Editing Bounded variation (section)

=== BV functions of one variable ===
{{EquationRef|1|Definition 1.1.}} The '''[[total variation]]''' of a [[real number|real]]-valued (or more generally [[complex number|complex]]-valued) [[function (mathematics)|function]] ''f'', defined on an [[interval (mathematics)|interval]] <math>[a,b] \subset \mathbb{R}</math> is the quantity

:<math> V_a^b(f)=\sup_{P \in \mathcal{P}} \sum_{i=0}^{n_{P}-1} | f(x_{i+1})-f(x_i) |. \,</math>

where the [[supremum]] is taken over the set <math display="inline"> \mathcal{P} =\left\{P=\{ x_0, \dots , x_{n_P}\} \mid P\text{ is a partition of } [a, b]\text{ satisfying } x_i\leq x_{i+1}\text{ for } 0\leq i\leq n_P-1 \right\} </math> of all [[partition of an interval|partitions]] of the interval considered.

If ''f'' is [[derivative|differentiable]] and its derivative is Riemann-integrable, its total variation is the vertical component of the [[arc length|arc-length]] of its graph, that is to say,

:<math> V_a^b(f) = \int _a^b |f'(x)|\,\mathrm{d}x.</math>

{{EquationRef|2|Definition 1.2.}} A  real-valued function <math> f </math> on the [[real line]] is said to be of '''bounded variation''' ('''BV function''') on a chosen [[interval (mathematics)|interval]] <math>[a,b] \subset \mathbb{R}</math>  if its total variation is finite, ''i.e.''
:<math> f \in \operatorname{BV}([a,b]) \iff V_a^b(f) < +\infty </math>

It can be proved that a real function <math>f</math> is of bounded variation in <math>[a,b]</math> if and only if it can be written as the difference <math>f=f_1-f_2</math> of two non-decreasing functions <math>f_1</math> and <math>f_2</math> on <math>[a,b]</math>: this result is known as the [https://www.encyclopediaofmath.org/index.php/Jordan_decomposition_(of_a_function) Jordan decomposition of a function] and it is related to the [[Hahn decomposition theorem#Jordan measure decomposition|Jordan decomposition of a measure]].

Through the [[Stieltjes integral]], any function of bounded variation on a closed interval <math>[a, b]</math> defines a [[bounded linear functional]] on <math>C([a, b])</math>. In this special case,<ref>See for example {{harvtxt|Kolmogorov|Fomin|1969|pp=374–376}}.</ref> the [[Riesz–Markov–Kakutani representation theorem]] states that every bounded linear functional arises uniquely in this way. The normalized positive functionals or [[probability measure]]s correspond to positive non-decreasing lower [[semicontinuous function]]s. This point of view has been important in
[[spectral theory]],<ref>For a general reference on this topic, see {{harvtxt|Riesz|Szőkefalvi-Nagy|1990}}</ref> in particular in its application to [[spectral theory of ordinary differential equations|ordinary differential equations]].