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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.

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Convergence in Total Variation vs implies Uniform Convergence

Let $(f_n)_{n\in \mathbb{N}}$ and $f$ real function defined on an interval $[a,b]$ s.t. $f, f_n$ are bounded variation function on $I$, and suppose that $TV(f_n) \to TV(f)$. I should prove that it ...
Shiva's user avatar
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22 views

Convergence of the derivative of a BV function in sense of measures

Suppose I have a smooth sequence $f_{n} : \mathbb{R} \to \mathbb{R}$ with $f_{n} \to f$ strongly in $L^{1}_{loc}(\mathbb{R})$ and $f \in BV_{loc}(\mathbb{R})$. Is there any way to justify that $\...
duelspace's user avatar
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0 answers
40 views

Integration with respect to finite Radon measure

Let $ u \in BV( \mathbb{R}^N). $ We know that $$ \int_{\mathbb{R}^N} \left|Du\right| = \sup\left\{ \int_{\mathbb{R}^N} u\ div \varphi\ dx,\ \varphi \in C_c^1( \mathbb{R}^N, \mathbb{R}^N),\ \left|\...
SemiMath's user avatar
  • 177
2 votes
1 answer
23 views

Equality of the $L^1$ norm and measure norm of an corresponding absolutely continuous measure for BV functions

Let $u \in BV(I;\mathbb{R}^d)$ be a vector valued function of bounded variation, (in particular Bochner integrable) on an open interval $I \subset \mathbb{R}$. We can then define the Radon measure $u \...
ThommyAC's user avatar
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1 answer
22 views

Sequences of bounded variation have convergent series

A sequence $\{a_n\}$ has bounded variation if $\sum_{k=1}^\infty |a_{k+1}-a_k|$ converges. I am trying to prove that, if $\{a_n\}$ has bounded variation then $\sum_{k=1}^\infty a_n$ converges. This ...
Addem's user avatar
  • 5,706
13 votes
2 answers
377 views

Prove that if $\{a_{k}\}$ is a sequence of real numbers such that $\sum_{k=1}^{\infty} \frac{|a_{k}|}{k} = \infty$,

Prove that if $\{a_{k}\}$ is a sequence of real numbers such that $$\sum_{k=1}^{\infty} \frac{|a_{k}|}{k} = \infty$$ and $$\sum_{n=1}^{\infty} \left( \sum_{k=2^{n-1}}^{2^n-1} k(a_k - a_{k+1})^2 \right)...
Martin.s's user avatar
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1 vote
0 answers
52 views

Equality involving an SBV (Special Bounded Variation) function and an $L^{\infty}$ function

The notations are mainly those of Functions of Bounded Variation and Free Discontinuity Problems by Ambrosio, Fusco and Pallara. Hi, In a problem I am considering, I have reached the following ...
C.G.'s user avatar
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0 answers
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Continuous bounded variation path that fails to be Lipschitz

Let $C^{1\text{-var}}([0,T];\mathbb{R}^d)$ denote the space of continuous bounded variation paths taking values in $\mathbb{R}^d$. Similarly, let $C^{1\text{-Höl}}([0,T];\mathbb{R}^d)$ denote the ...
Oscar's user avatar
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Connection between variation of a function and its weighted integral

The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan: My question is that how Eq. D.10 is derived from the previous one? (The ...
Ali's user avatar
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2 votes
1 answer
62 views

Is a measure, whose distributional derivative is a measure, absolutely continuous wrt Lebesgue?

Suppose $\mu$ is a finite Borel measure on $\mathbb{R}^n$ with the property that it's distributional gradient $\nabla\mu$ is a vector-valued finite Borel measure. Does it follows then that $\mu$ ...
Kiliroy's user avatar
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2 votes
0 answers
43 views

A question regarding bounded variation and differentialibility as well as integration

Let $a, b \in \mathbb{R}$ with $a < b$, and $f: [a, b] \to \mathbb{R}$ be a monotonically increasing, right-continuous function. Show that there exists a monotonically increasing function $g: [a, b]...
MathGeek's user avatar
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1 answer
49 views

Determine bounded variation and absolute continuity for different parameters

Determine for which parameters $\alpha, \beta \in [0, +\infty[$ the function $$f_{\alpha,\beta}:[0,1] \to \mathbb{R}, \quad f_{\alpha, \beta}(x) := \begin{cases} x^{\alpha}\ \text{sin}(\frac{1}{x^{\...
MathGeek's user avatar
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0 answers
29 views

Smooth approximation of BV functions, Proof clarification

Im reading Evans&Gariepy book measure theory and fine properties of functions second edition. I have a question about the proof of theorem 5.3 that I don't understand. The theorem states that for ...
Franlezana's user avatar
1 vote
0 answers
121 views

Understanding finitely additive (signed) measures

I am currently trying to understand finitely additive (signed) measures, mostly reading Dunford & Schwarz and Rao & Rao. Knowing only the normal measure theory, the finitely additive measures ...
guest1's user avatar
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0 answers
70 views

Is the measure of continuous injective image of $\mathbb{S}^1\rightarrow\mathbb{R}^2$ always 0? [duplicate]

Summarized: There are various conditions under which the image of a one-dimensional manifold has measure 0 in $\mathbb{R}^2$. These can be formulated in different little lemmas which are usually easy ...
cnikbesku's user avatar
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1 vote
1 answer
53 views

Relationship between $p$ and $q$ variation.

Let $T>0$, $p>0$ and $f: [0,T]\to \mathbb{R}$ continuous. The $p$-variation of $f$ on $[0,T]$ is given by:$$\operatorname{Var}^p(f)_T:=\lim\limits_{n\to\infty}\sum \limits_{k=1}^{2^n}\left|f(k/2^...
math_undergrad_questions's user avatar
2 votes
1 answer
28 views

Function of bounded variation in terms of a sequence

Let $(a_n)$ be a sequence of positive numbers and define $$f(x) = \begin{cases}a_n, \text{ if } x = 1/n, \ \ n \geq 1\\ 0, \text{otherwise}\end{cases}.$$Show that $f$ is of bounded variation on $[0,1]$...
user57's user avatar
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0 votes
1 answer
65 views

Equality of continuous functions of bounded and unbounded variations [closed]

Suppose $f:[0,1] \to \mathbb{R}$ is continuous and has infinite total variation on any interval $[a,b] \subset [0,1]$ with $a <b$. Suppose $g$ is a function of bounded variation on $[0,1]$ and is ...
Soumya Ganguly's user avatar
3 votes
1 answer
35 views

For an arbitrary continuous function $f$, is the Stieltjes integral $\int_0^1(df(x))^3=0$?

Suppose $f:[0,1]\to\mathbb R$ is continuous, possibly with unbounded variation. We consider sums of the form $$\sum_{i=1}^n\Big(f(x_i)-f(x_{i-1})\Big)^3$$ where $0=x_0<x_1<x_2<\cdots<x_{n-...
mr_e_man's user avatar
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5 votes
2 answers
278 views

Bounding $L^1$ norm of the difference between a function $f:\mathbb R^n\to\mathbb R$ of bounded variation and a piecewise constant approximation

As a follow up to this question, which deals with univariate functions, I assume that we are given a function $f:\mathbb R^n\to\mathbb R$ which is of bounded variation on bounded sets, meaning, ...
Václav Mordvinov's user avatar
3 votes
1 answer
72 views

If $f$ is bounded variation then $|f|$ is bounded variation

I want to show that if $f$ is of bounded variation in $[a, b]$ then $|f|$ is of bounded variation in $[a, b]$. Suppose that $f$ is of bounded variation then if $P=\{x_{0}, \dots, x_{n} \}$ is a ...
Wrloord's user avatar
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1 vote
0 answers
44 views

Bounded Variation In Compact set

Let $D$ be any finite collection interval on $E$ and function $f:E\to\mathbb{R}$. If $E$ is a compact set, show that $$V(f,E)=\sup\{V(f,D) \mid \text{for } D \text{ is any finite collection interval ...
CPKTNWT's user avatar
  • 11
3 votes
1 answer
82 views

Naturalness of definition of line integral

Let $I = [a,b]$ denote some interval, and $f : I \rightarrow \mathbb{C}$ a continuous function of bounded variation, in other words, $f$ is a parameterisation of $\gamma = f(I)$, a rectifiable curve. ...
porridgemathematics's user avatar
2 votes
2 answers
126 views

Bounded variation of functions and sequences

I would like to know if there is a relation between bounded variation of functions and bounded variation in sequences. I know one can not directly correlate the concepts but if we consider the Fourier ...
Hap's user avatar
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1 vote
0 answers
111 views

How pathological can a function of bounded variation be?

For $f: [0,1] \to [0,1] $, Define $A_f = \{y : |\{f^{-1}(y)\}| > \aleph_0\}$. It is known that $A_f$ must have null measure if $f$ is continuous and of bounded variation. This thread explains the ...
shyb's user avatar
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6 votes
1 answer
182 views

Approximating a function of bounded variation function by a stepfunction obtained by averaging over measurable sets

Inspired by this question, suppose that we have a function $f\in\mathrm{BV}([0,1])\cap L^1([0,1])$, take a measurable partition $\mathcal P^N$ of $[0,1]=\bigcup_{i=1}^NI_i$, and obtain a new function $...
Václav Mordvinov's user avatar
4 votes
1 answer
133 views

Proof request: Extending vector valued function of unit outer normals to the whole space $\Bbb R^n$.

I was wondering if anyone could provide a proof or a reference of the following proposition: Let $C$ be an open subset of $\mathbb{R}^n$ with $C^2$ boundary so that the unit outer normal vector $v(x)$ ...
Franlezana's user avatar
3 votes
1 answer
196 views

$\sum_{n=1}^\infty 2^{-n} f_n$ has unbounded variation on every $[a,b] \subset [0,1]$ with $a < b$

Consider $n\in \Bbb N$, and define a piecewise linear function $f_n: [0,1] \to \Bbb R$ so that the graph of $f_n$ contains the points $\{(k \cdot 2^{-n}, (-1)^k)\}_{0\le k\le 2^n}$. Define $f: =\sum_{...
stoic-santiago's user avatar
2 votes
1 answer
60 views

Why if p-th variation is finite, then q-th variation is also finite for q>p?

Consider function $L^p(X)$ and $L^q(X)$ space with $p<q$ where $X$ is a finite measure space. Then from Holder, I have $L^q(X)$ embedding into $L^p(X)$. In particular, if $\|f\|_q<\infty$, then $...
user45765's user avatar
  • 8,520
0 votes
0 answers
145 views

What is the positive variation of a function?

I am having trouble understanding what the positive variation of a function $\frac{ 1}{2} ( T_F + F)$ represents. (Where, in this case, $T_F$ is the total variation of $F$ from $[0,x]$.) For consider $...
kam's user avatar
  • 302
0 votes
0 answers
71 views

How to show $f(x) = \begin{cases} \frac{\sin(x)}{x}, & \text{when } x \neq 0,\\ 0, & \text{when } x = 0. \end{cases}$ is of bounded variation?

Prove or disprove: $f(x) = \begin{cases} \frac{\sin(x)}{x}, & \text{when } x \neq 0,\\ 0, & \text{when } x = 0. \end{cases}$ is of bounded variation on $[0,1]$ and $\sup_{p \in P} V(f,p)...
Mahmoud albahar's user avatar
0 votes
0 answers
91 views

Let $f$ be a real-valued, everywhere differentiable function on $[0,1]$. Suppose $f \in BV[0,1]$. Prove $f \in AC[0,1]$. [duplicate]

I tried to prove this by showing that the derivative $f'$ must be bounded. But I am not sure this is true, as the Extreme Value Theorem cannot be invoked ($f'$ is not guaranteed to be continuous) nor ...
kam's user avatar
  • 302
4 votes
0 answers
91 views

Riemann-Stieljes integration and Total variation

Let $\alpha:[a,b]\to\mathbb{R}$ be a function of bounded variation on $[a,b]$ and $f:[a,b]\to\mathbb{R}$ a bounded function. It is well known that if $f$ is Riemann-Stieljes integrable respect to $\...
Tito Eliatron's user avatar
2 votes
1 answer
91 views

$L^1$ equicontinuity for functions with uniformly bounded total variation

I've been trying to prove that the space $BV[0, 1]$ of elements of $L^1 [0, 1]$ with bounded variation, is compactly embedded in $L^1[0, 1]$, using the Fréchet-Kolmogorov theorem. For context, a ...
KCJV's user avatar
  • 186
1 vote
1 answer
64 views

Total variation of a class in $L^1 [0, 1]$ and lower semicontinuity

Consider a function $f \in \mathcal L^1 [0, 1]$: we define the total variation of $f$ as usual by $$ V_0^1 (f) = \sup \sum_{k = 0}^{n - 1} | f(x_{k + 1}) - f(x_k) |, $$ where the supremum is taken ...
KCJV's user avatar
  • 186
1 vote
0 answers
35 views

boundedness of signed measures

Let us consider signed charges, these are finitely additive signed measures (I suppose this would also work with sigma additive signed measures). We work on a measure space $(\Omega, \mathcal{A})$, ...
guest1's user avatar
  • 351
1 vote
0 answers
51 views

How to prove by contradiction that a function of bounded variation is bounded

I would like to prove that a function defined $[a,b]$ of bounded variation is bounded by contradiction : suppose that $f$ is of bounded variation while note bounded and come up with a contradiction. ...
G2MWF's user avatar
  • 1,339
1 vote
0 answers
31 views

Decay rate of a functional norm.

I’ve been working on a project lately that has resulted in needing a following fact: if $\chi_{B_r(x)}$ denotes the characteristic function of the ball of radius $r$ about $x$ in $\mathbb{R}^2$, then ...
Nick's user avatar
  • 468
0 votes
1 answer
30 views

A norm-decreasing and right-continuous mapping $f: [0,1] \to H$ is of bounded-variation with respect to the norm of $H$?

Let $H$ be a separable Hilbert space and $f:[0,1] \to H$ be a norm-decreasing map in the sense that \begin{equation} \lVert f(t_2) \rVert_H \leq \lVert f(t_1) \rVert_H \end{equation} if $0 \leq t_1 \...
Keith's user avatar
  • 7,715
1 vote
0 answers
25 views

Proving that for $u \in BV([a,b])$, $\frac{d}{dx} (\text{Var}_{[a,x]} u) = |u'(x)|$.

I am trying to work through Exercise 2.29 of Leoni's A First Course in Sobolev Spaces, which gives an alternative proof of the theorem in the title in the real-valued case. The exercise is broken up ...
maxematician's user avatar
  • 1,477
2 votes
1 answer
139 views

Conditions on Radon measure for it to be a measure of a BV function

Functions of bounded variation (BV) on $\mathbb{R}^d$ are functions with $$ \|u\|_{BV} := \|u\|_{L^1(\mathbb{R}^d)} + TV(u) $$ finite, where $$ TV(u) := \sup\left\{ \int_{\mathbb{R}^d}u(x)div(\phi)(...
ChocolateRain's user avatar
6 votes
1 answer
371 views

Is $x^a $sin$(x^{-a})$ Hölder continuous?

I'm currently trying the following Exercise #11 of Chapter 3 in Stein's Real Analysis. Exercise 11. If $a, b>0$, let $$ f(x)=\begin{cases} x^a \sin \left(x^{-b}\right) & \text { for } 0<x \...
jason 1's user avatar
  • 769
1 vote
0 answers
32 views

Bounded variation of derivatives when fitting a function from one interval into another

I pose this question to seek to prove a statement like the following: Let $f(x)$ and $g(x)$ be continuous on the interval $[0, 1]$, and let $h(y)=f((y+1)/2)$ where $-1\le y\le 1$. If $f^{(r)}$ is of ...
Peter O.'s user avatar
  • 941
5 votes
1 answer
112 views

Riemann-Stieltjes integral with respect to functions equal almost everywhere [closed]

Let $f, g$ be Lebesgue integrable, real-valued functions on $[0, 1]$ with bounded variation, and let $\phi : [0, 1] \to \mathbb R$ be continuous. Assuming that $f = g$ almost everywhere, does one have ...
KCJV's user avatar
  • 186
0 votes
1 answer
52 views

The sequence of function $f_k(x) = \frac{x^k}{(1+x)^2},~x\ge 0$

For non-negative integers $k\geq 1$ define $$ f_k(x) = \frac{x^k}{(1+x)^2},~x\geq 0. $$ Which of the following statements are true? For each $k$, $f_k$ is a function of bounded variation on ...
I am pi's user avatar
  • 617
0 votes
0 answers
66 views

Dealing with preimages of a function of bounded variation

The problem is the following: If $f$ is a real-valued function of bounded variation on an interval $[a,b] \subset \mathbb{R} $. Show that the set $$ A=\{y \in \mathbb{R} : \{x \in [a, b] : f(x)=y \}\:...
gumo1234's user avatar
2 votes
0 answers
46 views

Variation of a Brownian Motion

Let $B_t$ be a standard brownian motion. The variation is not bounded on any interval. What if I can only detect variation of a,ie I only observe the process $[B/a]_t$. Then this is a bounded ...
percojazz's user avatar
  • 411
0 votes
1 answer
111 views

Show that $\forall\varepsilon>0$, there exists a bounded continuous function $g(x)$ on $E$, such that $m(\{x\in E\mid f(x)\neq g(x)\})<\varepsilon$.

Let $E$ be a measurable subset of $\mathbb{R}^n$, and $mE<\infty$. $f(x)$ is a bounded measurable function almost everywhere. I want to find such a function $g$. For $n=1$, i.e., $E\in\mathbb{R}$. $...
一団和気's user avatar
4 votes
1 answer
42 views

Densitity of Locally Lipschitz functions on the space of locally integrable functions

A metric measure space is a tuple $(X,d,\mu)$ where $(X,d)$ is a metric space and $\mu$ is a measure on $(X,\mathcal{B}_X)$ ($\mathcal{B}_X$ is the notation for the Borel $\sigma$-algebra associated ...
Raúl Filigrana Villalba's user avatar
10 votes
0 answers
135 views

Can we prove Weierstrass function is not of bounded variation according to the definition of bounded variation rather than its indifferentiability?

As we know, the Weierstrass function defined by $$ f(x) = \sum_{n=0}^{\infty} a^n \cos(b^n \pi x), 0<a<1,b \in 2\mathbb N+1,ab > 1+\frac{3\pi}{2} $$ is differentiable nowhere (see here for a ...
brushmonk's user avatar
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