Calculus How To

Bounded Variation

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A function of bounded variation (also called a BV function) “wiggles” or oscillates between bounds, much in the same way that a sine function wiggles between bounds of 1 and -1; The vertical (up and down movement) of these functions is restricted over an interval. In other words, the variation isn’t infinite: we can calculate a value for it.


These functions can be described as integrable functions with a derivative (in the sense of distributions) that is a signed measure with finite total variation [1]. The concept was originally developed in the context of Fourier series [2], when mathematicians were trying to prove the series convergence.

Examples of Functions of Bounded Variation

All monotonic functions and absolutely continuous functions are of bounded variation; Real‐valued functions with a variation on a compact interval can be expressed as the difference between two monotone (non-decreasing) functions [3], called a Jordan decomposition. Interestingly, these functions do not have to be continuous functions and can have a finite number of discontinuities (although they do have to be Riemann integrable). They can also be approximated by finite step functions, or decomposed to part continuous and part jump.

Normalized functions can be described as having bounded variation when on the interval [0,1] with h(0) = 0 and h(c) = h(c + 0) for 0 < c < 1.

More formally, a real-valued function α of bounded variation on the closed interval [a, b] has a constant M > 0 such that [4]:
bounded variation definition

It’s not always necessary to specific the interval, especially when the interval in question is obvious [5].


References

[1] Ziemer W.P. (1989) Functions of Bounded Variation. In: Weakly Differentiable Functions. Graduate Texts in Mathematics, vol 120. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1015-3_5
[2] Monteiro, G. et al. Series in Real Analysis. Volume 15-Kurzweil–Stieltjes Integral: Theory and Applications. World Scientific.
[3] Bridges, D. (2016). A Constructive Look at Functions of BV. Bulletin of the London Mathematical Society. Volume 32, Issue 3 p. 316-324
[4] Bridges, D. (2016). Functions of Bounded Variation. retrieved April 8, 2021 from:
http://www.math.ubc.ca/~feldman/m321/variation.pdf
[5] Functions of BV. Retrieved April 8, 2021 from: https://www.diva-portal.org/smash/get/diva2:5850/FULLTEXT01.pdf

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Stephanie Glen. "Bounded Variation" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/bounded-variation/
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