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.
865 questions
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Verification of formula for positive and negative variation in Folland's
Let $F\in BV$ be real-valued, and denote by $T_F(x)$ the total variation function of $F$, $$T_F(x)=\sup\left\{\sum_1^n |F(x_j)-F(x_{j-1})|:n\in\mathbb N,-\infty<x_0<\cdots<x_n=x\right\}.$$ ...
- 1,322
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Brownian motion has unbonded variation
We recently state in a course on probability and stochastic processes that a Wiener process (Brownian motion) is nowhere differentiable almost surely.
My teacher recall the following lemma that is :
$...
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Why is it not generally possible to extend the measure associated with a locally bounded variation function to all of $\mathbb R_+$?
Let $g: [0,+\infty) \rightarrow \mathbb R$ be a right-continuous function of locally bounded variation. It is well-known that on every compact interval $[0,T]$, we can define a unique Radon (signed) ...
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33
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Lyapunov function for double integrator with saturated input and saturated states
Consider the closed-loop double-integrator system
$\begin{aligned}& \dot{x}(t)=A x(t)+B \sigma(u(t)), \quad x(0)=x_0 \\& u(t)=K x(t)\end{aligned}$
$A=\left[\begin{array}{ll}0 & 1 \\0 & ...
2
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Exercise 2.34 in the book A first course in Sobolev spaces - Leoni
Let $u:[a,b] \rightarrow \mathbb R^N$
i) Let $g:[a,b] \rightarrow \mathbb R^N$ and take one $x_0 \in [a,b]$. Prove that
$$ \left\| \int_a^b g(x)dx\right\| \geq \int_a^b \|g(x) \| dx -2 \int_a^b \| g(x)...
- 900
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2
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How to prove that the Blancmange function doesent have a Bounded variation?
Given the Blancmange function
$$B(x)=\sum^{\infty}_{n=1} f_{n}(x)=\sum^{\infty}_{n=1}\frac{f_{1}(2^{n-1}x)}{2^{n-1}}.$$
where $f_{1}$,is the saw tooth function. Now a bounded variation is defined by
$$...
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1
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Does a Lipschitz continuous pdf necessarily have bounded variation?
Does a probability density function $f(x)$ that satisfies the Lipschitz condition on the entire real line $\mathbb{R}$ have bounded variation?
Undoubtedly, this holds on compact subsets of $\mathbb{R}...
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How is called L1 norm of L2 norm of components of the vector?
In TV regularization relatively important role has a norm, which in the discrete case of e.g. one object $v \in R^n \times R^n$ has the form
$$\|w\|_1, \mbox{where } w^i = \| v^i \|_2= \sqrt{(v^i_1)^2+...
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Clarification on covariation between semimartingale and finite variation process
Let $X$ be a semimartingale and $V$ be a finite variation process.
Lemma 3 of this page of almostsuremath proves that
$$[X,V]_t=\int_0^t\Delta X_s\,dV_s=\sum_{s\leq t}\Delta X_s\,\Delta V_s$$
This is ...
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On real functions with the property that for each $\varepsilon>0,$ every interval contains two points whose secant line is $>\varepsilon.$
Let $f:\mathbb{R}\to\mathbb{R}$ have the following property:
$$\forall\varepsilon>0,\text{ no matter how large },\forall a<b,\ \exists a<x_1<x_2<b\text{ such that }\left\lvert\frac{f(...
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Question about the Weak-* Limit of the Incremental Ratio
I am studying Michael E. Taylor's book Measure Theory and Integration and revising Chapter 13 on Radon Measures.
Let $ f \in L^\infty(\mathbb{R}) $. Show that the following are equivalent:
(a) $ \...
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Continuity + Banach (S) property implies bounded variation?
Are the following implications true:
(I) if $g$ is continuous on $[a,b]$, and if $g$ satisfies the Banach (S) property on $[a,b]$, then $g$ has bounded variation on $[a,b]$? Here we assume that $a,b\...
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If $f:\mathbb{R}\to\mathbb{R}$ is continuous, does there exists a nonzero continuous function $g:\mathbb{R}\to\mathbb{R}$ s.t. $g(x)f(x)$ is monotone?
Is the following proposition true or false?
If $f:\mathbb{R}\to\mathbb{R}$ is continuous, then there exists a not-everywhere-zero (i.e. $g\not\equiv 0$)
continuous function $g:\mathbb{R}\to\mathbb{R},...
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1
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1
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Bounded variation + injective implies piecewise strictly monotonic almost everywhere?
The question is can we prove the following: Consider $a,b\in {\bf R}$, $a<b$.
(1) If $u\in {\rm BV}([a,b])$, and
(2) if $u$ is an injection on $[a,b]$,
then there exists $u_0:[a,b]\rightarrow {\bf ...
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4
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1
answer
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Bounded variation + injective implies strictly monotonic?
The question is are the following assertions true:
I.(strong version) Consider $a,b\in {\bf R}$, $a<b$.
(1) If $u\in {\rm BV}([a,b])$, and
(2) if $u$ is an injection on $[a,b]$,
then $u$ is ...
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1
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1
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Proof: The Variation of $V_{F_{\mu}}$ is finite.
Background Information
Suppose that $\mu$ is a finite signed measure on $(\mathbb{R},\mathscr{B}(\mathbb{R}))$. Define a function $F_{\mu}:\mathbb{R}\to\mathbb{R}$ by letting
$$
F_{\mu}(x) = \mu((-\...
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Show $V_{F_{\mu}}(-\infty,x]=|\mu|((-\infty,x])$ for all $x\in\mathbb{R}$ ($\mu$ a finite signed measure on $(\mathbb{R},\mathscr{B}(\mathbb{R}))$).
My Question
I want to prove the following:
Claim$\quad$ Let $\mu$ be a finite signed measure on $(\mathbb{R},\mathscr{B}(\mathbb{R}))$. Show that $V_{F_{\mu}}(-\infty,x]=|\mu|((-\infty,x])$ for all $...
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1
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1
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Radon measure, passage to limit in BV space
Let $ \Omega $ be a bounded domain of $ \mathbb{R}^N,\ N \geq 2 $ and $ (u_n)_n $ a bounded sequence of $ W^{1,p_n}( \Omega), $ where $ p_n = 1 + \frac{1}{n}, n =1,2,\ldots. $ We also assume that $ (...
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1
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Derivative of $f \in BV_{loc}(\mathbb{R})$ with $f =0$ a.e.
Suppose I have $f \in BV_{loc}(\mathbb{R})$ and some open set $K \subset \mathbb{R}$ with $f = 0$ a.e. on $K$. Then is it true that $\partial_{x}f$ must necessarily be a measure (and not $L^p$) if $f$...
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1
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1
answer
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Gaussian pdf unbounded variation on $\mathbb{R}$？
Let a real-valued function $f(x)$ defined on $\mathbb{R}$.
For a bounded interval $[a,b]\subset \mathbb{R},$ taking a partition $\mathcal{P} =\{x_0, x_1, \ldots, x_n\},$ where $a\le x_0<x_1<\...
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Question about $f$ of bounded variation [closed]
Def.
The variation of $f$ on $[a, b]$ corresponding to a partition $P = \{x_0, x_1, \ldots, x_n\},$
$V^P(f):=\sum_{i=1}^{n} |f(x_i) - f(x_{i-1})|.$
Obviously, if $Q$ is a refinement of $P$, then $V^P(...
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1
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Continuous function which satisfies the Luzin N property, but which does not satisfy the Banach S property
My question is: can we find the function $g\in{\rm C}([a,b])$ which satisfies the Luzin N property on $[a,b]$, but which does not satisfy the Banach S property on [a,b]? Here $[a,b]$ is a compact non-...
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1
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Continuity of $T(f) = \int (h\circ f)\operatorname df$
On the space $BV:=BV(\mathbb R, [0,1])$ of $[0,1]$-valued functions on $\mathbb R$ that are of bounded variation equipped with the norm $$\Vert f-g\Vert_{BV} := \Vert f-g\Vert_{L_1}+V(f-g),$$ where $$...
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1
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When is function of bounded variation piecewise monotone on finite number of subintervals?
Given is a function $f:I \rightarrow \mathbb{R}$ on a closed and bounded interval $I \subset \mathbb{R}$. If $f$ is of bounded variation on $I$, that is $\text{Var}_I(f) < \infty$, then we know ...
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BV function $f$ is $\|Df\|$-measurable
Let $f:\mathbb R\to\mathbb R$ be a $L^1$-function of bounded variation where the total variation measure $\|Df\|(\mathbb R)$ is bounded, i.e. $f\in BV(\mathbb R)$.
Since $f\in L^1(\mathbb R)$, we have ...
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1
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Is there a short proof of the Second Mean Value Theorm for Integrals (strong, preferably asymmetric version)
The parenthesis in the title comes from the fact that there are essentially six versions of the conclusion in what may be called 2nd Mean Value Th. for Int. - not including special variants like those ...
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1
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Integral function of bounded variation function derivative
Let $f: [a,b] \to \mathbb{R}$ be bounded variation. So $f’$ exists almost everywhere, and let
$g(x):=\int_a^x f’(y)dy$.
(Due to the fact that it is possible that $f\notin AC([a,b])$ it is not ...
user1286545
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Integration with respect to finite Radon measure
Let $ u \in BV( \mathbb{R}^N). $ We know that
$$
\DeclareMathOperator{\Div}{div}
\DeclareMathOperator{\dm}{d\!}
\int_{\mathbb{R}^N} \left|Du\right| = \sup\left\{ \int_{\mathbb{R}^N} u\Div \varphi\dm ...
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1
answer
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Equality of the $L^1$ norm and measure norm of an corresponding absolutely continuous measure for BV functions
Let $u \in BV(I;\mathbb{R}^d)$ be a vector valued function of bounded variation, (in particular Bochner integrable) on an open interval $I \subset \mathbb{R}$. We can then define the Radon measure
$u \...
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Sequences of bounded variation have convergent series
A sequence $\{a_n\}$ has bounded variation if $\sum_{k=1}^\infty |a_{k+1}-a_k|$ converges. I am trying to prove that, if $\{a_n\}$ has bounded variation then $\sum_{k=1}^\infty a_n$ converges. This ...
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Prove that if $\{a_{k}\}$ is a sequence of real numbers such that $\sum_{k=1}^{\infty} \frac{|a_{k}|}{k} = \infty$,
Prove that if $\{a_{k}\}$ is a sequence of real numbers such that
$$\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)...
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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 ...
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0
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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 ...
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0
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38
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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 ...
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1
answer
82
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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 $\mu$ ...
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0
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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]...
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1
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55
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Determine bounded variation and absolute continuity for different parameters
Determine for which parameters $\alpha, \beta \in [0, +\infty[$ the function
$$f_{\alpha,\beta}:[0,1] \to \mathbb{R}, \quad f_{\alpha, \beta}(x) :=
\begin{cases}
x^{\alpha}\ \text{sin}(\frac{1}{x^{\...
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0
answers
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Smooth approximation of BV functions, Proof clarification
Im reading Evans&Gariepy book measure theory and fine properties of functions second edition. I have a question about the proof of theorem 5.3 that I don't understand.
The theorem states that for ...
- 532
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0
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156
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Understanding finitely additive (signed) measures
I am currently trying to understand finitely additive (signed) measures, mostly reading Dunford & Schwarz and Rao & Rao. Knowing only the normal measure theory, the finitely additive measures ...
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0
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0
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Is the measure of continuous injective image of $\mathbb{S}^1\rightarrow\mathbb{R}^2$ always 0? [duplicate]
Summarized:
There are various conditions under which the image of a one-dimensional manifold has measure 0 in $\mathbb{R}^2$. These can be formulated in different little lemmas which are usually easy ...
- 2,376
1
vote
1
answer
75
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Relationship between $p$ and $q$ variation.
Let $T>0$, $p>0$ and $f: [0,T]\to \mathbb{R}$ continuous. The $p$-variation of $f$ on $[0,T]$ is given by:$$\operatorname{Var}^p(f)_T:=\lim\limits_{n\to\infty}\sum \limits_{k=1}^{2^n}\left|f(k/2^...
2
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1
answer
36
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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 $[0,1]$...
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1
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76
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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 ...
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3
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1
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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-...
- 5,939
5
votes
2
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315
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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, ...
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3
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1
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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 ...
- 1,798
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0
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56
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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 ...
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1
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86
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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. ...
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2
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2
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144
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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 ...
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0
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147
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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 ...
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