For questions about functions $f$ defined on an interval $[a,b]$ such that there exists a constant $M>0$, such that if $a=x_0

learn more… | top users | synonyms

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 $?
3
votes
1answer
202 views

Weak derivatives

How can I prove that $|\nabla u|=|\nabla|u||$ when $u$ is regular enough for example Lipschitz or $W^{1,1}_{loc}$. Other question is about the pointwise derivative when $f:[0,1]\to R$ is BV is that ...
4
votes
1answer
733 views

Is the total variation function uniform continuous or continuous?

I have been doing some excercises on total variation when the following questions came up to my mind: (1) Let $f$ be continuous on the interval $[0,1]$ and be of bounded variation. Is it true that ...
2
votes
4answers
98 views

Variation of periodic function

If $f\colon\Bbb R\to\Bbb R$ then $\operatorname{var}(f, [a,b]):=\sup \{\sum_{k=1}^n |f(x_k)-f(x_{k-1})| \}$, where supremum is taken over all finite sequences $(x_k)$ such that ...
3
votes
1answer
1k views

Bounded variation, difference of two increasing functions

Prove that if $f$ is of bounded variation in $[a,b]$, it is the difference of two positive, monotonic increasing functions; and the difference of two bounded monotonic increasing functions is a ...
0
votes
0answers
71 views

About regularity of the maximal operator of Hardy-Littlewood

What difficulties arise when you consider the centered maximal operator, such that you can't prove that it maps BV into BV? Does someone have some reference which that maps BV into BV? PS: The ...
1
vote
1answer
140 views

looking for a Poincare-type lemma for BV functions

Given a smooth bounded open subset $\Omega$ of $\mathbb{R}^n$, does there exist $A >0$ such that if $f\in BV(\Omega)$ with zero trace on $\partial \Omega$, and $\int_\Omega |Df| = 1$, then ...
2
votes
0answers
99 views

Showing uniqueness

I have a problem which asks to show that a function $f$ of bounded variation can be expressed uniquely except for addition of constants as the sum of an absolutely continuous function and a singular ...
2
votes
1answer
245 views

BV functions and absolute continuity short inquiry

This comes as a complement to: Relation between total variation and absolute continuity; I was wondering if the following holds: Let $F$ be a function of bounded variation on $[a,b]$, then ...
3
votes
2answers
494 views

Exercise from Stein again - characterization of BV functions

I find this pretty hard and it would be awesome if someone could help me. The problem is the following (Problem 6/Chapter 3 from S&S's Real Analysis). Suppose $F$ is a bounded measurable ...
3
votes
1answer
781 views

Total Variation and Integral

Let $f$ be a function of bounded variation on $[a, b]$ and $T_{a}^{b}(f)$ its total variation. We do not assume that $f$ is continuous. Show that $$\int_{a}^{b}|f'(t)|\, dt \leq T_{a}^{b}(f).$$ ...
4
votes
1answer
215 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
3answers
323 views

Total variation of (weakly) differentiable functions

the total variation of a function $u\in L^1(\Omega)$, $\Omega\subset \mathbb{R}^n$, can be defined as $$ \sup \{ \int_\Omega u \; \mathrm{div} g \; dx:\; g \in C_c^1(\Omega,\mathbb{R}^n), \; \lvert ...
3
votes
1answer
259 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) ...
5
votes
1answer
1k views

Prove the normed space of bounded variation functions is complete

Let $\Vert f \Vert = |f(0)| + \mathrm{Var}f$ for all $f \in BV([0,1])$; we are given that it is a norm. Show that $BV([0,1])$ is a complete normed space with this norm. I have shown that any ...
4
votes
0answers
922 views

Integral of the derivative of a function of bounded variation

Let $f\colon [a,b] \to \mathbb R$ be of bounded variation. Must it be the case that $|\int_a ^b f' (x) |\leq |TV(f)|$, where $TV(f)$ is the total variation of $f$ over $[a,b]$? If so, how can one ...
1
vote
1answer
95 views

Please help me find the maxima of this expression

I want to find $p$ which maximizes the given functional. $p$ is a function of the form $\mathbb{R}^2 \to \mathbb{R}$. $\Omega$ is a region in the 2-d plane. $\underset{p}{\sup} \int_\Omega \{ ...
2
votes
0answers
104 views

Perturbations of a function of bounded variation

Suppose $f,g:[0,1]\rightarrow[0,1]$, $g>0$, and $f/g$ is of bounded variation. If $f_n, f \in C^1[0,1]$ and $f_n \rightarrow f$ in $C^1$. Does it follow that $\exists N$ such that $f_n/g$ is of ...