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

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4
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1answer
109 views

Bounded variation problem

Let $x=0.a_1a_2\dots$ be the decimal expansion of a number $x$, $0<x<1$. If two decimal expansions of $x$ exist, the one that ends with $0$’s is taken. For what values of $q > 1$ is the ...
3
votes
1answer
70 views

A basic problem on bounded variation

If $a > 0$ let $$f(x) =\left\{\begin{array}{ll} x^{a} \sin (x^{-a})&\text{if } 0 < x \leq 1\\ 0&\text {if }x=0 \end{array}\right.$$ Is it true that for each $0 < \alpha < 1$ ...
3
votes
1answer
111 views

Definition of total variation

According to wikipedia, the total variation of the real-valued function $f$, defined on an interval $[a,b]\subset \mathbb{R}$, is the quantity $$V_b^a=\sup_{P\in\mathcal{P}}\sum_{i=0}^{n_P-1}\left | ...
2
votes
1answer
30 views

$\int_0^\pi g(x)\sin(nx) dx$ coverges to $0$ for any function $g$ is bounded variation

I have a homework as follows : Prove that $\int_0^\pi g(x)\sin(nx) dx$ coverges to $0$ for any function $g$ is bounded variation on $[0,1]$. my attempt: for any bounded variation function $g$, ...
2
votes
1answer
29 views

Question about finite preimeter in $BV$ space

Given $\Omega\subset R^N$ is open bounded, we say $E\subset \Omega$ has finite perimeter in $\Omega$ if $\chi_E\in BV(\Omega)$. Follows from Evans & Gapriep's book, we write for any $\varphi\in ...
2
votes
1answer
240 views

Fourier transform of derivative for the bounded function

I was going through the derivation of Wigner distribution properties, and encountered a certain step in the proof I could not justify. Namely, the step requires the following equality to be true: ...
1
vote
1answer
25 views

Compute total variation when discontinuities are given bounds

Say you have a function such as $f(x)=1+\sin(x)$ that is defined from $-\frac{\pi}{4}$ to $\frac{\pi}{4}$. Everywhere else, the function takes on the value $-\frac{1}{2}$. How do you compute the ...
1
vote
1answer
17 views

Semimartingale jumps question

I am reading a statement which contains $\Delta X \cdot Y$ where $X$ is a semimartingale and $Y$ is a finite variation process and the notation means the lebesgue stieltjes integral. My problem is ...
1
vote
1answer
45 views

Why is this sequence relatively compact in $L^1$?

I am currently reading this paper from 1973. In short, one has given a linear continuous operator $P : L^1([0,1]) \to L^1([0,1])$ with ||P||=1 and for $f \in L^1$ a family of functions ...
1
vote
1answer
56 views

Does bounded variation and continuous means total variation continuous

$F$ is of bounded variation and continuous. Is it true that total variation is continuous ? In case, $F$ is absolutely continuous it is trivial to see. But for the above case how to proceed ?
1
vote
1answer
64 views

One inequality involving Total variation function

If $F$ is of bounded variation in $[a,b]$, then I need to prove that $$ \int_{a}^{b}|F'(x)| dx \leq T_F(a,b)$$ If $F'$ were Riemann integrable then it was easy to prove (in fact we can prove ...
1
vote
1answer
111 views

Conditions for $o(|u|^{-1})$ decay of the Fourier transform of a bounded variation function

As the question suggests I am looking for a (not very restrictive) condition on a function of bounded variation so that its Fourier transform is $o(|u|^{-1})$ as $|u| \to \infty$. Let me elaborate on ...
0
votes
1answer
20 views

If $S$ is any linear space which contains all real valued monotone functions on $[a,b]$ , then $S$ contains all functions of bounded variation

Let $[a,b]$ be a closed bounded interval in real line and $S$ be any set containing all real valued monotone functions on $[a,b]$ such that $f,g \in S \implies f+g \in S$ and $f \in S , c \in \mathbb ...
0
votes
1answer
63 views

When the function is continuous, bounded of variations, absolutely continuous?

Let the function $f_a:[0,1] \to \Bbb R$ be defined by $$f_a(x)=\begin{cases} x^a \cdot \cos(\frac{1}{x}) & 0 < x \leq 1 ;\\ 0 & x=0.\end{cases}$$ Find all values $a\ge 0$ such that ...
0
votes
1answer
31 views

Calculate total variation of g on a given interval.

I am dealing with the following function: $$g(x) = \left\{ \begin{array}{lr} 1+\sin(x) & -\frac{\pi}{4} < x < \frac{\pi}{4} \\ -\frac{1}{2} & otherwise ...
0
votes
1answer
101 views

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 ...
0
votes
1answer
49 views

Big Omega Proof for $5^n = \Omega(6^n)$

I got the Big-O ($\mathcal{O}$) proof but I am having troubles with the Big Omega ($\Omega$) proof. I am trying to prove $6^n$ is not the tightest bound for $5^n$. Is this logic true: $7^n$, $8^n$ ...
0
votes
1answer
36 views

A question related to bounded variation

Let $f\in C^{BV}([0,1])$ (i.e. continuous and has bounded variation). Let the intervals $I$ and $T$ satisfy the following: $I\subset T\subset [0,1]$ and for sufficient small $\delta>0$, ...
0
votes
1answer
39 views

On functions that are bounded by other certain functions

I am trying to address a specific, but also rather abstract, problem, which briefly can be stated as follows: Let $f$, $g$ be real-valued functions defined for all column vectors in $\Re^n$, i.e., ...
0
votes
1answer
152 views

exponential upper bound on sum of exponentials

For what value of $c_3$ can I guarantee that $(a+b)\exp(-c_3\theta)>a\exp(-c_1\theta)+b\exp(-c_2\theta)$ where $a,b,c_i>0$
4
votes
0answers
1k 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 ...
3
votes
0answers
33 views

The best constant in Poincare-liked inequality in BV space

Let $\Omega\subset \mathbb R^N$ open bounded smooth boundary. Then we could prove that for any $u\in BV(\Omega)$ and $\omega\in \operatorname{ker}\mathcal E $, where $\mathcal E$ denotes the ...
3
votes
0answers
61 views

Understanding the duals of $L^{\infty}(K)$ and $C(K)$ and weak-* compactness

Let $(K\subset \mathbb{R}^n,\mathcal{B}(K),\lambda)$ be a measure space, where $\lambda$ is the Lebesgue measure and $K$ is compact. According to Wikipedia (with adapted notation), The dual ...
3
votes
0answers
88 views

Discontinuous local martingales with finite variations

I would like to know if a discontinuous local martingale with paths of finite variations almost surely is a martingale. I feel that it should be the case but can't find a straightforward argument. As ...
2
votes
0answers
143 views

The Co-area formula for $BV$ function V.S. the co-area formula for $C^\infty$ functions

I am working on the proof of the co-area formula for $BV$ functions. Suppose $u\in BV(\Omega)$ then the co-area formula states that $$ \|Du\|(\Omega)=\int_{-\infty}^\infty \|\partial ...
2
votes
0answers
158 views

Arzela Ascoli variation theorem

I am trying to prove the following variation of the Arzel-Ascoli theorem: Let $(f_n)n$ be a sequence of differentiable functions on $[a,b]$ whose derivatives are uniformly bounded and there is an ...
2
votes
0answers
208 views

Functions of bounded variation and continuity

Suppose that $f:[a, b] \to \mathbb{R}$ is a function of bounded variation. Define $g:[a, b] \to \mathbb{R}$ by $g(x) = V_a^x f$. Show that $f$ is continuous at $x \in [a, b]$ iff $g$ is continuous at ...
2
votes
0answers
554 views

Show that $f$ is of Bounded Variation by $f'$' being integrable.

For $\alpha$, $\beta$ > 0, define the function $f$ on $[0, 1]$ by $f(x) = x^{\alpha}\sin(1/x^{\beta})$ for $0 < x \leq 1$ and $f(x) = 0$ for $x=0$. Show that if $\alpha > \beta$, then $f$ is of ...
2
votes
0answers
203 views

Jordan decomposition measure and its distribution function

Let $f: \mathbb{R} \rightarrow \mathbb{R}$, and for $x \in \mathbb{R}$, let $T_f(x)=\sup \left\{\sum_{j=1}^N \left|f(x_j)-f(x_{j-1})\right|\right\}$, where the supremum is taken over all $N \in ...
2
votes
0answers
103 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
0answers
118 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 ...
1
vote
0answers
70 views

Absolutely continuous iff continuous of bounded variation

I have the following problem (taken from q1 p341 of Kolmogorov and Fomin's Introductory Real Analysis), which I am struggling to prove completely. I think I know how to show the only if part, but not ...
1
vote
0answers
41 views

p-variation of semimartingales

Does every (particularly continuous) semi-martingale have bounded 2+$\epsilon$-variation for all $\epsilon>0$? Note that I am not asking, whether they have finite quadratic variation - that is ...
1
vote
0answers
29 views

Bound of a fourier series when coefficients are bounded

Let $f(x)$ be a finite fourier series with $$f(x)=a_0+\sum_{n=1}^N\left(a_n\sin{\left(2\pi nx/P\right)}+b_n\cos{\left(2\pi nx/P\right)}\right)$$ and bounded coefficients ...
1
vote
0answers
43 views

Counterexample for Trace operator in BV space.

Suppose $\Omega\subset R^N$ is open bounded with Lipschitz boundary. Then for $u\in W^{1,p}(\Omega)$, there exists a linear operator $T$: $W^{1,p}(\Omega)\to L^{p}(\partial\Omega)$ such that the ...
1
vote
0answers
42 views

Distributional Representation of Perimeter in Chan-Vese

While re-implementing the classic Chan-Vese Algorithm for image segmentation, I stumbled upon the following statement, which I have problems to understand: Let $H: \mathbb{R} \to \mathbb{R}$ be the ...
1
vote
0answers
44 views

Change of variables formula with integrator of bounded variation

Let $G$ be be continuous with bounded variation on finite intervals. If $f$ is continuous then it is well known that $\int_a^bf(G(s))dG(s)=\int_{G(a)}^{G(b)}f(x)dx$. How general can $f$ be so that ...
1
vote
0answers
54 views

Question on Bounded Variation involving partitions

If $f\colon[a,b]\rightarrow \mathbb{R}$ and $P=(a=x_0, x_1, ...,x_n=b)$ is any partition of $[a,b]$ let $v_f(P)$ be defined as $$v_f(P) = \sum_{k=1}^n |f(x_k) - f(x_{k-1})| $$ and the total variation ...
1
vote
0answers
114 views

Does the fundamental theorem of calculus hold for BV functions?

I am a bit confused and I hope you can help me in understanding a bit better these things. Let us start by considering one dimensional case. Let $f\colon \mathbb (a,b) \to \mathbb R$ be a function. ...
1
vote
0answers
204 views

Every polynomial bounded on [a,b] is of bounded variation.

Question: Prove that every polynomial bounded on the compact interval [a,b] is of bounded variationMy proof: Let $f(x)$ be a bounded polynomial of n degree. Then $f '(x)$ is also a polynomial bounded ...
1
vote
0answers
94 views

How to show that the first derivative is bounded in a function

\begin{equation} y=\arccos\left( -\frac{1}{2\left(Dr^{\dfrac {|\sin(2x+\theta)|}{M\sin x\sqrt{A+2B\cos(2x+\theta)}}}+1\right)} \right) \nonumber \end{equation} How to show in above function the ...
1
vote
0answers
263 views

Solving $\int_{0}^{2\pi} \cos (x) \cos (z(x)) \, dx$ with a bounded version of $z(x)=a \cos (x)+b$

I am trying to solve the following equation: $$\int_{0}^{2 \pi } \cos (x) \cos z(x) \, dx \hspace{20 mm} (1)$$ where $z(x)$ is a bounded version of $z(x)=a \cos (x)+b$ that reads as: ...
1
vote
0answers
96 views

What is Mountain Pass theorem?

The title says it all, what is mountain pass theorem, how does it work is variational problems with doubly differential constraint?
0
votes
0answers
34 views

Exercise 3.32 from Real Analysis of Folland

Can someone give me some hint on how to solve this problem? Thanks a lot If $F_1, F_2, ..., F \in NBV$ and $F_j \rightarrow F$ pointwise, then $T_F \le \liminf T_{F_j}$ Here, NBV is the ...
0
votes
0answers
9 views

$d \log f'$ is a compactly supported radon measure for $f\in C_c^{1}$ with bounded variation

In a note of John N. Mather, "Commutators of Diffeomorphisms, III: a group which is not perfect," he says that it is well known that if $f$ is a compactly supported $C^1$ diffeomorphism on ...
0
votes
0answers
87 views

How to prove $\int_{a}^{b} f dα$ = $\int_{a}^{b} f dβ$

I'm working on an exercise from Carothers' Real Analysis: where $BV[a,b]$ denotes the set of all functions on [a,b] with bounded variation and $C[a,b]$ is the set of all continuous functions on ...
0
votes
0answers
64 views

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 ...
0
votes
0answers
47 views

Finite Variation Function.

Let $V$ be a right continuous BV function and put $V_t = \int_0^t a_s dC_s$ where $C$ is increasing and right continuous. Is it true that if $V$ is continuous then $\int_0^t |f_s a_s| dC_s < ...
0
votes
0answers
27 views

reference request about sobolev space and BV space

I am studying Sobolev Space and BV space by using Leoni's and Evans & Gariepy's book. I was wondering that where can I find some explicit example and some computational question of those space. A ...
0
votes
0answers
23 views

A question on a function of bounded semivariation(part 2)

Let $(X,\tau)$ be a Hausdorff locally convex topological vector space and let $P(X)$ be a family of continuous seminorms on $X$ that generates the topology $\tau$. I got the following definition from ...