Tagged Questions

1
vote
1answer
44 views

Show $\left| \int_a^b f d\alpha \right| \le \int_a^b|f|dV$ if $V$ is variation of $\alpha$ on $[a, b]$

Let $\alpha$ be of bounded variation on $[a, b]$ and assume that $V(x)$ be the total variation of $\alpha$ on $[a, x]$, $a<x\le b$ and $V(a) = 0$. Let $f$ be defined and bounded on $[a, b]$. If ...
1
vote
0answers
66 views

Riemann integrals with respect to bounded variation.

Let $\alpha$ be of bounded variation on $[a,b]$. let $V(x)$ be the total variation of $\alpha$ on $[a,x]$ and let $V(a)=0$. If $f \in R(\alpha)$, prove that $f \in R(V)$? Can anyone help me with the ...
12
votes
5answers
1k views

Question about Riemann integral and total variation

Let $g$ be Riemann integrable on $[a,b]$, $f(x)=\int_a^x g(t)dt $ for $x \in[a,b]$. Can I show that the total variation of $f$ is equal to $\int_a^b |g(x)| dx $?
4
votes
1answer
135 views

Does the total variation of a function bound its numerical integration error, much like its first derivative?

When estimating the convergence of a Riemann sum to its integral, or equivalently the error in numerical integration, the commonly used bound is by upper bounding it's first derivative (see, for ...
3
votes
1answer
229 views

Functions of bounded variation on all $\mathbb{R}$

Consider $F:\mathbb{R}\rightarrow\mathbb{R}$ such that $\sup_{a,b}T_F (a,b)<\infty$ where $T_F (a,b)$ is the total variation of $F$ on the interval $[a,b]$. Then we have i) ...