Is there a good reference for facts and theorems about BV real valued functions? I’m looking for something with much more than say Stein and Shakarchi 3, or Evans and Gariepy. Thanks!
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2$\begingroup$ What kind of result are you interested in? in 1D the theory of BV functions essentially reduces to the theory of monotone functions, which is pretty restrictive $\endgroup$– Piero D'AnconaCommented Jan 14, 2019 at 8:56
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$\begingroup$ Just everything there might be to know about them. Does BV pretty much imply monotone? $\endgroup$– James BaxterCommented Jan 14, 2019 at 9:42
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$\begingroup$ @JamesBaxter: BV means "difference of two increasing". (There's a lovely old book in Polish on that subject, by Łojasiewicz. I do not thing it has been translated into English, though). $\endgroup$– Mateusz KwaśnickiCommented Jan 14, 2019 at 12:27
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$\begingroup$ The following book is worth looking at if you're interested in classical results (i.e. early to mid 20th century): Rangachary Kannan and Carole King Krueger, Advanced Analysis on the Real Line, Universitext series, Springer-Verlag, 1996, x + 259 pages. $\endgroup$– Dave L RenfroCommented Jan 15, 2019 at 20:24
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2 Answers
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The following book has a nice and long chapter of BV functions in one variable:
G. Leoni, A First Course in Sobolev Spaces Graduate Studies in Mathematics Volume: 181.
Also an excellent book covering a great deal of the material is:
I. P. Natanson, Theory of functions of a real variable. Translated by Leo F. Boron with the collaboration of Edwin Hewitt. Frederick Ungar Publishing Co., New York, 1955.
answered Jan 15, 2019 at 13:28
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The book Gordon, Russell A., The integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Mathematics. 4. Providence, RI: American Mathematical Society (AMS). xi, 395 p. (1994). ZBL0807.26004. has a lot of material on BV functions in one variable and various generalizations of BV.
answered Jan 17, 2019 at 9:07