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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66 views

$\| f \|_{BV} = | f(a) | + V_{a}^{b} f$ defines a norm in the space $BV[a,b]$

I'm learning about functions of bounded variations and need to verify my work for this problem: Show that $\| f \|_{BV} = | f(a) | + V_{a}^{b} f$ defines a norm in the space $BV[a,b]$. My ...
3
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0answers
23 views

Is it okay if I considered $f$ as uniformly continuous and bounded on specific interval if $f$ is continuous?

Problem Suppose $f$ is continuous and $\phi$ is of bounded variation on $[a, b]$. Show that the function $\psi(x)=\int_a^xfd\phi$ is of bounded variation on $[a, b]$. If $g$ is continous on $[a, b]$...
3
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0answers
50 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
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0answers
217 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. ...
3
votes
0answers
102 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
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0answers
66 views

Pointwise convergence of a sequence of approximate limits of BV functions.

So, let $\Omega\subset \mathbb R^2$ bounded and consider a sequence of functions $\{u_k\}_{k\in\mathbb N}\subset BV(\Omega)$ and $u\in BV(\Omega)$ such that $u_k\rightarrow u$ weakly* in $BV(\Omega)$. ...
2
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0answers
112 views

Definition of a bounded complex function and how to apply Liouville's theorem?

The definition of a bounded function is: $$\exists M\in\mathbb{R} \quad st \quad |f(x)| \leq M \quad\forall x\in Domain $$ So consider the complex entire function $f(z)$ such that $Re(f(z))<0$ ...
2
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0answers
58 views

Approximation of Sets of Finite Perimeter

Fix an open set $\Omega \subset \mathbb{R}^n$. If $E$ is a measurable subset of $\Omega$, we may define the perimeter of $E$ in $\Omega$, denoted by $P(E;\Omega)$, to be $$P(E;\Omega) = \sup_{\...
2
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0answers
75 views

Concerning the ring of all real valued functions of bounded variation on $[a,b]$

Let $B[a,b]$ the ring of all real valued functions of bounded variation on $[a,b]$ . What is the cardinality of $B[a,b]$ ? How does the maximal ideals of $B[a,b]$ look like ? How does the prime ideals ...
2
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0answers
288 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 E_t(u)\|(\Omega)\...
2
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0answers
220 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
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0answers
257 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
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0answers
648 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
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246 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
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0answers
109 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
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0answers
131 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
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34 views

Functions of bounded variations and Riemann-Stieltjes integral

In my work, I used recently the classical Riesz theorem. It has lead me to study functions of bounded variations and Riemann-Stieltjes integrals. Unfortunately, even if there exist a lot of books and ...
1
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0answers
21 views

Can someone explain the detail of the proof (rectifiable curve)

Curve $C$ in the plain, \begin{align} C:= \begin{cases} x=\phi(t) \\ y=\psi(t) \end{cases} \quad, \quad a \le t \le b \end{align} The graph of C is $\{(x, y): x=\phi(t), y=\psi(t), a\le t\le b\}$ $\...
1
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0answers
39 views

When is the Stieltjes integral of bounded variations?

I was trying to figure out when a Riemann or Lebsgue Stieltjes integral is of bounded variation. For simplicity let $f$ be a increasing RCLL function; when is that $$\int_0^t g(x) df(x)$$ is of ...
1
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0answers
46 views

Wrong result: a continuous function has zero $p$-variation, for every $p$. Where's the error?

Let $\Pi_n$ be a sequence of partitions with $|\Pi_n| \to 0$. Then the $p$-variation of a continuous function $g$ along the partitions $\Pi_n$ is defined as $$V_T^p(g) = \lim_{n \to \infty} V_T^p(g, \...
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0answers
44 views

Relationship between functions of bounded variation and signed measures

Let $f$ be a left continuous function of bounded variation on $[a,b]$ s.t. $f(a)=0$. $f$ can be written as the difference $f=f_1-f_2$ of two non-decreasing functions on $[a,b]$. $f_1=\frac{1}{2}(v(x)...
1
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0answers
25 views

Guess quadratic variation of 2 processes, with same/different BM.

I have 2 processes with stochastic parts R, S. $$dR = \mu_1 dt + \sigma_1 dW_{t1}$$ $$dS = \mu_2 dt + \sigma_2 dW_{t2}$$ I am trying to show what precisely quadratic variation $[R,S]$ means for 2 ...
1
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0answers
16 views

RCLL (cadlag) But Not Bounded Variation

In probability theory, we often require that functions and/or processes be RCLL (i.e. cadlag) and of bounded variation (usually bounded on any finite interval). I'm having trouble coming up with an ...
1
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0answers
32 views

Bounding fluctuations on a random variable

I have some discrete random variable $w$ that has values (in decreasing order) $\underline{w}^\downarrow = \left( w_1, w_2, \dots , w_d \right)$ with corresponding probabilities $\underline{x}=(x_1, ...
1
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0answers
124 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
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0answers
473 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 $P=\{x_0=a,x_1,\ldots,x_{n-1}...
1
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0answers
73 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
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0answers
70 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
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0answers
47 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
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0answers
64 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
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0answers
70 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
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0answers
315 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
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0answers
97 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
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0answers
274 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: $$z(x)=\min\...
1
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0answers
102 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?
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7 views

How to indicate which is the best bound for accuracy either the max or min bounds?

A Few lists of precipitation data (P-data at different stations) in descending order is used to estimate streamflow at its corresponding stations. For every P data, 2 streamflow values are ...
0
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0answers
13 views

Proof Function is Bounded/Unbounded

How can I prove that the function $$\sigma_i\left(t\right) = k_i\left[\left(a+b\left(T_i-t\right)\right)e^{-c\left(T_i-t\right)}+d\right]$$ is bounded/unbounded? Note: $\sigma_i\left(t\right)$ is ...
0
votes
0answers
26 views

Minimum surface area soap film variational principles

An axisymmetric soap film $y(x)$ is formed between two circular wires at $x = ±l$. The wires both have radius $r$. Show that the shape that minimises the surface area takes the form $$y(x) = ...
0
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0answers
17 views

Characterization of the Jordan decomposition of a real-valued function of bounded variation from Folland.

This is a characterization of the Jordan decomposition of $F$ from Folland's Real Analysis. However, I can't see how the characterization makes sense. Let $F\in BV$ be a real valued function and $T_F$...
0
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0answers
34 views

Prove that if $f$ is of bounded variation on $[a, b]$, then $f$ is bounded on $[a, b]$.

Prove that if $f$ is of bounded variation on $[a, b]$, then $f$ is bounded on $[a, b]$. My professor proved the proposition like the following processes: Choose $x$ between $a$ and $b$, that is, $...
0
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0answers
47 views

Can someone prove this bounded variation problem?

I am doing my homework, which is solving all even number problems in textbook. Finally, I have almost finished.... but I failed to solve one problem. I am not sure that it is okay if I post the whole ...
0
votes
0answers
28 views

Construct a continuous function $f$ with $0\leq f\leq 1$ such that $\int_a^bf d\alpha\geq\alpha(d)-\alpha(c)-\epsilon$

Suppose that $\alpha$ is right continuous and increasing. Given $\epsilon > 0$ and $[c,d]\subset[a,b]$, construct a continuous function $f$ with $0\leq f\leq 1$ such that $\int_a^bf d\alpha\geq\...
0
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0answers
36 views

Given $f(x) = x^{1/3}$, show that $f \in BV[0, 1]$

I'm learning about functions of bounded variations and need some help with this problem: Given $f(x) = x^{1/3}, x \in [0, 1]$ show that $f \in BV[0, 1]$. My work and thoughts: We know that ...
0
votes
0answers
25 views

Show that $V_{a}^{b}(X_\mathbb{Q}) = +\infty$ on any interval [a, b]

I'm self-studying the theory of functions of bounded variations from the book of Neal Carother (Real Analysis) and need some help with this problem: Show that $V_{a}^{b}(X_\mathbb{Q}) = +\infty$ ...
0
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0answers
75 views

Upper bound on successive difference inequalities

I would like to know tight upper bounds of the following equations perhaps some might have telescopic sum which can result in very tight bound.In the following equations assume $B\geq a_i \geq 0$,$\...
0
votes
0answers
40 views

Total variation function

For a function $f : [a,b] \times [c,d] \rightarrow R$, we define $v_f : [a,b] \times [c,d] \rightarrow R$ as $(x,y) \rightarrow V_{f_{[a,x] \times [c,y]}}$. Here $V_f$ is the total variation of $f$ ...
0
votes
0answers
28 views

Under Given conditions, is f absolutely continuous on $[0,1]$?

1)Let $f: [0,1] \to R$ be a continuous function such that $\mid f(x)-f(y) \mid \le \mid \sqrt{x}-\sqrt{y} \mid $ for all $ x,y \in [0,1]$. Then is $f$ absolutely continuous on $[0,1]$? 2) what about ...
0
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0answers
27 views

Difference of increasing functions differentiable a.e.

I'm working through Royden & Fitzpatrick's Real Analysis, in the beginning of section 6.3 it reads and I quote: "Lebesgue's theorem tells us that a monotone function on an open interval is ...
0
votes
0answers
19 views

The normalization of gradient in weak convergence.

Given $\Omega\subset \mathbb R^2$ is open bounded, smooth boundary, $u_n\in BV(\Omega)$ is bounded in $BV$ norm and in addition we have $$0<\inf |u_n|_{TV}\leq \sup |u_n|_{TV}<+\infty$$ where $|\...
0
votes
0answers
33 views

Weak chain rule the other way round.

If $f: \mathbb{R} \to \mathbb{R}$ is weakly differentiable and $u: \Omega \to \mathbb{R}$ is smooth, where $\Omega \subset \mathbb{R}^n$, can we show that for any test function $\varphi \in C_c^1(\...