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Bounded Variation


A function f(x) is said to have bounded variation if, over the closed interval x in [a,b], there exists an M such that

 |f(x_1)-f(a)|+|f(x_2)-f(x_1)|+... 
 +|f(b)-f(x_(n-1))|<=M
(1)

for all a<x_1<x_2<...<x_(n-1)<b.

The space of functions of bounded variation is denoted "BV," and has the seminorm

 Phi(f)=supintf(dphi)/(dx),
(2)

where phi ranges over all compactly supported functions bounded by -1 and 1. The seminorm is equal to the supremum over all sums above, and is also equal to int|df/dx|dx (when this expression makes sense).

BoundedVariation

On the interval [0,1], the function x^2sin(1/x) (purple) is of bounded variation, but xsin1/x (red) is not. More generally, a function f is locally of bounded variation in a domain U if f is locally integrable, f in L_(loc)^1, and for all open subsets W, with compact closure in U, and all smooth vector fields g compactly supported in W,

 int_Wfdivgdx<=c(W)sup|g|,
(3)

div denotes divergence and c is a constant which only depends on the choice of W and f.

Such functions form the space BV_(loc)(U). They may not be differentiable, but by the Riesz representation theorem, the derivative of a BV_(loc)-function f is a regular Borel measure Df. Functions of bounded variation also satisfy a compactness theorem.

Given a sequence f_n of functions in BV_(loc)(U), such that

 sup_(n)[||f_n||_(L^1(W))+int_W|Df_n|dx]<infty,
(4)

that is the total variation of the functions is bounded, in any compactly supported open subset W, there is a subsequence f_(n_k) which converges to a function f in BV_(loc) in the topology of L_(loc)^1. Moreover, the limit satisfies

 int_W|Df|dx<=lim infint_W|Df_(n_k)|dx.
(5)

They also satisfy a version of Poincaré's lemma.


See also

Compact Support, Differentiable, Weakly Differentiable

Portions of this entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd and Weisstein, Eric W. "Bounded Variation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BoundedVariation.html

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