Category:Definitions/Bounded Variation
This category contains definitions related to Bounded Variation.
Related results can be found in Category:Bounded Variation.
Definition
Closed Bounded Interval
For each finite subdivision $P$ of $\closedint a b$, write:
- $P = \set {x_0, x_1, \ldots, x_n}$
with:
- $a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$
Also write:
- $\ds \map {V_f} {P ; \closedint a b} = \sum_{i \mathop = 1}^n \size {\map f {x_i} - \map f {x_{i - 1} } }$
We say $f$ is of bounded variation if and only if there exists an $M \ge 0$ such that:
- $\map {V_f} {P ; \closedint a b} \le M$
for all finite subdivisions $P$.
Closed Unbounded Interval
For each finite non-empty subset $\mathcal S$ of $I$, write:
- $\mathcal S = \set {x_0, x_1, \ldots, x_n}$
with:
- $x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n$
Also write:
- $\ds \map {V_f^\ast} {\mathcal S; I} = \sum_{i \mathop = 1}^n \size {\map f {x_i} - \map f {x_{i - 1} } }$
for $n \ge 1$, and $\map {V_f^\ast} {\mathcal S ; I} = 0$ otherwise.
We say $f$ is of bounded variation if and only if there exists an $M \ge 0$ such that:
- $\map {V_f^\ast} {\mathcal S; I} \le M$
for all finite non-empty subsets $\mathcal S$ of $I$.
Also known as
Some sources say that $f$ is of finite variation rather than bounded variation.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): bounded variation
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): bounded variation
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): variation: 2.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): bounded variation
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): variation: 2.
Subcategories
This category has the following 2 subcategories, out of 2 total.
B
D
Pages in category "Definitions/Bounded Variation"
The following 7 pages are in this category, out of 7 total.
B
- Definition:Bounded Variation
- Definition:Bounded Variation/Also known as
- Definition:Bounded Variation/Closed Bounded Interval
- Definition:Bounded Variation/Closed Bounded Interval/Definition 1
- Definition:Bounded Variation/Closed Bounded Interval/Definition 2
- Definition:Bounded Variation/Closed Unbounded Interval