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- Bounded_variation abstract "In mathematical analysis, a function of bounded variation, also known as a BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the y-axis, neglecting the contribution of motion along x-axis, traveled by a point moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function (which is a hypersurface in this case), but can be every intersection of the graph itself with a hyperplane (in the case of functions of two variables, a plane) parallel to a fixed x-axis and to the y-axis.Functions of bounded variation are precisely those with respect to which one may find Riemann–Stieltjes integrals of all continuous functions.Another characterization states that the functions of bounded variation on a closed interval are exactly those f which can be written as a difference g − h, where both g and h are bounded monotone.In the case of several variables, a function f defined on an open subset of ℝn is said to have bounded variation if its distributional derivative is a finite vector Radon measure.One of the most important aspects of functions of bounded variation is that they form an algebra of discontinuous functions whose first derivative exists almost everywhere: due to this fact, they can and frequently are used to define generalized solutions of nonlinear problems involving functionals, ordinary and partial differential equations in mathematics, physics and engineering. Considering the problem of multiplication of distributions or more generally the problem of defining general nonlinear operations on generalized functions, function of bounded variation are the smallest algebra which has to be embedded in every space of generalized functions preserving the result of multiplication.".
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- Bounded_variation author "Rowland, Todd and Weisstein, Eric W.".
- Bounded_variation author2Link "Anatolii Georgievich Vitushkin".
- Bounded_variation first "Anatolii G.".
- Bounded_variation first "Boris I.".
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- Bounded_variation id "6969".
- Bounded_variation id "V/v096110".
- Bounded_variation last "Golubov".
- Bounded_variation last "Vitushkin".
- Bounded_variation title "BV function".
- Bounded_variation title "Bounded Variation".
- Bounded_variation title "Variation of a function".
- Bounded_variation urlname "BoundedVariation".
- Bounded_variation subject Category:Calculus_of_variations.
- Bounded_variation subject Category:Measure_theory.
- Bounded_variation subject Category:Real_analysis.
- Bounded_variation comment "In mathematical analysis, a function of bounded variation, also known as a BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the y-axis, neglecting the contribution of motion along x-axis, traveled by a point moving along the graph has a finite value.".
- Bounded_variation label "Beschränkte Variation".
- Bounded_variation label "Bounded variation".
- Bounded_variation label "Fonction à variation bornée".
- Bounded_variation label "Funzione a variazione limitata".
- Bounded_variation label "有界变差".
- Bounded_variation sameAs Beschränkte_Variation.
- Bounded_variation sameAs Fonction_à_variation_bornée.
- Bounded_variation sameAs Funzione_a_variazione_limitata.
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