{TV}=\mu _{+}(X)+\mu _{-}(X)~,}
1357:{\displaystyle {\overline {\mathrm {W} }}(\mu ,\cdot )}
1250:
7191:
6439:
6396:
6355:
2636:, p. 139)) and coincides with the one defined by
691:{\displaystyle a=x_{0}<x_{1}<...<x_{n_{P}}=b}
6893:
6800:
6773:
6701:
6662:
6581:
6518:
6004:
5974:
5942:
5910:
5883:
5863:
5831:
5712:
5687:
5502:
5420:
5312:
5211:
5135:
5077:
5054:
5032:
4936:
4900:
4830:
4748:
4643:
4619:
4590:
4567:
4532:
4506:
4463:
4119:
4084:
4064:
4044:
3973:
3871:
3839:
3819:
3735:
3708:
3611:
3583:
3543:
3520:
3458:
3415:
3307:
3272:
3180:
3145:
3009:
2959:
2927:
2809:
2699:
2672:
2648:
2570:
2547:
2524:
2414:
2388:
2358:
2326:
2251:
2204:
2170:
2143:
2067:
1996:
1916:
1784:
1761:
1654:
1537:
1425:
1370:
1326:
1291:
1268:
1209:
1174:
1152:{\displaystyle \Vert \;\Vert _{L^{\infty }(\Omega )}}
1118:
1096:
1025:
824:
620:
505:
341:
298:
275:
7164:
6648:, the link between the total variation of a measure
3000:, we eventually arrive at the equivalent definition
2564:
This definition coincides with the above definition
1245:
7641:). English translation from the original French by
1068:{\displaystyle C_{c}^{1}(\Omega ,\mathbb {R} ^{n})}
6911:
6872:
6779:
6752:
6684:
6587:
6561:
6495:
6425:
6382:
6338:
5987:
5960:
5928:
5896:
5869:
5849:
5817:
5695:
5670:
5477:
5405:
5297:
5196:
5120:
5060:
5040:
5015:
4920:{\displaystyle \mathbf {R} :=f\mathbf {\varphi } }
4919:
4883:
4801:
4710:
4625:
4603:
4576:
4553:
4512:
4492:
4438:
4102:
4070:
4058:is locally monotonic, then the total variation of
4050:
4030:
3959:
3857:
3825:
3802:
3714:
3691:
3594:
3569:
3526:
3464:
3444:
3388:
3290:
3255:
3151:
3128:
2992:
2945:
2815:
2799:, p. 138) since it requires only to consider
2784:
2678:
2654:
2612:
2553:
2530:
2503:
2394:
2364:
2332:
2293:
2236:
2183:
2156:
2126:
2052:
1953:
1892:
1767:
1733:
1634:
1522:
1400:
1356:
1309:
1274:
1230:
1180:
1151:
1102:
1067:
1005:
690:
602:
477:
324:
281:
247:Total variation for functions of one real variable
7493:Transactions of the American Mathematical Society
6961:, where the solution to a certain problem has to
6426:{\textstyle \int _{\Omega }\left|\nabla f\right|}
4554:{\displaystyle \Omega \subseteq \mathbb {R} ^{n}}
1231:{\displaystyle \Omega \subseteq \mathbb {R} ^{n}}
7892:
7796:
7593:A First Course in Sobolev Spaces: Second Edition
7367:
7152:
7062:Total variation distance of probability measures
7015:
6763:Then, the total variation of the signed measure
6753:{\displaystyle \varphi (t)=\mu ((-\infty ,t])~.}
6192:
6073:
6013:
3161:total variation distance of probability measures
3062:
2918:total variation distance of probability measures
2903:Total variation distance of probability measures
2726:
2441:
1568:
1456:
846:
370:
32:Total variation distance of probability measures
6602:
3476:involving the given function instead of as the
7194:"On Choosing and Bounding Probability Metrics"
3298:by halving the previous definition as follows
2624:Total variation norm of vector-valued measures
7489:
5936:with the same integral. We can do this since
2620:for the case of real-valued signed measures.
7827:
7023:
6970:Numerical analysis of differential equations
6808:
6801:
6599:precisely if its total variation is finite.
6349:This means we have a convergent sequence of
6253:
6238:
6108:
6093:
5782:
5767:
4493:{\displaystyle C^{1}({\overline {\Omega }})}
3445:{\displaystyle C^{1}({\overline {\Omega }})}
3022:
3010:
2940:
2928:
2766:
2751:
1923:
1917:
1124:
1119:
964:
957:
566:
527:
7454:Atti dell'Accademia delle Scienze di Torino
3405:Total variation of differentiable functions
3139:and its values are non-trivial. The factor
2869:. Unsourced material may be challenged and
744:. Unsourced material may be challenged and
7475:. The paper containing the first proof of
3249:
1122:
7505:
7192:Gibbs, Alison; Francis Edward Su (2002).
7132:Learn how and when to remove this message
6678:
6670:
6115:
5995:. Now again substituting into the lemma:
5762:
5731:
4541:
4384:
4380:
4234:
4230:
3563:
3120:
3070:
2889:Learn how and when to remove this message
2304:The last measure is sometimes called, by
2120:
2046:
1965:Modern definition of total variation norm
1218:
1052:
938:
894:
764:Learn how and when to remove this message
318:
127:Learn how and when to remove this message
7214:
7171:. Oxford University Press. p. 119.
7095:This article includes a list of general
5488:
3803:{\displaystyle V_{a}^{b}(f)=|f(a)-f(b)|}
3266:It may also be normalized to values in
2613:{\displaystyle |\mu |=\mu ^{+}+\mu ^{-}}
2316:Total variation norm of complex measures
2294:{\displaystyle |\mu |=\mu ^{+}+\mu ^{-}}
7348:
7329:
7310:
7291:
7272:
7253:
3833:, we can decompose the domain interval
2831:Total variation of probability measures
1987:. Using a more modern notation, define
14:
7893:
7538:
7443:
7389:
7220:
2237:{\displaystyle \mu =\mu ^{+}-\mu ^{-}}
220:
185:of the curve with parametric equation
63:Please improve this article by adding
7830:IEEE Transactions on Image Processing
7727:
7652:
7590:
7549:Annali della Scuola Normale Superiore
7020:Caselles, Chambolle & Novaga 2007
4812:
2796:
2633:
2349:
27:Measure of local oscillation behavior
7609:
7421:
7081:
3570:{\displaystyle \subset \mathbb {R} }
3167:can assign to the same event. For a
2867:adding citations to reliable sources
2834:
1969:
1317:: then it is possible to define two
1256:
1251:Classical total variation definition
1181:{\displaystyle \operatorname {div} }
742:adding citations to reliable sources
709:
325:{\displaystyle \subset \mathbb {R} }
36:
7869:and Jackie (Jianhong) Shen (2005),
5696:{\displaystyle \mathbf {\varphi } }
5041:{\displaystyle \mathbf {\varphi } }
3399:
24:
7101:it lacks sufficient corresponding
6903:
6729:
6543:
6531:
6482:
6472:
6445:
6412:
6402:
6361:
6321:
6311:
6283:
6271:
6259:
6241:
6202:
6168:
6156:
6144:
6096:
6083:
6033:
6023:
5802:
5790:
5770:
5657:
5647:
5628:
5602:
5583:
5570:
5551:
5538:
5508:
5469:
5456:
5426:
5384:
5366:
5327:
5318:
5261:
5235:
5217:
5147:
5141:
5083:
5055:
4982:
4979:
4942:
4863:
4860:
4836:
4787:
4782:
4754:
4701:
4680:
4670:
4656:
4571:
4568:
4533:
4479:
3682:
3577:, has the following expression if
3431:
3117:
2779:
2770:
2498:
2489:
2096:
2022:
1954:{\displaystyle \|\mu \|=|\mu |(X)}
1887:
1878:
1848:
1813:
1728:
1719:
1694:
1658:
1629:
1620:
1597:
1541:
1517:
1508:
1485:
1429:
1374:
1330:
1301:
1210:
1141:
1133:
1097:
1044:
981:
973:
930:
896:
859:
837:
508:
375:
210:
25:
7912:
7764:
7681:
7639:Polish Virtual Library of Science
7507:10.1090/S0002-9947-1933-1501718-2
6927:Total variation can be seen as a
1246:Total variation in measure theory
702:Total variation for functions of
7086:
5009:
4902:
4877:
4869:
4848:
4577:{\displaystyle \partial \Omega }
3813:For any differentiable function
2993:{\displaystyle (\mu -\nu )(X)=0}
2839:
714:
41:
7645:, with two additional notes by
6922:
5850:{\displaystyle \theta _{N}^{*}}
5706:On the other hand, we consider
3726:, then the above simplifies to
2769:
2488:
1974:Hahn–Jordan decomposition
1877:
1718:
1619:
1507:
7799:Physica D: Nonlinear Phenomena
7428:Theorie der reellen Funktionen
7185:
7153:Golubov & Vitushkin (2001)
7145:
7016:Rudin, Osher & Fatemi 1992
6906:
6894:
6861:
6855:
6839:
6833:
6789:Jordan's decomposition theorem
6741:
6738:
6723:
6720:
6711:
6705:
6674:
6522:
6199:
6080:
6020:
4692:
4686:
4659:
4647:
4487:
4474:
4428:
4424:
4418:
4409:
4396:
4389:
4373:
4369:
4356:
4347:
4334:
4327:
4319:
4315:
4302:
4293:
4287:
4280:
4266:
4260:
4224:
4218:
4183:
4177:
4145:
4139:
4097:
4085:
3954:
3935:
3923:
3897:
3891:
3872:
3852:
3840:
3796:
3792:
3786:
3777:
3771:
3764:
3757:
3751:
3677:
3673:
3667:
3655:
3633:
3627:
3556:
3544:
3439:
3426:
3378:
3372:
3363:
3357:
3323:
3311:
3285:
3273:
3241:
3235:
3226:
3220:
3196:
3184:
3100:
3094:
3085:
3079:
3053:
3047:
3043:
3029:
2981:
2975:
2972:
2960:
2763:
2757:
2719:
2713:
2709:
2701:
2628:The variation so defined is a
2580:
2572:
2484:
2480:
2474:
2467:
2434:
2428:
2424:
2416:
2382:of the complex-valued measure
2261:
2253:
2117:
2105:
2084:
2078:
2043:
2031:
2013:
2007:
1948:
1942:
1938:
1930:
1869:
1857:
1834:
1822:
1804:
1798:
1794:
1786:
1715:
1703:
1679:
1667:
1585:
1579:
1562:
1550:
1473:
1467:
1450:
1438:
1395:
1383:
1351:
1339:
1304:
1292:
1144:
1138:
1062:
1041:
984:
978:
948:
927:
891:
885:
873:
867:
840:
828:
592:
580:
468:
464:
451:
442:
423:
416:
363:
357:
311:
299:
241:
204:functions of bounded variation
13:
1:
7711:Function of bounded variation
7483:
4634:has the following expression
2946:{\displaystyle \|\mu -\nu \|}
2518:is taken over all partitions
577: is a partition of
65:secondary or tertiary sources
7819:10.1016/0167-2789(92)90242-F
7232:, Nuova serie (in Italian),
6603:Total variation of a measure
5870:{\displaystyle \varepsilon }
4482:
3434:
2026:
1817:
1662:
1433:
1334:
151:or global) structure of the
7:
7759:Encyclopedia of Mathematics
7715:Encyclopedia of Mathematics
7379:Encyclopedia of Mathematics
7356:Encyclopedia of Mathematics
7349:Golubov, Boris I. (2001) ,
7337:Encyclopedia of Mathematics
7330:Golubov, Boris I. (2001) ,
7318:Encyclopedia of Mathematics
7311:Golubov, Boris I. (2001) ,
7299:Encyclopedia of Mathematics
7292:Golubov, Boris I. (2001) ,
7280:Encyclopedia of Mathematics
7273:Golubov, Boris I. (2001) ,
7261:Encyclopedia of Mathematics
7254:Golubov, Boris I. (2001) ,
7047:Total variation diminishing
7030:
6979:total variation diminishing
6912:{\displaystyle (X,\Sigma )}
5897:{\displaystyle \theta _{N}}
4454:
3504:
3492:
3486:
2907:The total variation of any
2638:
2375:
1744:
1310:{\displaystyle (X,\Sigma )}
1238:of the given function be a
1081:continuously differentiable
776:
262:-valued (or more generally
251:
10:
7917:
4819:Gauss–Ostrogradsky theorem
4729:Gauss–Ostrogradsky theorem
2900:
1259:, p. 10), consider a
29:
7658:Real and Complex Analysis
7395:"Sur la série de Fourier"
7374:"Variation of a function"
7351:"Tonelli plane variation"
7052:Total variation denoising
7011:Total variation denoising
6607:The total variation is a
5961:{\displaystyle C_{c}^{1}}
5929:{\displaystyle C_{c}^{1}}
5048:is zero on the border of
3960:{\displaystyle ,,\dots ,}
3409:The total variation of a
3165:probability distributions
7591:Leoni, Giovanni (2017),
7077:
7024:Blomgren & Chan 1998
6938:defined on the space of
6780:{\displaystyle \varphi }
6433:as well as we know that
4734:
4722:
3169:categorical distribution
2825:finite-additive measures
2184:{\displaystyle \mu ^{-}}
2157:{\displaystyle \mu ^{+}}
1755:) of the signed measure
30:Not to be confused with
7643:Laurence Chisholm Young
7477:Vitali covering theorem
7116:more precise citations.
7067:Kolmogorov–Smirnov test
6883:for any signed measure
6644:For finite measures on
5061:{\displaystyle \Omega }
3513:differentiable function
3472:can be expressed as an
2310:total variation measure
1103:{\displaystyle \Omega }
7705:One and more variables
7616:Theory of the Integral
6974:differential equations
6959:calculus of variations
6913:
6887:on a measurable space
6874:
6781:
6754:
6686:
6589:
6563:
6497:
6427:
6384:
6340:
5989:
5962:
5930:
5898:
5871:
5851:
5819:
5697:
5672:
5479:
5407:
5299:
5198:
5122:
5062:
5042:
5017:
4921:
4885:
4803:
4712:
4627:
4605:
4578:
4555:
4514:
4494:
4440:
4104:
4072:
4052:
4032:
3961:
3859:
3827:
3804:
3722:is differentiable and
3716:
3693:
3602:is Riemann integrable
3596:
3571:
3528:
3466:
3446:
3390:
3292:
3257:
3153:
3130:
2994:
2947:
2817:
2786:
2680:
2656:
2614:
2555:
2532:
2505:
2396:
2366:
2334:
2295:
2238:
2185:
2158:
2128:
2054:
1955:
1894:
1769:
1735:
1636:
1524:
1408:, respectively called
1402:
1358:
1311:
1276:
1232:
1182:
1153:
1104:
1069:
1007:
692:
604:
479:
414:
326:
283:
52:relies excessively on
7901:Mathematical analysis
7370:Vitushkin, Anatoli G.
7215:Historical references
7072:Anisotropic diffusion
6914:
6875:
6782:
6755:
6687:
6590:
6564:
6498:
6428:
6385:
6341:
5990:
5988:{\displaystyle L^{1}}
5963:
5931:
5899:
5872:
5852:
5820:
5698:
5673:
5489:Proof of the equality
5480:
5408:
5300:
5199:
5123:
5063:
5043:
5018:
4922:
4886:
4804:
4713:
4628:
4606:
4604:{\displaystyle C^{1}}
4579:
4556:
4515:
4495:
4441:
4105:
4073:
4053:
4033:
3962:
3860:
3828:
3805:
3717:
3694:
3597:
3572:
3529:
3467:
3447:
3391:
3293:
3258:
3154:
3131:
2995:
2948:
2818:
2787:
2681:
2657:
2615:
2556:
2533:
2506:
2397:
2367:
2335:
2296:
2239:
2191:are two non-negative
2186:
2159:
2129:
2055:
1956:
1895:
1770:
1736:
1637:
1525:
1403:
1359:
1312:
1277:
1233:
1183:
1154:
1105:
1070:
1008:
706:> 1 real variables
693:
605:
480:
381:
327:
284:
7755:Jordan decomposition
7745:Jordan decomposition
7637:. (available at the
7313:"Pierpont variation"
6947:integrable functions
6891:
6798:
6771:
6699:
6660:
6656:, define a function
6579:
6516:
6437:
6394:
6353:
6002:
5972:
5940:
5908:
5881:
5861:
5829:
5710:
5685:
5500:
5418:
5310:
5209:
5133:
5075:
5052:
5030:
4934:
4898:
4828:
4746:
4641:
4617:
4588:
4565:
4530:
4504:
4461:
4117:
4082:
4062:
4042:
3971:
3869:
3865:, into subintervals
3837:
3817:
3733:
3706:
3609:
3581:
3541:
3518:
3456:
3413:
3305:
3270:
3178:
3143:
3007:
2957:
2925:
2913:probability measures
2863:improve this section
2807:
2697:
2679:{\displaystyle \mu }
2670:
2655:{\displaystyle \mu }
2646:
2568:
2545:
2531:{\displaystyle \pi }
2522:
2412:
2395:{\displaystyle \mu }
2386:
2365:{\displaystyle \mu }
2356:
2333:{\displaystyle \mu }
2324:
2249:
2202:
2168:
2141:
2065:
1994:
1914:
1782:
1775:is the set function
1768:{\displaystyle \mu }
1759:
1652:
1535:
1423:
1368:
1324:
1289:
1275:{\displaystyle \mu }
1266:
1207:
1172:
1116:
1094:
1023:
822:
738:improve this section
618:
503:
339:
296:
273:
18:Total variation norm
7842:1998ITIP....7..304B
7811:1992PhyD...60..259R
7552:, II (in Italian),
7368:Golubov, Boris I.;
7275:"Fréchet variation"
7057:Quadratic variation
6061:
5957:
5925:
5857:which is the up to
5846:
4613:total variation of
4259:
4217:
4176:
4138:
3750:
3653:
3626:
2909:probability measure
1040:
926:
791:. Given a function
614:. Which means that
356:
234:whose variation is
168:continuous function
7332:"Vitali variation"
7256:"Arzelà variation"
6955:numerical analysis
6909:
6870:
6777:
6750:
6682:
6585:
6559:
6493:
6423:
6380:
6336:
6334:
6206:
6087:
6047:
6027:
5985:
5958:
5943:
5926:
5911:
5894:
5867:
5847:
5832:
5815:
5693:
5668:
5475:
5403:
5295:
5194:
5118:
5058:
5038:
5013:
4917:
4881:
4813:Proof of the lemma
4799:
4708:
4623:
4601:
4574:
4551:
4510:
4490:
4436:
4434:
4238:
4189:
4155:
4124:
4100:
4068:
4048:
4028:
3957:
3855:
3823:
3800:
3736:
3712:
3689:
3639:
3612:
3595:{\displaystyle f'}
3592:
3567:
3524:
3462:
3442:
3386:
3348:
3288:
3253:
3211:
3149:
3126:
2990:
2943:
2921:can be defined as
2813:
2782:
2750:
2734:
2676:
2652:
2610:
2551:
2528:
2501:
2465:
2449:
2392:
2362:
2330:
2291:
2234:
2181:
2154:
2124:
2103:
2050:
1951:
1890:
1855:
1765:
1753:absolute variation
1731:
1701:
1632:
1548:
1520:
1398:
1381:
1354:
1307:
1272:
1228:
1178:
1161:essential supremum
1149:
1100:
1065:
1026:
1003:
912:
688:
600:
475:
380:
342:
322:
279:
232:periodic functions
7850:10.1109/83.661180
7730:"Total Variation"
7602:978-1-4704-2921-8
7294:"Hardy variation"
7142:
7141:
7134:
7037:Bounded variation
7002:data transmission
6866:
6746:
6631:distance function
6626:finitely additive
6597:bounded variation
6595:is said to be of
6588:{\displaystyle f}
6554:
6294:
6191:
6179:
6072:
6012:
5877:approximation of
5813:
5681:in the last part
4626:{\displaystyle f}
4513:{\displaystyle f}
4485:
4071:{\displaystyle f}
4051:{\displaystyle f}
3826:{\displaystyle f}
3715:{\displaystyle f}
3527:{\displaystyle f}
3465:{\displaystyle f}
3437:
3339:
3337:
3202:
3152:{\displaystyle 2}
2899:
2898:
2891:
2816:{\displaystyle X}
2801:finite partitions
2735:
2725:
2554:{\displaystyle E}
2450:
2440:
2306:abuse of notation
2094:
2029:
1846:
1820:
1692:
1665:
1603:
1539:
1491:
1436:
1372:
1337:
956:
774:
773:
766:
578:
369:
282:{\displaystyle f}
137:
136:
129:
111:
76:"Total variation"
16:(Redirected from
7908:
7860:
7821:
7789:image processing
7786:
7740:
7739:
7676:
7636:
7605:
7580:
7556:(3–4): 299–313,
7540:Cesari, Lamberto
7534:
7509:
7474:
7469:, archived from
7445:Vitali, Giuseppe
7439:
7413:
7386:
7363:
7344:
7325:
7306:
7287:
7268:
7249:
7244:, archived from
7209:
7208:
7206:
7204:
7198:
7189:
7183:
7182:
7162:
7156:
7149:
7137:
7130:
7126:
7123:
7117:
7112:this article by
7103:inline citations
7090:
7089:
7082:
6990:image processing
6918:
6916:
6915:
6910:
6879:
6877:
6876:
6871:
6864:
6854:
6853:
6832:
6831:
6819:
6818:
6786:
6784:
6783:
6778:
6759:
6757:
6756:
6751:
6744:
6691:
6689:
6688:
6683:
6681:
6673:
6623:, consisting of
6594:
6592:
6591:
6586:
6568:
6566:
6565:
6560:
6555:
6553:
6549:
6537:
6526:
6502:
6500:
6499:
6494:
6492:
6488:
6476:
6475:
6463:
6449:
6448:
6432:
6430:
6429:
6424:
6422:
6418:
6406:
6405:
6389:
6387:
6386:
6381:
6379:
6365:
6364:
6345:
6343:
6342:
6337:
6335:
6331:
6327:
6315:
6314:
6299:
6295:
6293:
6289:
6277:
6269:
6258:
6257:
6256:
6233:
6229:
6205:
6184:
6180:
6178:
6174:
6162:
6154:
6143:
6142:
6141:
6137:
6118:
6112:
6111:
6086:
6065:
6060:
6055:
6037:
6036:
6026:
6008:
5994:
5992:
5991:
5986:
5984:
5983:
5967:
5965:
5964:
5959:
5956:
5951:
5935:
5933:
5932:
5927:
5924:
5919:
5903:
5901:
5900:
5895:
5893:
5892:
5876:
5874:
5873:
5868:
5856:
5854:
5853:
5848:
5845:
5840:
5824:
5822:
5821:
5816:
5814:
5812:
5808:
5796:
5788:
5786:
5785:
5765:
5759:
5758:
5757:
5753:
5734:
5722:
5721:
5702:
5700:
5699:
5694:
5692:
5677:
5675:
5674:
5669:
5667:
5663:
5651:
5650:
5638:
5634:
5619:
5615:
5606:
5605:
5593:
5589:
5579:
5574:
5573:
5547:
5542:
5541:
5526:
5512:
5511:
5484:
5482:
5481:
5476:
5465:
5460:
5459:
5444:
5430:
5429:
5412:
5410:
5409:
5404:
5399:
5398:
5397:
5396:
5382:
5381:
5376:
5370:
5369:
5354:
5353:
5348:
5342:
5341:
5340:
5339:
5322:
5321:
5304:
5302:
5301:
5296:
5288:
5287:
5282:
5276:
5275:
5274:
5273:
5250:
5249:
5248:
5247:
5233:
5232:
5227:
5221:
5220:
5203:
5201:
5200:
5195:
5187:
5183:
5182:
5181:
5176:
5162:
5161:
5160:
5159:
5145:
5144:
5127:
5125:
5124:
5119:
5111:
5107:
5106:
5087:
5086:
5067:
5065:
5064:
5059:
5047:
5045:
5044:
5039:
5037:
5022:
5020:
5019:
5014:
5012:
5004:
5000:
4999:
4986:
4985:
4970:
4966:
4965:
4946:
4945:
4926:
4924:
4923:
4918:
4916:
4905:
4894:by substituting
4890:
4888:
4887:
4882:
4880:
4872:
4867:
4866:
4851:
4840:
4839:
4808:
4806:
4805:
4800:
4786:
4785:
4758:
4757:
4717:
4715:
4714:
4709:
4704:
4699:
4695:
4674:
4673:
4632:
4630:
4629:
4624:
4610:
4608:
4607:
4602:
4600:
4599:
4583:
4581:
4580:
4575:
4560:
4558:
4557:
4552:
4550:
4549:
4544:
4519:
4517:
4516:
4511:
4499:
4497:
4496:
4491:
4486:
4478:
4473:
4472:
4456:
4445:
4443:
4442:
4437:
4435:
4431:
4408:
4407:
4392:
4376:
4368:
4367:
4346:
4345:
4330:
4322:
4314:
4313:
4283:
4272:
4258:
4253:
4252:
4251:
4216:
4215:
4214:
4204:
4203:
4202:
4175:
4174:
4173:
4163:
4137:
4132:
4109:
4107:
4106:
4103:{\displaystyle }
4101:
4077:
4075:
4074:
4069:
4057:
4055:
4054:
4049:
4037:
4035:
4034:
4029:
4021:
4020:
4002:
4001:
3989:
3988:
3966:
3964:
3963:
3958:
3947:
3946:
3922:
3921:
3909:
3908:
3890:
3889:
3864:
3862:
3861:
3858:{\displaystyle }
3856:
3832:
3830:
3829:
3824:
3809:
3807:
3806:
3801:
3799:
3767:
3749:
3744:
3721:
3719:
3718:
3713:
3698:
3696:
3695:
3690:
3685:
3680:
3666:
3658:
3652:
3647:
3625:
3620:
3601:
3599:
3598:
3593:
3591:
3576:
3574:
3573:
3568:
3566:
3534:, defined on an
3533:
3531:
3530:
3525:
3506:
3471:
3469:
3468:
3463:
3451:
3449:
3448:
3443:
3438:
3430:
3425:
3424:
3400:Basic properties
3395:
3393:
3392:
3387:
3385:
3381:
3347:
3338:
3330:
3297:
3295:
3294:
3291:{\displaystyle }
3289:
3262:
3260:
3259:
3254:
3248:
3244:
3210:
3158:
3156:
3155:
3150:
3135:
3133:
3132:
3127:
3125:
3121:
3107:
3103:
3046:
3032:
2999:
2997:
2996:
2991:
2952:
2950:
2949:
2944:
2894:
2887:
2883:
2880:
2874:
2843:
2835:
2822:
2820:
2819:
2814:
2791:
2789:
2788:
2783:
2749:
2733:
2712:
2704:
2685:
2683:
2682:
2677:
2661:
2659:
2658:
2653:
2630:positive measure
2619:
2617:
2616:
2611:
2609:
2608:
2596:
2595:
2583:
2575:
2560:
2558:
2557:
2552:
2537:
2535:
2534:
2529:
2510:
2508:
2507:
2502:
2487:
2470:
2464:
2448:
2427:
2419:
2401:
2399:
2398:
2393:
2377:
2371:
2369:
2368:
2363:
2339:
2337:
2336:
2331:
2300:
2298:
2297:
2292:
2290:
2289:
2277:
2276:
2264:
2256:
2243:
2241:
2240:
2235:
2233:
2232:
2220:
2219:
2190:
2188:
2187:
2182:
2180:
2179:
2163:
2161:
2160:
2155:
2153:
2152:
2133:
2131:
2130:
2125:
2104:
2099:
2077:
2076:
2059:
2057:
2056:
2051:
2030:
2025:
2020:
2006:
2005:
1960:
1958:
1957:
1952:
1941:
1933:
1899:
1897:
1896:
1891:
1876:
1872:
1856:
1851:
1821:
1816:
1811:
1797:
1789:
1774:
1772:
1771:
1766:
1746:
1740:
1738:
1737:
1732:
1702:
1697:
1666:
1661:
1656:
1641:
1639:
1638:
1633:
1618:
1614:
1604:
1601:
1549:
1544:
1529:
1527:
1526:
1521:
1506:
1502:
1492:
1489:
1437:
1432:
1427:
1407:
1405:
1404:
1399:
1382:
1377:
1363:
1361:
1360:
1355:
1338:
1333:
1328:
1316:
1314:
1313:
1308:
1284:measurable space
1281:
1279:
1278:
1273:
1237:
1235:
1234:
1229:
1227:
1226:
1221:
1198:does not require
1196:This definition
1187:
1185:
1184:
1179:
1158:
1156:
1155:
1150:
1148:
1147:
1137:
1136:
1109:
1107:
1106:
1101:
1084:vector functions
1074:
1072:
1071:
1066:
1061:
1060:
1055:
1039:
1034:
1012:
1010:
1009:
1004:
999:
995:
988:
987:
977:
976:
954:
947:
946:
941:
925:
920:
899:
863:
862:
778:
769:
762:
758:
755:
749:
718:
710:
697:
695:
694:
689:
681:
680:
679:
678:
649:
648:
636:
635:
609:
607:
606:
601:
599:
595:
579:
576:
565:
564:
563:
562:
539:
538:
512:
511:
484:
482:
481:
476:
471:
463:
462:
441:
440:
419:
413:
406:
405:
395:
379:
378:
355:
350:
332:is the quantity
331:
329:
328:
323:
321:
289:, defined on an
288:
286:
285:
280:
253:
173:, defined on an
132:
125:
121:
118:
112:
110:
69:
45:
37:
21:
7916:
7915:
7911:
7910:
7909:
7907:
7906:
7905:
7891:
7890:
7767:
7728:Rowland, Todd.
7694:Total variation
7684:
7611:Saks, Stanisław
7603:
7581:. Available at
7486:
7391:Jordan, Camille
7217:
7212:
7202:
7200:
7196:
7190:
7186:
7179:
7163:
7159:
7150:
7146:
7138:
7127:
7121:
7118:
7108:Please help to
7107:
7091:
7087:
7080:
7033:
6986:Image denoising
6951:optimal control
6925:
6892:
6889:
6888:
6849:
6845:
6827:
6823:
6811:
6807:
6799:
6796:
6795:
6772:
6769:
6768:
6700:
6697:
6696:
6677:
6669:
6661:
6658:
6657:
6605:
6580:
6577:
6576:
6542:
6538:
6527:
6525:
6517:
6514:
6513:
6481:
6477:
6471:
6467:
6459:
6444:
6440:
6438:
6435:
6434:
6411:
6407:
6401:
6397:
6395:
6392:
6391:
6375:
6360:
6356:
6354:
6351:
6350:
6333:
6332:
6320:
6316:
6310:
6306:
6297:
6296:
6282:
6278:
6270:
6268:
6237:
6216:
6212:
6211:
6207:
6195:
6182:
6181:
6167:
6163:
6155:
6153:
6124:
6120:
6119:
6114:
6113:
6092:
6088:
6076:
6063:
6062:
6056:
6051:
6032:
6028:
6016:
6005:
6003:
6000:
5999:
5979:
5975:
5973:
5970:
5969:
5952:
5947:
5941:
5938:
5937:
5920:
5915:
5909:
5906:
5905:
5888:
5884:
5882:
5879:
5878:
5862:
5859:
5858:
5841:
5836:
5830:
5827:
5826:
5801:
5797:
5789:
5787:
5766:
5761:
5760:
5740:
5736:
5735:
5730:
5729:
5717:
5713:
5711:
5708:
5707:
5688:
5686:
5683:
5682:
5656:
5652:
5646:
5642:
5627:
5623:
5611:
5607:
5601:
5597:
5575:
5569:
5565:
5564:
5560:
5543:
5537:
5533:
5522:
5507:
5503:
5501:
5498:
5497:
5491:
5461:
5455:
5451:
5440:
5425:
5421:
5419:
5416:
5415:
5392:
5388:
5387:
5383:
5377:
5372:
5371:
5365:
5361:
5349:
5344:
5343:
5335:
5331:
5330:
5326:
5317:
5313:
5311:
5308:
5307:
5283:
5278:
5277:
5269:
5265:
5264:
5260:
5243:
5239:
5238:
5234:
5228:
5223:
5222:
5216:
5212:
5210:
5207:
5206:
5177:
5172:
5171:
5167:
5163:
5155:
5151:
5150:
5146:
5140:
5136:
5134:
5131:
5130:
5102:
5098:
5094:
5082:
5078:
5076:
5073:
5072:
5068:by definition:
5053:
5050:
5049:
5033:
5031:
5028:
5027:
5008:
4995:
4991:
4987:
4978:
4974:
4961:
4957:
4953:
4941:
4937:
4935:
4932:
4931:
4912:
4901:
4899:
4896:
4895:
4876:
4868:
4859:
4855:
4847:
4835:
4831:
4829:
4826:
4825:
4815:
4781:
4777:
4753:
4749:
4747:
4744:
4743:
4737:
4725:
4700:
4679:
4675:
4669:
4665:
4642:
4639:
4638:
4618:
4615:
4614:
4595:
4591:
4589:
4586:
4585:
4566:
4563:
4562:
4545:
4540:
4539:
4531:
4528:
4527:
4505:
4502:
4501:
4477:
4468:
4464:
4462:
4459:
4458:
4452:
4433:
4432:
4427:
4403:
4399:
4388:
4372:
4363:
4359:
4341:
4337:
4326:
4318:
4309:
4305:
4279:
4270:
4269:
4254:
4247:
4243:
4242:
4210:
4206:
4205:
4198:
4194:
4193:
4169:
4165:
4164:
4159:
4148:
4133:
4128:
4120:
4118:
4115:
4114:
4083:
4080:
4079:
4063:
4060:
4059:
4043:
4040:
4039:
4016:
4012:
3997:
3993:
3984:
3980:
3972:
3969:
3968:
3942:
3938:
3917:
3913:
3904:
3900:
3885:
3881:
3870:
3867:
3866:
3838:
3835:
3834:
3818:
3815:
3814:
3795:
3763:
3745:
3740:
3734:
3731:
3730:
3707:
3704:
3703:
3681:
3676:
3659:
3654:
3648:
3643:
3621:
3616:
3610:
3607:
3606:
3584:
3582:
3579:
3578:
3562:
3542:
3539:
3538:
3519:
3516:
3515:
3509:total variation
3502:
3484:of definitions
3457:
3454:
3453:
3429:
3420:
3416:
3414:
3411:
3410:
3407:
3402:
3353:
3349:
3343:
3329:
3306:
3303:
3302:
3271:
3268:
3267:
3216:
3212:
3206:
3179:
3176:
3175:
3144:
3141:
3140:
3075:
3071:
3069:
3065:
3042:
3028:
3008:
3005:
3004:
2958:
2955:
2954:
2926:
2923:
2922:
2905:
2895:
2884:
2878:
2875:
2860:
2844:
2833:
2808:
2805:
2804:
2739:
2729:
2708:
2700:
2698:
2695:
2694:
2671:
2668:
2667:
2647:
2644:
2643:
2626:
2604:
2600:
2591:
2587:
2579:
2571:
2569:
2566:
2565:
2546:
2543:
2542:
2523:
2520:
2519:
2483:
2466:
2454:
2444:
2423:
2415:
2413:
2410:
2409:
2387:
2384:
2383:
2376:Definition 1.4.
2357:
2354:
2353:
2346:complex measure
2325:
2322:
2321:
2320:If the measure
2318:
2285:
2281:
2272:
2268:
2260:
2252:
2250:
2247:
2246:
2228:
2224:
2215:
2211:
2203:
2200:
2199:
2175:
2171:
2169:
2166:
2165:
2148:
2144:
2142:
2139:
2138:
2095:
2093:
2072:
2068:
2066:
2063:
2062:
2021:
2019:
2001:
1997:
1995:
1992:
1991:
1967:
1937:
1929:
1915:
1912:
1911:
1905:total variation
1847:
1845:
1844:
1840:
1812:
1810:
1793:
1785:
1783:
1780:
1779:
1760:
1757:
1756:
1745:Definition 1.3.
1693:
1691:
1657:
1655:
1653:
1650:
1649:
1602: and
1600:
1575:
1571:
1540:
1538:
1536:
1533:
1532:
1490: and
1488:
1463:
1459:
1428:
1426:
1424:
1421:
1420:
1414:lower variation
1410:upper variation
1373:
1371:
1369:
1366:
1365:
1329:
1327:
1325:
1322:
1321:
1290:
1287:
1286:
1267:
1264:
1263:
1253:
1248:
1222:
1217:
1216:
1208:
1205:
1204:
1173:
1170:
1169:
1132:
1128:
1127:
1123:
1117:
1114:
1113:
1095:
1092:
1091:
1088:compact support
1056:
1051:
1050:
1035:
1030:
1024:
1021:
1020:
972:
968:
967:
963:
942:
937:
936:
921:
916:
895:
858:
854:
853:
849:
823:
820:
819:
805:total variation
777:Definition 1.2.
770:
759:
753:
750:
735:
719:
708:
674:
670:
669:
665:
644:
640:
631:
627:
619:
616:
615:
575:
558:
554:
553:
549:
534:
530:
520:
516:
507:
506:
504:
501:
500:
467:
458:
454:
430:
426:
415:
401:
397:
396:
385:
374:
373:
351:
346:
340:
337:
336:
317:
297:
294:
293:
274:
271:
270:
256:total variation
252:Definition 1.1.
249:
244:
213:
211:Historical note
145:total variation
133:
122:
116:
113:
70:
68:
62:
58:primary sources
46:
35:
28:
23:
22:
15:
12:
11:
5:
7914:
7904:
7903:
7889:
7888:
7863:
7862:
7824:
7823:
7793:
7792:
7766:
7763:
7762:
7761:
7752:
7742:
7721:Measure theory
7718:
7717:
7702:
7701:
7683:
7682:External links
7680:
7679:
7678:
7650:
7607:
7601:
7587:
7586:
7536:
7500:(4): 824–854,
7485:
7482:
7481:
7480:
7457:(in Italian),
7441:
7419:
7414:(available at
7387:
7365:
7346:
7327:
7308:
7289:
7270:
7251:
7236:(4): 100–107,
7224:(7 May 1905),
7222:Arzelà, Cesare
7216:
7213:
7211:
7210:
7184:
7177:
7157:
7143:
7140:
7139:
7094:
7092:
7085:
7079:
7076:
7075:
7074:
7069:
7064:
7059:
7054:
7049:
7044:
7039:
7032:
7029:
7028:
7027:
6983:
6924:
6921:
6908:
6905:
6902:
6899:
6896:
6881:
6880:
6869:
6863:
6860:
6857:
6852:
6848:
6844:
6841:
6838:
6835:
6830:
6826:
6822:
6817:
6814:
6810:
6806:
6803:
6776:
6761:
6760:
6749:
6743:
6740:
6737:
6734:
6731:
6728:
6725:
6722:
6719:
6716:
6713:
6710:
6707:
6704:
6680:
6676:
6672:
6668:
6665:
6604:
6601:
6584:
6570:
6569:
6558:
6552:
6548:
6545:
6541:
6536:
6533:
6530:
6524:
6521:
6491:
6487:
6484:
6480:
6474:
6470:
6466:
6462:
6458:
6455:
6452:
6447:
6443:
6421:
6417:
6414:
6410:
6404:
6400:
6390:that tends to
6378:
6374:
6371:
6368:
6363:
6359:
6347:
6346:
6330:
6326:
6323:
6319:
6313:
6309:
6305:
6302:
6300:
6298:
6292:
6288:
6285:
6281:
6276:
6273:
6267:
6264:
6261:
6255:
6252:
6249:
6246:
6243:
6240:
6236:
6232:
6228:
6225:
6222:
6219:
6215:
6210:
6204:
6201:
6198:
6194:
6190:
6187:
6185:
6183:
6177:
6173:
6170:
6166:
6161:
6158:
6152:
6149:
6146:
6140:
6136:
6133:
6130:
6127:
6123:
6117:
6110:
6107:
6104:
6101:
6098:
6095:
6091:
6085:
6082:
6079:
6075:
6071:
6068:
6066:
6064:
6059:
6054:
6050:
6046:
6043:
6040:
6035:
6031:
6025:
6022:
6019:
6015:
6011:
6009:
6007:
5982:
5978:
5955:
5950:
5946:
5923:
5918:
5914:
5891:
5887:
5866:
5844:
5839:
5835:
5811:
5807:
5804:
5800:
5795:
5792:
5784:
5781:
5778:
5775:
5772:
5769:
5764:
5756:
5752:
5749:
5746:
5743:
5739:
5733:
5728:
5725:
5720:
5716:
5691:
5679:
5678:
5666:
5662:
5659:
5655:
5649:
5645:
5641:
5637:
5633:
5630:
5626:
5622:
5618:
5614:
5610:
5604:
5600:
5596:
5592:
5588:
5585:
5582:
5578:
5572:
5568:
5563:
5559:
5556:
5553:
5550:
5546:
5540:
5536:
5532:
5529:
5525:
5521:
5518:
5515:
5510:
5506:
5490:
5487:
5486:
5485:
5474:
5471:
5468:
5464:
5458:
5454:
5450:
5447:
5443:
5439:
5436:
5433:
5428:
5424:
5413:
5402:
5395:
5391:
5386:
5380:
5375:
5368:
5364:
5360:
5357:
5352:
5347:
5338:
5334:
5329:
5325:
5320:
5316:
5305:
5294:
5291:
5286:
5281:
5272:
5268:
5263:
5259:
5256:
5253:
5246:
5242:
5237:
5231:
5226:
5219:
5215:
5204:
5193:
5190:
5186:
5180:
5175:
5170:
5166:
5158:
5154:
5149:
5143:
5139:
5128:
5117:
5114:
5110:
5105:
5101:
5097:
5093:
5090:
5085:
5081:
5057:
5036:
5024:
5023:
5011:
5007:
5003:
4998:
4994:
4990:
4984:
4981:
4977:
4973:
4969:
4964:
4960:
4956:
4952:
4949:
4944:
4940:
4915:
4911:
4908:
4904:
4892:
4891:
4879:
4875:
4871:
4865:
4862:
4858:
4854:
4850:
4846:
4843:
4838:
4834:
4814:
4811:
4810:
4809:
4798:
4795:
4792:
4789:
4784:
4780:
4776:
4773:
4770:
4767:
4764:
4761:
4756:
4752:
4736:
4733:
4724:
4721:
4720:
4719:
4707:
4703:
4698:
4694:
4691:
4688:
4685:
4682:
4678:
4672:
4668:
4664:
4661:
4658:
4655:
4652:
4649:
4646:
4622:
4598:
4594:
4573:
4570:
4548:
4543:
4538:
4535:
4509:
4489:
4484:
4481:
4476:
4471:
4467:
4451:
4448:
4447:
4446:
4430:
4426:
4423:
4420:
4417:
4414:
4411:
4406:
4402:
4398:
4395:
4391:
4387:
4383:
4379:
4375:
4371:
4366:
4362:
4358:
4355:
4352:
4349:
4344:
4340:
4336:
4333:
4329:
4325:
4321:
4317:
4312:
4308:
4304:
4301:
4298:
4295:
4292:
4289:
4286:
4282:
4278:
4275:
4273:
4271:
4268:
4265:
4262:
4257:
4250:
4246:
4241:
4237:
4233:
4229:
4226:
4223:
4220:
4213:
4209:
4201:
4197:
4192:
4188:
4185:
4182:
4179:
4172:
4168:
4162:
4158:
4154:
4151:
4149:
4147:
4144:
4141:
4136:
4131:
4127:
4123:
4122:
4099:
4096:
4093:
4090:
4087:
4067:
4047:
4027:
4024:
4019:
4015:
4011:
4008:
4005:
4000:
3996:
3992:
3987:
3983:
3979:
3976:
3956:
3953:
3950:
3945:
3941:
3937:
3934:
3931:
3928:
3925:
3920:
3916:
3912:
3907:
3903:
3899:
3896:
3893:
3888:
3884:
3880:
3877:
3874:
3854:
3851:
3848:
3845:
3842:
3822:
3811:
3810:
3798:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3770:
3766:
3762:
3759:
3756:
3753:
3748:
3743:
3739:
3711:
3700:
3699:
3688:
3684:
3679:
3675:
3672:
3669:
3665:
3662:
3657:
3651:
3646:
3642:
3638:
3635:
3632:
3629:
3624:
3619:
3615:
3590:
3587:
3565:
3561:
3558:
3555:
3552:
3549:
3546:
3523:
3501:
3498:
3461:
3441:
3436:
3433:
3428:
3423:
3419:
3406:
3403:
3401:
3398:
3397:
3396:
3384:
3380:
3377:
3374:
3371:
3368:
3365:
3362:
3359:
3356:
3352:
3346:
3342:
3336:
3333:
3328:
3325:
3322:
3319:
3316:
3313:
3310:
3287:
3284:
3281:
3278:
3275:
3264:
3263:
3252:
3247:
3243:
3240:
3237:
3234:
3231:
3228:
3225:
3222:
3219:
3215:
3209:
3205:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3148:
3137:
3136:
3124:
3119:
3116:
3113:
3110:
3106:
3102:
3099:
3096:
3093:
3090:
3087:
3084:
3081:
3078:
3074:
3068:
3064:
3061:
3058:
3055:
3052:
3049:
3045:
3041:
3038:
3035:
3031:
3027:
3024:
3021:
3018:
3015:
3012:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2942:
2939:
2936:
2933:
2930:
2901:Main article:
2897:
2896:
2847:
2845:
2838:
2832:
2829:
2812:
2793:
2792:
2781:
2778:
2775:
2772:
2768:
2765:
2762:
2759:
2756:
2753:
2748:
2745:
2742:
2738:
2732:
2728:
2724:
2721:
2718:
2715:
2711:
2707:
2703:
2688:vector measure
2675:
2664:signed measure
2651:
2625:
2622:
2607:
2603:
2599:
2594:
2590:
2586:
2582:
2578:
2574:
2550:
2540:measurable set
2527:
2512:
2511:
2500:
2497:
2494:
2491:
2486:
2482:
2479:
2476:
2473:
2469:
2463:
2460:
2457:
2453:
2447:
2443:
2439:
2436:
2433:
2430:
2426:
2422:
2418:
2391:
2361:
2342:complex-valued
2329:
2317:
2314:
2302:
2301:
2288:
2284:
2280:
2275:
2271:
2267:
2263:
2259:
2255:
2244:
2231:
2227:
2223:
2218:
2214:
2210:
2207:
2178:
2174:
2151:
2147:
2135:
2134:
2123:
2119:
2116:
2113:
2110:
2107:
2102:
2098:
2092:
2089:
2086:
2083:
2080:
2075:
2071:
2060:
2049:
2045:
2042:
2039:
2036:
2033:
2028:
2024:
2018:
2015:
2012:
2009:
2004:
2000:
1966:
1963:
1962:
1961:
1950:
1947:
1944:
1940:
1936:
1932:
1928:
1925:
1922:
1919:
1901:
1900:
1889:
1886:
1883:
1880:
1875:
1871:
1868:
1865:
1862:
1859:
1854:
1850:
1843:
1839:
1836:
1833:
1830:
1827:
1824:
1819:
1815:
1809:
1806:
1803:
1800:
1796:
1792:
1788:
1764:
1742:
1741:
1730:
1727:
1724:
1721:
1717:
1714:
1711:
1708:
1705:
1700:
1696:
1690:
1687:
1684:
1681:
1678:
1675:
1672:
1669:
1664:
1660:
1643:
1642:
1631:
1628:
1625:
1622:
1617:
1613:
1610:
1607:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1574:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1547:
1543:
1530:
1519:
1516:
1513:
1510:
1505:
1501:
1498:
1495:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1462:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1435:
1431:
1397:
1394:
1391:
1388:
1385:
1380:
1376:
1353:
1350:
1347:
1344:
1341:
1336:
1332:
1306:
1303:
1300:
1297:
1294:
1271:
1261:signed measure
1252:
1249:
1247:
1244:
1225:
1220:
1215:
1212:
1194:
1193:
1177:
1167:
1146:
1143:
1140:
1135:
1131:
1126:
1121:
1111:
1099:
1090:contained in
1064:
1059:
1054:
1049:
1046:
1043:
1038:
1033:
1029:
1014:
1013:
1002:
998:
994:
991:
986:
983:
980:
975:
971:
966:
962:
959:
953:
950:
945:
940:
935:
932:
929:
924:
919:
915:
911:
908:
905:
902:
898:
893:
890:
887:
884:
881:
878:
875:
872:
869:
866:
861:
857:
852:
848:
845:
842:
839:
836:
833:
830:
827:
815:is defined as
772:
771:
754:September 2022
722:
720:
713:
707:
700:
687:
684:
677:
673:
668:
664:
661:
658:
655:
652:
647:
643:
639:
634:
630:
626:
623:
598:
594:
591:
588:
585:
582:
574:
571:
568:
561:
557:
552:
548:
545:
542:
537:
533:
529:
526:
523:
519:
515:
510:
492:runs over the
486:
485:
474:
470:
466:
461:
457:
453:
450:
447:
444:
439:
436:
433:
429:
425:
422:
418:
412:
409:
404:
400:
394:
391:
388:
384:
377:
372:
368:
365:
362:
359:
354:
349:
345:
320:
316:
313:
310:
307:
304:
301:
278:
248:
245:
243:
240:
225:Fourier series
219:in the paper (
217:Camille Jordan
212:
209:
135:
134:
49:
47:
40:
26:
9:
6:
4:
3:
2:
7913:
7902:
7899:
7898:
7896:
7886:
7885:0-89871-589-X
7882:
7878:
7874:
7873:
7868:
7865:
7864:
7859:
7855:
7851:
7847:
7843:
7839:
7835:
7831:
7826:
7825:
7820:
7816:
7812:
7808:
7804:
7800:
7795:
7794:
7790:
7785:on 2011-09-27
7784:
7780:
7776:
7775:
7769:
7768:
7760:
7756:
7753:
7750:
7746:
7743:
7737:
7736:
7731:
7725:
7724:
7723:
7722:
7716:
7712:
7709:
7708:
7707:
7706:
7699:
7695:
7691:
7690:
7689:
7688:
7675:
7671:
7667:
7663:
7659:
7655:
7654:Rudin, Walter
7651:
7648:
7647:Stefan Banach
7644:
7640:
7634:
7630:
7626:
7622:
7618:
7617:
7612:
7608:
7604:
7598:
7594:
7589:
7588:
7584:
7579:
7575:
7571:
7567:
7563:
7559:
7555:
7551:
7550:
7545:
7541:
7537:
7533:
7529:
7525:
7521:
7517:
7513:
7508:
7503:
7499:
7495:
7494:
7488:
7487:
7478:
7473:on 2009-03-31
7472:
7468:
7464:
7460:
7456:
7455:
7450:
7446:
7442:
7438:
7434:
7430:
7429:
7424:
7420:
7417:
7412:
7408:
7404:
7401:(in French),
7400:
7396:
7392:
7388:
7385:
7381:
7380:
7375:
7371:
7366:
7362:
7358:
7357:
7352:
7347:
7343:
7339:
7338:
7333:
7328:
7324:
7320:
7319:
7314:
7309:
7305:
7301:
7300:
7295:
7290:
7286:
7282:
7281:
7276:
7271:
7267:
7263:
7262:
7257:
7252:
7248:on 2007-08-07
7247:
7243:
7239:
7235:
7231:
7227:
7223:
7219:
7218:
7195:
7188:
7180:
7178:9780198502456
7174:
7170:
7169:
7161:
7154:
7151:According to
7148:
7144:
7136:
7133:
7125:
7122:February 2012
7115:
7111:
7105:
7104:
7098:
7093:
7084:
7083:
7073:
7070:
7068:
7065:
7063:
7060:
7058:
7055:
7053:
7050:
7048:
7045:
7043:
7040:
7038:
7035:
7034:
7025:
7021:
7017:
7013:
7012:
7007:
7003:
6999:
6995:
6991:
6987:
6984:
6981:
6980:
6975:
6971:
6968:
6967:
6966:
6964:
6960:
6956:
6952:
6948:
6944:
6941:
6937:
6933:
6930:
6920:
6900:
6897:
6886:
6867:
6858:
6850:
6846:
6842:
6836:
6828:
6824:
6820:
6815:
6812:
6804:
6794:
6793:
6792:
6790:
6774:
6766:
6747:
6735:
6732:
6726:
6717:
6714:
6708:
6702:
6695:
6694:
6693:
6666:
6663:
6655:
6651:
6647:
6642:
6640:
6636:
6632:
6628:
6627:
6622:
6618:
6615:, called the
6614:
6610:
6600:
6598:
6582:
6575:
6556:
6550:
6546:
6539:
6534:
6528:
6519:
6512:
6511:
6510:
6507:
6506:
6489:
6485:
6478:
6468:
6464:
6460:
6456:
6453:
6450:
6441:
6419:
6415:
6408:
6398:
6376:
6372:
6369:
6366:
6357:
6328:
6324:
6317:
6307:
6303:
6301:
6290:
6286:
6279:
6274:
6265:
6262:
6250:
6247:
6244:
6234:
6230:
6226:
6223:
6220:
6217:
6213:
6208:
6196:
6188:
6186:
6175:
6171:
6164:
6159:
6150:
6147:
6138:
6134:
6131:
6128:
6125:
6121:
6105:
6102:
6099:
6089:
6077:
6069:
6067:
6057:
6052:
6048:
6044:
6041:
6038:
6029:
6017:
6010:
5998:
5997:
5996:
5980:
5976:
5953:
5948:
5944:
5921:
5916:
5912:
5889:
5885:
5864:
5842:
5837:
5833:
5809:
5805:
5798:
5793:
5779:
5776:
5773:
5754:
5750:
5747:
5744:
5741:
5737:
5726:
5723:
5718:
5714:
5704:
5689:
5664:
5660:
5653:
5643:
5639:
5635:
5631:
5624:
5620:
5616:
5612:
5608:
5598:
5594:
5590:
5586:
5580:
5576:
5566:
5561:
5557:
5554:
5548:
5544:
5534:
5530:
5527:
5523:
5519:
5516:
5513:
5504:
5496:
5495:
5494:
5472:
5466:
5462:
5452:
5448:
5445:
5441:
5437:
5434:
5431:
5422:
5414:
5400:
5393:
5389:
5378:
5373:
5362:
5358:
5355:
5350:
5345:
5336:
5332:
5323:
5314:
5306:
5292:
5289:
5284:
5279:
5270:
5266:
5257:
5254:
5251:
5244:
5240:
5229:
5224:
5213:
5205:
5191:
5188:
5184:
5178:
5173:
5168:
5164:
5156:
5152:
5137:
5129:
5115:
5112:
5108:
5103:
5099:
5095:
5091:
5088:
5079:
5071:
5070:
5069:
5034:
5005:
5001:
4996:
4992:
4988:
4975:
4971:
4967:
4962:
4958:
4954:
4950:
4947:
4938:
4930:
4929:
4928:
4913:
4909:
4906:
4873:
4856:
4852:
4844:
4841:
4832:
4824:
4823:
4822:
4820:
4796:
4793:
4790:
4778:
4774:
4771:
4768:
4765:
4762:
4759:
4750:
4742:
4741:
4740:
4732:
4730:
4705:
4696:
4689:
4683:
4676:
4666:
4662:
4653:
4650:
4644:
4637:
4636:
4635:
4633:
4620:
4596:
4592:
4546:
4536:
4526:
4523:
4520:defined on a
4507:
4469:
4465:
4421:
4415:
4412:
4404:
4400:
4393:
4385:
4381:
4377:
4364:
4360:
4353:
4350:
4342:
4338:
4331:
4323:
4310:
4306:
4299:
4296:
4290:
4284:
4276:
4274:
4263:
4255:
4248:
4244:
4239:
4235:
4231:
4227:
4221:
4211:
4207:
4199:
4195:
4190:
4186:
4180:
4170:
4166:
4160:
4156:
4152:
4150:
4142:
4134:
4129:
4125:
4113:
4112:
4111:
4094:
4091:
4088:
4065:
4045:
4025:
4022:
4017:
4013:
4009:
4006:
4003:
3998:
3994:
3990:
3985:
3981:
3977:
3974:
3951:
3948:
3943:
3939:
3932:
3929:
3926:
3918:
3914:
3910:
3905:
3901:
3894:
3886:
3882:
3878:
3875:
3849:
3846:
3843:
3820:
3789:
3783:
3780:
3774:
3768:
3760:
3754:
3746:
3741:
3737:
3729:
3728:
3727:
3725:
3709:
3686:
3670:
3663:
3660:
3649:
3644:
3640:
3636:
3630:
3622:
3617:
3613:
3605:
3604:
3603:
3588:
3585:
3559:
3553:
3550:
3547:
3537:
3521:
3514:
3510:
3497:
3495:
3494:
3489:
3488:
3483:
3479:
3475:
3459:
3421:
3417:
3382:
3375:
3369:
3366:
3360:
3354:
3350:
3344:
3340:
3334:
3331:
3326:
3320:
3317:
3314:
3308:
3301:
3300:
3299:
3282:
3279:
3276:
3250:
3245:
3238:
3232:
3229:
3223:
3217:
3213:
3207:
3203:
3199:
3193:
3190:
3187:
3181:
3174:
3173:
3172:
3170:
3166:
3162:
3146:
3122:
3114:
3111:
3108:
3104:
3097:
3091:
3088:
3082:
3076:
3072:
3066:
3059:
3056:
3050:
3039:
3036:
3033:
3025:
3019:
3016:
3013:
3003:
3002:
3001:
2987:
2984:
2978:
2969:
2966:
2963:
2937:
2934:
2931:
2920:
2919:
2914:
2910:
2904:
2893:
2890:
2882:
2872:
2868:
2864:
2858:
2857:
2853:
2848:This section
2846:
2842:
2837:
2836:
2828:
2826:
2810:
2803:of the space
2802:
2798:
2776:
2773:
2760:
2754:
2746:
2743:
2740:
2736:
2730:
2722:
2716:
2705:
2693:
2692:
2691:
2689:
2673:
2665:
2649:
2641:
2640:
2635:
2631:
2621:
2605:
2601:
2597:
2592:
2588:
2584:
2576:
2562:
2548:
2541:
2525:
2517:
2495:
2492:
2477:
2471:
2461:
2458:
2455:
2451:
2445:
2437:
2431:
2420:
2408:
2407:
2406:
2405:
2389:
2381:
2373:
2359:
2351:
2347:
2343:
2327:
2313:
2311:
2307:
2286:
2282:
2278:
2273:
2269:
2265:
2257:
2245:
2229:
2225:
2221:
2216:
2212:
2208:
2205:
2198:
2197:
2196:
2194:
2176:
2172:
2149:
2145:
2121:
2114:
2111:
2108:
2100:
2090:
2087:
2081:
2073:
2069:
2061:
2047:
2040:
2037:
2034:
2016:
2010:
2002:
1998:
1990:
1989:
1988:
1986:
1983:
1979:
1975:
1971:
1945:
1934:
1926:
1920:
1910:
1909:
1908:
1906:
1884:
1881:
1873:
1866:
1863:
1860:
1852:
1841:
1837:
1831:
1828:
1825:
1807:
1801:
1790:
1778:
1777:
1776:
1762:
1754:
1751:(also called
1750:
1725:
1722:
1712:
1709:
1706:
1698:
1688:
1685:
1682:
1676:
1673:
1670:
1648:
1647:
1646:
1626:
1623:
1615:
1611:
1608:
1605:
1594:
1591:
1588:
1582:
1576:
1572:
1565:
1559:
1556:
1553:
1545:
1531:
1514:
1511:
1503:
1499:
1496:
1493:
1482:
1479:
1476:
1470:
1464:
1460:
1453:
1447:
1444:
1441:
1419:
1418:
1417:
1416:, as follows
1415:
1411:
1392:
1389:
1386:
1378:
1348:
1345:
1342:
1320:
1319:set functions
1298:
1295:
1285:
1269:
1262:
1258:
1243:
1241:
1223:
1213:
1203:
1199:
1191:
1175:
1168:
1165:
1162:
1129:
1112:
1089:
1085:
1082:
1078:
1057:
1047:
1036:
1031:
1027:
1019:
1018:
1017:
1000:
996:
992:
989:
969:
960:
951:
943:
933:
922:
917:
913:
909:
906:
903:
900:
888:
882:
879:
876:
870:
864:
855:
850:
843:
834:
831:
825:
818:
817:
816:
814:
810:
806:
802:
798:
795:belonging to
794:
790:
786:
782:
768:
765:
757:
747:
743:
739:
733:
732:
728:
723:This section
721:
717:
712:
711:
705:
699:
685:
682:
675:
671:
666:
662:
659:
656:
653:
650:
645:
641:
637:
632:
628:
624:
621:
613:
610:of the given
596:
589:
586:
583:
572:
569:
559:
555:
550:
546:
543:
540:
535:
531:
524:
521:
517:
513:
499:
495:
491:
472:
459:
455:
448:
445:
437:
434:
431:
427:
420:
410:
407:
402:
398:
392:
389:
386:
382:
366:
360:
352:
347:
343:
335:
334:
333:
314:
308:
305:
302:
292:
276:
269:
265:
261:
257:
239:
237:
233:
230:
229:discontinuous
226:
222:
218:
208:
206:
205:
200:
196:
192:
188:
184:
180:
176:
172:
169:
166:
162:
158:
154:
150:
146:
142:
131:
128:
120:
117:February 2012
109:
106:
102:
99:
95:
92:
88:
85:
81:
78: –
77:
73:
72:Find sources:
66:
60:
59:
55:
50:This article
48:
44:
39:
38:
33:
19:
7871:
7867:Tony F. Chan
7833:
7829:
7802:
7798:
7783:the original
7773:
7765:Applications
7733:
7720:
7719:
7704:
7703:
7687:One variable
7686:
7685:
7657:
7615:
7592:
7553:
7547:
7497:
7491:
7471:the original
7458:
7452:
7427:
7402:
7398:
7377:
7354:
7335:
7316:
7297:
7278:
7259:
7246:the original
7233:
7229:
7201:. Retrieved
7187:
7167:
7160:
7147:
7128:
7119:
7100:
7009:
6985:
6977:
6969:
6929:non-negative
6926:
6923:Applications
6884:
6882:
6764:
6762:
6653:
6649:
6645:
6643:
6638:
6634:
6624:
6613:Banach space
6606:
6571:
6508:
6348:
5968:is dense in
5705:
5680:
5492:
5025:
4893:
4816:
4738:
4726:
4612:
4453:
3812:
3701:
3508:
3503:
3491:
3485:
3408:
3265:
3138:
2916:
2906:
2885:
2876:
2861:Please help
2849:
2800:
2794:
2637:
2627:
2563:
2513:
2404:set function
2379:
2374:
2319:
2309:
2303:
2136:
1982:non-positive
1978:non-negative
1968:
1904:
1902:
1752:
1748:
1743:
1644:
1413:
1409:
1254:
1197:
1195:
1015:
812:
808:
804:
800:
796:
792:
788:
780:
775:
760:
751:
736:Please help
724:
703:
487:
255:
250:
214:
202:
198:
194:
190:
186:
178:
170:
144:
138:
123:
114:
104:
97:
90:
83:
71:
51:
7405:: 228–230,
7199:. p. 7
7114:introducing
7042:p-variation
6940:real-valued
4927:, we have:
4038:) in which
3482:functionals
2797:Rudin (1966
2634:Rudin (1966
2372:as follows
2350:Rudin (1966
1240:bounded set
785:open subset
242:Definitions
221:Jordan 1881
165:real-valued
141:mathematics
7749:PlanetMath
7698:PlanetMath
7674:0142.01701
7633:0017.30004
7625:63.0183.05
7578:0014.29605
7562:62.0247.03
7532:0008.00602
7516:59.0285.01
7484:References
7467:39.0101.05
7437:48.0261.09
7423:Hahn, Hans
7411:13.0184.01
7242:36.0491.02
7097:references
6936:functional
4455:Theorem 2.
3505:Theorem 1.
2514:where the
2344:i.e. is a
2195:such that
1970:Saks (1937
1257:Saks (1937
1255:Following
1190:divergence
498:partitions
488:where the
87:newspapers
54:references
7735:MathWorld
7461:: 75–92,
7447:(1908) ,
7384:EMS Press
7372:(2001) ,
7361:EMS Press
7342:EMS Press
7323:EMS Press
7304:EMS Press
7285:EMS Press
7266:EMS Press
6943:functions
6904:Σ
6851:−
6847:μ
6825:μ
6809:‖
6805:μ
6802:‖
6775:φ
6730:∞
6727:−
6718:μ
6703:φ
6675:→
6667::
6664:φ
6544:∇
6532:∇
6529:−
6523:→
6520:φ
6483:∇
6473:Ω
6469:∫
6465:≤
6461:φ
6457:
6446:Ω
6442:∫
6413:∇
6403:Ω
6399:∫
6377:φ
6373:
6362:Ω
6358:∫
6322:∇
6312:Ω
6308:∫
6284:∇
6272:∇
6266:⋅
6260:∇
6248:≠
6242:∇
6235:∩
6218:−
6209:∫
6203:∞
6200:→
6169:∇
6157:∇
6151:⋅
6145:∇
6126:−
6103:≠
6097:∇
6090:∫
6084:∞
6081:→
6058:∗
6049:θ
6045:
6034:Ω
6030:∫
6024:∞
6021:→
5886:θ
5865:ε
5843:∗
5834:θ
5803:∇
5791:∇
5777:≠
5771:∇
5742:−
5727:−
5715:θ
5690:φ
5658:∇
5648:Ω
5644:∫
5640:≤
5629:∇
5621:⋅
5613:φ
5603:Ω
5599:∫
5595:≤
5584:∇
5581:⋅
5577:φ
5571:Ω
5567:∫
5558:≤
5552:∇
5549:⋅
5545:φ
5539:Ω
5535:∫
5531:−
5524:φ
5520:
5509:Ω
5505:∫
5470:∇
5467:⋅
5463:φ
5457:Ω
5453:∫
5449:−
5442:φ
5438:
5427:Ω
5423:∫
5385:∂
5374:φ
5367:Ω
5363:∫
5359:−
5346:φ
5328:∂
5319:Ω
5315:∫
5280:φ
5262:∂
5236:∂
5225:φ
5218:Ω
5214:∫
5174:φ
5148:∂
5142:Ω
5138:∫
5104:φ
5092:
5084:Ω
5080:∫
5056:Ω
5035:φ
5006:⋅
4997:φ
4983:Ω
4980:∂
4976:∫
4963:φ
4951:
4943:Ω
4939:∫
4914:φ
4874:⋅
4864:Ω
4861:∂
4857:∫
4845:
4837:Ω
4833:∫
4817:From the
4797:φ
4794:⋅
4788:∇
4783:Ω
4779:∫
4775:−
4769:φ
4766:
4755:Ω
4751:∫
4681:∇
4671:Ω
4667:∫
4657:Ω
4584:of class
4572:Ω
4569:∂
4537:⊆
4534:Ω
4500:function
4483:¯
4480:Ω
4413:−
4382:⋯
4351:−
4297:−
4232:⋯
4007:⋯
3930:…
3781:−
3724:monotonic
3641:∫
3560:⊂
3452:function
3435:¯
3432:Ω
3370:ν
3367:−
3355:μ
3341:∑
3321:ν
3315:μ
3309:δ
3233:ν
3230:−
3218:μ
3204:∑
3194:ν
3188:μ
3182:δ
3118:Σ
3115:∈
3092:ν
3089:−
3077:μ
3040:ν
3037:−
3034:μ
3023:‖
3020:ν
3017:−
3014:μ
3011:‖
2970:ν
2967:−
2964:μ
2941:‖
2938:ν
2935:−
2932:μ
2929:‖
2850:does not
2780:Σ
2777:∈
2771:∀
2767:‖
2755:μ
2752:‖
2747:π
2744:∈
2737:∑
2731:π
2706:μ
2674:μ
2650:μ
2606:−
2602:μ
2589:μ
2577:μ
2526:π
2499:Σ
2496:∈
2490:∀
2472:μ
2462:π
2459:∈
2452:∑
2446:π
2421:μ
2390:μ
2380:variation
2360:μ
2328:μ
2287:−
2283:μ
2270:μ
2258:μ
2230:−
2226:μ
2222:−
2213:μ
2206:μ
2177:−
2173:μ
2146:μ
2115:⋅
2109:μ
2101:_
2091:−
2082:⋅
2074:−
2070:μ
2041:⋅
2035:μ
2027:¯
2011:⋅
1999:μ
1935:μ
1924:‖
1921:μ
1918:‖
1888:Σ
1885:∈
1879:∀
1861:μ
1853:_
1826:μ
1818:¯
1791:μ
1763:μ
1749:variation
1729:Σ
1726:∈
1720:∀
1707:μ
1699:_
1689:≥
1683:≥
1671:μ
1663:¯
1630:Σ
1627:∈
1621:∀
1609:⊂
1598:Σ
1595:∈
1589:∣
1577:μ
1554:μ
1546:_
1518:Σ
1515:∈
1509:∀
1497:⊂
1486:Σ
1483:∈
1477:∣
1465:μ
1442:μ
1434:¯
1393:⋅
1387:μ
1379:_
1349:⋅
1343:μ
1335:¯
1302:Σ
1270:μ
1214:⊆
1211:Ω
1200:that the
1192:operator.
1142:Ω
1134:∞
1125:‖
1120:‖
1098:Ω
1045:Ω
990:≤
982:Ω
974:∞
965:‖
961:ϕ
958:‖
931:Ω
910:∈
907:ϕ
904::
883:ϕ
880:
860:Ω
856:∫
838:Ω
725:does not
570:∣
544:…
446:−
408:−
383:∑
315:⊂
266:-valued)
183:arclength
7895:Category
7858:18276250
7656:(1966),
7613:(1937).
7542:(1936),
7425:(1921),
7393:(1881),
7031:See also
6963:minimize
6934:-valued
6621:ba space
6617:ca space
6574:function
4525:open set
4457:Given a
3664:′
3589:′
3536:interval
3478:supremum
3474:integral
2879:May 2012
2516:supremum
2193:measures
1903:and its
1645:clearly
1016:where
612:interval
490:supremum
291:interval
268:function
175:interval
163:. For a
157:function
153:codomain
7838:Bibcode
7807:Bibcode
7666:0210528
7570:1556778
7524:1501718
7416:Gallica
7203:8 April
7110:improve
7018:) and (
7006:sensing
4611:, the
4561:, with
4522:bounded
3480:of the
2871:removed
2856:sources
2402:is the
1985:measure
1188:is the
1159:is the
1075:is the
803:), the
746:removed
731:sources
496:of all
264:complex
236:bounded
197:), for
161:measure
101:scholar
7883:
7856:
7672:
7664:
7631:
7623:
7599:
7583:Numdam
7576:
7568:
7560:
7530:
7522:
7514:
7465:
7435:
7409:
7240:
7175:
7099:, but
6996:in an
6957:, and
6865:
6745:
6505:Q.E.D.
5026:where
3967:(with
2915:, the
1980:and a
1202:domain
955:
783:be an
143:, the
103:
96:
89:
82:
74:
7696:" on
7197:(PDF)
7078:Notes
6998:image
6994:noise
6988:: in
4735:Lemma
4723:Proof
4078:over
3511:of a
2686:is a
2662:is a
2642:when
2632:(see
2538:of a
2137:Then
1282:on a
1166:, and
258:of a
159:or a
155:of a
149:local
108:JSTOR
94:books
7881:ISBN
7877:SIAM
7854:PMID
7779:SIAM
7597:ISBN
7205:2017
7173:ISBN
6932:real
6637:and
6609:norm
6572:The
5825:and
4023:<
4010:<
4004:<
3991:<
3978:<
3507:The
3490:and
2854:any
2852:cite
2378:The
2164:and
1747:The
1412:and
1364:and
1164:norm
779:Let
729:any
727:cite
663:<
651:<
638:<
260:real
254:The
80:news
7846:doi
7815:doi
7757:at
7747:at
7713:at
7670:Zbl
7629:Zbl
7621:JFM
7574:Zbl
7558:JFM
7528:Zbl
7512:JFM
7502:doi
7463:JFM
7433:JFM
7407:JFM
7238:JFM
7008:. "
7004:or
6791:by
6692:by
6454:div
6370:div
6193:lim
6074:lim
6042:div
6014:lim
5904:in
5517:div
5435:div
5089:div
4948:div
4842:div
4763:div
3702:If
3493:1.2
3487:1.1
3063:sup
2865:by
2727:sup
2639:1.3
2442:sup
2340:is
1569:inf
1457:sup
1176:div
1086:of
1079:of
1077:set
877:div
847:sup
811:in
807:of
787:of
740:by
494:set
371:sup
227:of
139:In
56:to
7897::
7879:,
7875:,
7852:,
7844:,
7832:,
7813:,
7803:60
7801:,
7791:).
7777:,
7751:..
7732:.
7668:,
7662:MR
7627:.
7572:,
7566:MR
7564:,
7546:,
7526:,
7520:MR
7518:,
7510:,
7498:35
7496:,
7459:43
7451:,
7403:92
7397:,
7382:,
7376:,
7359:,
7353:,
7340:,
7334:,
7321:,
7315:,
7302:,
7296:,
7283:,
7277:,
7264:,
7258:,
7234:IX
7228:,
7026:).
6953:,
6919:.
6641:.
6503:.
5724::=
4907::=
4821::
4731:.
3496:.
2827:.
2312:.
2308:,
1242:.
844::=
698:.
207:.
189:↦
177:⊂
67:.
7861:.
7848::
7840::
7834:7
7822:.
7817::
7809::
7741:.
7738:.
7700:.
7692:"
7677:.
7649:.
7635:.
7606:.
7585:.
7554:5
7535:.
7504::
7479:.
7440:.
7364:.
7345:.
7326:.
7307:.
7288:.
7269:.
7250:.
7207:.
7181:.
7155:.
7135:)
7129:(
7124:)
7120:(
7106:.
6982:"
6907:)
6901:,
6898:X
6895:(
6885:μ
6868:,
6862:)
6859:X
6856:(
6843:+
6840:)
6837:X
6834:(
6829:+
6821:=
6816:V
6813:T
6765:μ
6748:.
6742:)
6739:]
6736:t
6733:,
6724:(
6721:(
6715:=
6712:)
6709:t
6706:(
6679:R
6671:R
6654:μ
6650:μ
6646:R
6639:ν
6635:μ
6583:f
6557:.
6551:|
6547:f
6540:|
6535:f
6490:|
6486:f
6479:|
6451:f
6420:|
6416:f
6409:|
6367:f
6329:|
6325:f
6318:|
6304:=
6291:|
6287:f
6280:|
6275:f
6263:f
6254:}
6251:0
6245:f
6239:{
6231:]
6227:N
6224:,
6221:N
6214:[
6197:N
6189:=
6176:|
6172:f
6165:|
6160:f
6148:f
6139:]
6135:N
6132:,
6129:N
6122:[
6116:I
6109:}
6106:0
6100:f
6094:{
6078:N
6070:=
6053:N
6039:f
6018:N
5981:1
5977:L
5954:1
5949:c
5945:C
5922:1
5917:c
5913:C
5890:N
5838:N
5810:|
5806:f
5799:|
5794:f
5783:}
5780:0
5774:f
5768:{
5763:I
5755:]
5751:N
5748:,
5745:N
5738:[
5732:I
5719:N
5665:|
5661:f
5654:|
5636:|
5632:f
5625:|
5617:|
5609:|
5591:|
5587:f
5562:|
5555:f
5528:=
5514:f
5473:f
5446:=
5432:f
5401:f
5394:i
5390:x
5379:i
5356:=
5351:i
5337:i
5333:x
5324:f
5293:0
5290:=
5285:i
5271:i
5267:x
5258:f
5255:+
5252:f
5245:i
5241:x
5230:i
5192:0
5189:=
5185:)
5179:i
5169:f
5165:(
5157:i
5153:x
5116:0
5113:=
5109:)
5100:f
5096:(
5010:n
5002:)
4993:f
4989:(
4972:=
4968:)
4959:f
4955:(
4910:f
4903:R
4878:n
4870:R
4853:=
4849:R
4791:f
4772:=
4760:f
4718:.
4706:x
4702:d
4697:|
4693:)
4690:x
4687:(
4684:f
4677:|
4663:=
4660:)
4654:,
4651:f
4648:(
4645:V
4621:f
4597:1
4593:C
4547:n
4542:R
4508:f
4488:)
4475:(
4470:1
4466:C
4429:|
4425:)
4422:b
4419:(
4416:f
4410:)
4405:N
4401:a
4397:(
4394:f
4390:|
4386:+
4378:+
4374:|
4370:)
4365:2
4361:a
4357:(
4354:f
4348:)
4343:1
4339:a
4335:(
4332:f
4328:|
4324:+
4320:|
4316:)
4311:1
4307:a
4303:(
4300:f
4294:)
4291:a
4288:(
4285:f
4281:|
4277:=
4267:)
4264:f
4261:(
4256:b
4249:N
4245:a
4240:V
4236:+
4228:+
4225:)
4222:f
4219:(
4212:2
4208:a
4200:1
4196:a
4191:V
4187:+
4184:)
4181:f
4178:(
4171:1
4167:a
4161:a
4157:V
4153:=
4146:)
4143:f
4140:(
4135:b
4130:a
4126:V
4098:]
4095:b
4092:,
4089:a
4086:[
4066:f
4046:f
4026:b
4018:N
4014:a
3999:2
3995:a
3986:1
3982:a
3975:a
3955:]
3952:b
3949:,
3944:N
3940:a
3936:[
3933:,
3927:,
3924:]
3919:2
3915:a
3911:,
3906:1
3902:a
3898:[
3895:,
3892:]
3887:1
3883:a
3879:,
3876:a
3873:[
3853:]
3850:b
3847:,
3844:a
3841:[
3821:f
3797:|
3793:)
3790:b
3787:(
3784:f
3778:)
3775:a
3772:(
3769:f
3765:|
3761:=
3758:)
3755:f
3752:(
3747:b
3742:a
3738:V
3710:f
3687:x
3683:d
3678:|
3674:)
3671:x
3668:(
3661:f
3656:|
3650:b
3645:a
3637:=
3634:)
3631:f
3628:(
3623:b
3618:a
3614:V
3586:f
3564:R
3557:]
3554:b
3551:,
3548:a
3545:[
3522:f
3460:f
3440:)
3427:(
3422:1
3418:C
3383:|
3379:)
3376:x
3373:(
3364:)
3361:x
3358:(
3351:|
3345:x
3335:2
3332:1
3327:=
3324:)
3318:,
3312:(
3286:]
3283:1
3280:,
3277:0
3274:[
3251:.
3246:|
3242:)
3239:x
3236:(
3227:)
3224:x
3221:(
3214:|
3208:x
3200:=
3197:)
3191:,
3185:(
3147:2
3123:}
3112:A
3109::
3105:|
3101:)
3098:A
3095:(
3086:)
3083:A
3080:(
3073:|
3067:{
3060:2
3057:=
3054:)
3051:X
3048:(
3044:|
3030:|
3026:=
2988:0
2985:=
2982:)
2979:X
2976:(
2973:)
2961:(
2892:)
2886:(
2881:)
2877:(
2873:.
2859:.
2811:X
2774:E
2764:)
2761:A
2758:(
2741:A
2723:=
2720:)
2717:E
2714:(
2710:|
2702:|
2598:+
2593:+
2585:=
2581:|
2573:|
2549:E
2493:E
2485:|
2481:)
2478:A
2475:(
2468:|
2456:A
2438:=
2435:)
2432:E
2429:(
2425:|
2417:|
2279:+
2274:+
2266:=
2262:|
2254:|
2217:+
2209:=
2150:+
2122:,
2118:)
2112:,
2106:(
2097:W
2088:=
2085:)
2079:(
2048:,
2044:)
2038:,
2032:(
2023:W
2017:=
2014:)
2008:(
2003:+
1949:)
1946:X
1943:(
1939:|
1931:|
1927:=
1882:E
1874:|
1870:)
1867:E
1864:,
1858:(
1849:W
1842:|
1838:+
1835:)
1832:E
1829:,
1823:(
1814:W
1808:=
1805:)
1802:E
1799:(
1795:|
1787:|
1723:E
1716:)
1713:E
1710:,
1704:(
1695:W
1686:0
1680:)
1677:E
1674:,
1668:(
1659:W
1624:E
1616:}
1612:E
1606:A
1592:A
1586:)
1583:A
1580:(
1573:{
1566:=
1563:)
1560:E
1557:,
1551:(
1542:W
1512:E
1504:}
1500:E
1494:A
1480:A
1474:)
1471:A
1468:(
1461:{
1454:=
1451:)
1448:E
1445:,
1439:(
1430:W
1396:)
1390:,
1384:(
1375:W
1352:)
1346:,
1340:(
1331:W
1305:)
1299:,
1296:X
1293:(
1224:n
1219:R
1145:)
1139:(
1130:L
1110:,
1063:)
1058:n
1053:R
1048:,
1042:(
1037:1
1032:c
1028:C
1001:,
997:}
993:1
985:)
979:(
970:L
952:,
949:)
944:n
939:R
934:,
928:(
923:1
918:c
914:C
901:x
897:d
892:)
889:x
886:(
874:)
871:x
868:(
865:f
851:{
841:)
835:,
832:f
829:(
826:V
813:Ω
809:f
801:Ω
799:(
797:L
793:f
789:R
781:Ω
767:)
761:(
756:)
752:(
748:.
734:.
704:n
686:b
683:=
676:P
672:n
667:x
660:.
657:.
654:.
646:1
642:x
633:0
629:x
625:=
622:a
597:}
593:]
590:b
587:,
584:a
581:[
573:P
567:}
560:P
556:n
551:x
547:,
541:,
536:0
532:x
528:{
525:=
522:P
518:{
514:=
509:P
473:,
469:|
465:)
460:i
456:x
452:(
449:f
443:)
438:1
435:+
432:i
428:x
424:(
421:f
417:|
411:1
403:P
399:n
393:0
390:=
387:i
376:P
367:=
364:)
361:f
358:(
353:b
348:a
344:V
319:R
312:]
309:b
306:,
303:a
300:[
277:f
199:x
195:x
193:(
191:f
187:x
179:R
171:f
130:)
124:(
119:)
115:(
105:·
98:·
91:·
84:·
61:.
34:.
20:)
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