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Questions tagged [bounded-variation]

For questions about functions $f$ defined on an interval $[a,b]$ such that there exists a constant $M>0$, such that if $a=x_0<x_1<\ldots<x_n=b$, $n\in\mathbb N^*$, then we have $\sum_{k=1}^n|f(x_k)-f(x_{k-1})|\leq M$.

This concept can be generalized to infinite intervals, requiring that the constant is uniform.

Let $[a,b]$ be a closed interval. A function $f\colon [a,b]\to \mathbb R$ is said to be of bounded variation if $$\sup_{n\in\mathbb N_{>0}}\sup_{a=x_0\lt x_1\lt\ldots\lt x_n=b}\sum_{j=1}^n|f(x_j)-f(x_{j-1})|<\infty.$$ We denote by $TV(f):=\sup_{n\in\mathbb N_{>0}}\sup_{a=x_0\lt x_1\lt\ldots\lt x_n=b}\sum_{j=1}^n|f(x_j)-f(x_{j-1})|$ the total variation of $f$, and we can endow the vector space of functions of bounded variation with the norm $\lVert f\rVert_{BV}:=TV(f)+|f(a)|$.

838 questions
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proof continuity of bounded variation functions

I would like to show that if $g:[a,b]\rightarrow R$ is continuous and has a bounded variation, then total Variation function is also continuous.
user562689
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Is this equivalent to bounded variation?

Let $\Omega \subset \mathbb{R}^n$ be open and bounded with smooth boundary. For $f \in L^1(\Omega)$, define $$ \|D_1 f\|_M(\Omega) =\inf\left\{\liminf_{k\to\infty}\int_\Omega |\nabla f_k|\,dx \mid f_k \to f \text{ in } L^1(\Omega),\ f_k \in \text{…
Stefan Smith
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Absolute continuity implies bounded variation

Let $f$ be absolutely continuous on $[a,b]$. I want to prove that $f$ is of bounded variation. I am reading Royden and Fitzpatrick so they use the following notations: Let $P=\{x_0=a,x_1,\ldots,x_{n-1},x_n=b\}$ be a partition of $[a,b]$. Then…
Laars Helenius
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Bounded variation condition if $f(0)=0$ and $f(x) =x^{\alpha}\sin(x^{-\beta})$ for $x\in(0,1]$

Let $f:[0,1] \rightarrow \Bbb R$ with $f(x) = \begin{cases} x^{\alpha}\sin(x^{-\beta})\;\; \text{if}\; x \neq 0 \\ f(x) =0 \;\;\text{if}\; x =0\end{cases}$ Find out for which value of$\;\alpha, \beta$ $f(x)$ would become of bounded variation on…
Daschin
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$f(z)=\sum_{k=0}^\infty a_kz^{k}$ with $\sum |a_k|<\infty$ is of bounded variation on circle $|z|=1$

Let $f(z)=\sum_{k=0}^\infty a_kz^{k}$ be a power series. Show that if $\sum |a_k|<\infty$, then $f(z)$ is of bounded variation on every radius of the circle $|z|=1$. (If, e.g., the radius is $0\leq x\leq 1$ and the $a_k$ are real, then $f(x)=\sum…
Valent
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Reconstruction of a function of bounded variation

The variation of a function $f:[a,b]\to\mathbb R$ is defined by \begin{align*} \text{Var}(f,[a,b]):=\sup_P\sum_{j=1}^n|f(t_{j-1})-f(t_j)|, \end{align*} where $P$ runs through all partitions $P=(a=t_0<\ldots
sranthrop
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Lebesgue decomposition of an increasing function

This problem asserts that any increasing function $f$ on $[a,b]$ can be decomposed as $$F=F_A+F_C+F_J,$$ with $F_A$ is absolutely continuous, $F_J$ is a jump function, and $F_C$ is a singular function, i.e., it is continuous and $F'_C=0$, a.e..…
user119758
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Are smooth bounded-variation functions in the Sobolev Space $W_{1,1}$?

Let $\Omega \subseteq \mathbb{R}^n$ open and $f \in BV(\Omega) \cap C^{\infty}(\Omega)$. Now I would like to prove that $f \in W_1^1(\Omega)$. I know that the distributional derivates of $f$ are the classical derivates, but I don't find a way to…
Matthis Stresemann
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How to prove that f/g is bounded variation?

I am trying to prove that when $f$ and $g$ are of bounded variation on $[a, b]$, $f/g$ is of bounded variation on [a, b] if there exists an $\varepsilon\gt 0$ such that $|g(x)|\ge \varepsilon$ for $x\in [a, b]$. Let $\Gamma$ be a partition of $[a,…
Danny_Kim
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Is there a function on $[a,b]$ that has a bounded derivative on $(a,b)$, is NOT continuous at $a$ and $b$, and is NOT of bounded variation on $[a,b]$?

It is well known that a continuous function on a compact interval $[a,b]$ that has a bounded derivative on $(a,b)$ is of bounded variation on $[a,b]$. I am curious that whether the continuity at the endpoints $a$ and $b$ is essential for the…
Jason
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Prove that the product of two BV([a,b]) functions is a bounded variation function (proof idea)

Show that if $f,g \in BV([a,b]) $ then $$fg \in BV([a,b]) $$ My idea was to write $f = f_1 - f_2$ and $g = g_1 - g_2$ (which are non decreasing functions) and write $fg$ as: $$ fg = (f_1 g_1 + f_2 g_2 ) - (f_1 g_2 + f_2 g_1)$$ which are two non…
Julian Vené
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There exists $(z_n)_{n\geq 1}$ monotone $z_n\to x_0$ such that $\sum |\phi(z_i)-\phi(z_{i-1})|=\infty$

I'm reading "Measure and integral" Wheeden & Zygmund. One of its exercises on bounded variation, asked me to prove : If $V[\phi; a,b]=\infty$. Show there exists a point $z\in[a,b]$ and a monotone sequence $(z_n)_{n\geq 1}$ such that $z_n\to z$ and…
Valent
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Can I make a BV function right-continuous this way?

Math people: This question is related to how can you "fix" one of the definitions of a BV function of one variable? . Suppose $f \in BV([0,1])$. I really have two-three questions. The second only can be answered if the answer to the first is…
Stefan Smith
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Bounded variation as a limit of sequence of partitions

Lemma 1 There is a sequence of partitions $(\Gamma_k)$, where each $\Gamma_{k+1}$ is a refinement of $\Gamma_k$, such that $$ \lim_{k\to\infty}S_{\Gamma_k}=V[f;a,b]. $$ Here there is a similar statement, we call it Lemma 2, without the word…
Mohamed Ali
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$f,g$ be of bounded variation $\Rightarrow$ $f\circ g$ is of bounded variation?

$f,g$ be of bounded variation $\Rightarrow$ $f \circ g$ is of bounded variation? if not, any counter example?
delog
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