Product of Functions of Bounded Variation is of Bounded Variation
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Theorem
Let $a, b$ be real numbers with $a < b$.
Let $f, g : \closedint a b \to \R$ be functions of bounded variation.
Let $\map {V_f} {\closedint a b}$ and $\map {V_g} {\closedint a b}$ be the total variations of $f$ and $g$ respectively.
Then the pointwise product $f \cdot g$ is of bounded variation with:
- $\map {V_{f \cdot g} } {\closedint a b} \le A \map {V_f} {\closedint a b} + B \map {V_g} {\closedint a b}$
where:
- $\map {V_{f \cdot g} } {\closedint a b}$ denotes the total variation of $f \cdot g$ on $\closedint a b$
- $A, B$ are non-negative real numbers.
Proof
For each finite subdivision $P$ of $\closedint a b$, write:
- $P = \set {x_0, x_1, \ldots, x_n }$
with:
- $a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$
By Function of Bounded Variation is Bounded:
- $f$ and $g$ are bounded.
So, there exists $A, B \in \R$ such that:
- $\size {\map f x} \le B$
- $\size {\map g x} \le A$
for all $x \in \closedint a b$.
Then:
\(\ds \map {V_{f \cdot g} } {P ; \closedint a b}\) | \(=\) | \(\ds \sum_{i \mathop = 1}^n \size {\map {\paren {f \cdot g} } {x_i} - \map {\paren {f \cdot g} } {x_{i - 1} } }\) | using the notation from the definition of bounded variation | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^n \size {\map f {x_i} \map g {x_i} - \map f {x_{i - 1} } \map g {x_{i - 1} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^n \size {\map f {x_i} \map g {x_i} - \map f {x_{i - 1} } \map g {x_i} + \map f {x_{i - 1} } \map g {x_i} - \map f {x_{i - 1} } \map g {x_{i - 1} } }\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \sum_{i \mathop = 1}^n \size {\map f {x_i} \map g {x_i} - \map f {x_{i - 1} } \map g {x_i} } + \sum_{i \mathop = 1}^n \size {\map f {x_{i - 1} } \map g {x_i} - \map f {x_{i - 1} } \map g {x_{i - 1} } }\) | Triangle Inequality | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^n \size {\map g {x_i} } \size {\map f {x_i} - \map f {x_{i - 1} } } + \sum_{i \mathop = 1}^n \size {\map f {x_{i - 1} } } \size {\map g {x_i} - \map g {x_{i - 1} } }\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds A \sum_{i \mathop = 1}^n \size {\map f {x_i} - \map f {x_{i - 1} } } + B \sum_{i \mathop = 1}^n \size {\map g {x_i} - \map g {x_{i - 1} } }\) | since $\size {\map g {x_i} } \le A$ and $\size {\map f {x_{i - 1} } } \le B$ | |||||||||||
\(\ds \) | \(=\) | \(\ds A \map {V_f} {P ; \closedint a b} + B \map {V_g} {P ; \closedint a b}\) |
Since $f$ and $g$ are of bounded variation, there exists $M, K \in \R$ such that:
- $\map {V_f} {P ; \closedint a b} \le M$
- $\map {V_g} {P ; \closedint a b} \le K$
for all finite subdivisions $P$.
We therefore have:
- $\map {V_{f \cdot g} } {P ; \closedint a b} \le A M + B K$
so $f \cdot g$ is of bounded variation.
Further, we have:
\(\ds \map {V_{f \cdot g} } {\closedint a b}\) | \(=\) | \(\ds \sup_P \paren {\map {V_{f \cdot g} } {P ; \closedint a b} }\) | Definition of Total Variation of Real Function on Closed Bounded Interval | |||||||||||
\(\ds \) | \(\le\) | \(\ds \sup_P \paren {A \map {V_f} {P ; \closedint a b} } + \sup_P \paren {B \map {V_g} {P ; \closedint a b} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds A \sup_P \paren {\map {V_f} {P ; \closedint a b} } + B \sup_P \paren {\map {V_g} {P ; \closedint a b} }\) | Multiple of Supremum | |||||||||||
\(\ds \) | \(=\) | \(\ds A \map {V_f} {\closedint a b} + B \map {V_g} {\closedint a b}\) | Definition of Total Variation of Real Function on Closed Bounded Interval |
$\blacksquare$
Sources
- 1973: Tom M. Apostol: Mathematical Analysis (2nd ed.) ... (previous) ... (next): $\S 6.4$: Total Variation: Theorem $6.9$