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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2answers
26 views

Establishing a bound in the complex plane

so my function is $$f(z)= \frac{e^{iz}}{z^2+a^2} $$ What is getting to me and probably I should've been comfortable with this fact is how they establish this upper bound: $$\bigg|\int_{0}^{\pi} ...
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1answer
68 views

Monotone increasing continuous function with $\int_a^b f' = f(b) - f(a)$ which is not absolutely continuous

If $f:[a, b] \to \mathbb{R}$ is continuous and real-valued, f' integrable on [a, b], and $\int_a^b f' = f(b) - f(a)$, must f be absolutely continuous? What if f is monotone increasing? For the ...
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0answers
38 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 ...
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1answer
35 views

Showing that functions of bounded variation are not closed under composition

Find functions $g: [a, b] \to [c, d]$ and $f: [c, d] \to \mathbb{R}$ both of bounded variation, with f continuous, so that $f \circ g$ is not of bounded variation. This occurs as a request for a ...
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1answer
25 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 $g$, ...
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1answer
155 views

Inequality on Functions of Bounded Variations

The following discussions were based on Schwabik's book, entitled "Generalized Ordinary Differential Equations", pp.26-29. Let $\delta$ be a positive function defined on $[a,b]$. We say that ...
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0answers
83 views

How to prove $\int_{a}^{b} f dΞ±$ = $\int_{a}^{b} f dΞ²$

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 ...
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0answers
32 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 ...
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1answer
32 views

A special approximation of BV functions

Suppose $\Omega:=B(0,1)\subset \mathbb R^2$. Let $u\in BV(\Omega)$ be a radially symmetric, i.e., $u(x)=u(Rx)$ for all $R\in SO(2)$. In addition, suppose $w$ to be an affine function,. i.e., ...
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0answers
20 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 ...
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2answers
52 views

Multiplication of absolutely continuous function and function of bounded variation

Is the product of an absolutely continuous function $f$ and a continuous function of bounded variation $g$ on $[0,1]$ for which $f(0)=0$ and $g(0)=0$, absolutely continuous?
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1answer
72 views

If a derivative $f^\prime$ is of bounded variation then prove that $f^\prime$ is continuous.

I am having trouble with the following problem: Let $f:[a,b]\to \mathbb R$ be differentiable on $[a,b]$ and $f^\prime$ is of bounded variation on $[a,b]$. Prove that $f^\prime$ is continuous ...
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1answer
42 views

Show that $\,f(x) = \sum_{n=0}^\infty a_nx^n$, for $x \in [0,1]$, is of bounded variation

Let $\{a_n\} \subset \mathbb{R}$, be such that $\sum_{n=0}^\infty \lvert a_n\rvert < \infty$. Define $$f(x) = \sum_{n=0}^\infty a_nx^n \quad \text{for } x \in [0,1]$$ Prove that $f$ is of bounded ...
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1answer
35 views

Boundedness of a modular form in $\mathbb{H}$

Let be $k>0$ and $f \in S_k(\Gamma)$. I want to show that the function $h(z)=Im(z)^{\frac{k}{2}}\cdot |f(z)|, \; z\in\mathbb{H}$ is bounded in $\mathbb{H}$. I have already shown, that $h$ is ...
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1answer
15 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 ...
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2answers
45 views

Finding upper and lower bounds on a trigonometric function

I've been tasked with finding the upper and lower bounds of the element: $A = sin(\frac{\pi.n}{2n+3}) | n\in\mathbb{N}$ I think I have found the upper bound by doing: $\lim_{n\to +\infty} ...
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1answer
41 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 ...
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0answers
25 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 < ...
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1answer
29 views

Prove convergence of this sequence $f(n)_{n \in \mathbb{N}}= \left(\frac{10+in}{n^2 + 2in}\right)^n$

I am having this sequence $f(n)_{n \in \mathbb{N}}= (\frac{10+in}{n^2 + 2in})^n$ Is this sequence bounded/ convergent? Thoughts: $lim_{n \to \infty}(\frac{10+in}{n^2 + 2in})=lim_{n \to ...
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1answer
61 views

Find $f \in B[0,2] $ and $g$ Riemann-Stieltjes integrable on $[0,1]$ and $[1,2]$ but not in $[0,2]$

I'm stuck at one exercise: Find $f \in BV[0,2]$ and $g$ such that $g$ is Riemann-Stieltjes integrable in $[0,1]$ and $[1,2]$, that is, $$\int_{0}^1 \ g \ df$$ and $$\int_{1}^2 g \ df$$ exist, but ...
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1answer
26 views

Alternate definition on bounded variation

If $g:[a,b]\to \mathbb{R}$ the $g$ is of bounded variation iff $$TV(g,[a,b])=\sup\sum_{i=1}^n |g(t_i)-g(t_{i-1})|<\infty$$ where the supremum is taken over all partitions ...
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1answer
56 views

Showing that $f$ is not Absolutely continuous

Frist:- I am not sure about what title this question should be. Suppose the function $f:[0,\frac{1}{2}]\rightarrow \mathbb{R}$ defined by $$ f(x) = \begin{cases} 0, & \text{if }x=0 \\ x ...
2
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0answers
74 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 ...
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4answers
196 views

A continuous function on $[0,1]$ not of bounded variation

I'm looking for a continuous function $f$ defined on the compact interval $[0,1]$ which is not of bounded variation. I think such function might exist. Any idea? Of course the function $f$ such ...
4
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1answer
85 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 ...
2
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1answer
67 views

Calculating explicitly the total variation of $x^2 \sin\left(\frac{\pi}{2x}\right)$ on $[0,1]$.

I am attempting to calculate the total variation of the function $f$ on [0,1] defined as $$f(x) = \begin{cases} x^2 \sin\left(\frac{\pi}{2x}\right) \text{ if } x\neq 0 \\ 0 \qquad \qquad \text{ if } ...
0
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1answer
74 views

Is $x^{3/2}\sin(\frac{1}{x})$ of bounded variation?

I'm trying to show that $f(x)$ is of bounded variation where $f(x)=x^{3/2}\sin(\frac{1}{x})$ on $[0,1]$. I think that it is but I can't show it explicitly. Any help will be appreciated.
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1answer
34 views

Bounded variation of function $(f-M)^+$ and the measure of the set where it is concentrated

I read this statement in the book by Evans & Gariepy, page 215, last two lines. Here $f\in BV(R^n)$ and for fixed $\epsilon>0$ and $N>0$, we define $$A_\epsilon^N :=\{x\in \mathbb ...
2
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1answer
38 views

The space $BPV(0,1)$ is a separable metric space under certain metric.

This is exercise 2.42 from Leoni's book A First Course in Sobolev Spaces. The BPV is defined as the space contain the function $u$ such that $$ \text{Var}[u]:=\sup\left\{ ...
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1answer
42 views

Total variation of a sequence of functions

I have this problem, Let $\{f_n\}$ be a sequence of real valued functions on $[a,b]$ that converges at each point of $[a,b]$ to a function $f$. Then $T_a^b(f)≀\liminf T_a^b(f_n)$. It is solved in, ...
2
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1answer
95 views

The derivative of total variation

Define function $u$ on $[0,1]$ such that $$ u(x)= \begin{cases} x^2\cos\frac{1}{x} &0<x\leq 1\\ 0 & x=0 \end{cases} $$ and by $V(x):= \text{Var}_{[0,x]}u$, i.e., the total variation of ...
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1answer
29 views

The behavior of BV functions at a point of approximate continuity

Given $u\in BV(\mathbb R^N)$, we say $u$ is approximate continuos at $x$ and the approximation limit is $l\in R$ if $$ \lim_{r\to 0}\frac{\mathcal{L}^N(B(x,r)\cap \{|u-l|>\epsilon\})}{r^N} =0 $$ ...
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0answers
27 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 ...
0
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1answer
26 views

If $\alpha,\beta \in BV[a,b]$, prove that $|a| \in BV[a,b]$ and $\min(\alpha,\beta),\max(\alpha,\beta) \in BV[a,b]$.

Question: If $\alpha,\beta \in BV[a,b]$, prove that $|a| \in BV[a,b]$ and $\min(\alpha,\beta),\max(\alpha,\beta) \in BV[a,b]$. Attempt: Since $\alpha \in BV[a,b]$, $$ \sum_{i=1}^n \left| ...
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1answer
71 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
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1answer
26 views

The algebraical properties of $BV(\Omega)$

We know in one dimension that $BV(I)$ is an algebra, for both $I$ is bounded or unbounded. Now suppose $\Omega\subset R^N$ is open bounded with Lipschitz boundary. My question is: is it $BV(\Omega)$ ...
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1answer
27 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 ...
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0answers
29 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 ...
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1answer
31 views

Find $K$ such that $|(x, y)| > K$ implies $(x - 1)^2 + (y + 2)^2 > C+ 4$.

For any 𝐢 ∈ ℝ, find 𝐾 such that |(π‘₯, 𝑦)| > 𝐾 β‡’ π‘₯2 + 𝑦2 - 2π‘₯ + 4𝑦 + 1 > 𝐢 i.e. (π‘₯ - 1)Β² + (𝑦 + 2)Β² > 𝐢 + 4 whenever |(π‘₯, 𝑦)| > 𝐾 NOTE: 𝐾 is a function of 𝐢 only, and does NOT depend ...
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2answers
75 views

Which of the following is true about $f(x)$?

If $f(x)=x+\sin x$, then which of the following is true about $f(x)$? $1.f(x)$ is uniformly continuous on $\mathbb{R}$. $2.f(x)$ has bounded variation on $\mathbb{R}$. $3.f(x)$ does not have ...
0
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1answer
22 views

First Order Variation Example

I am in a introductory stochastic calculus class and came across and example from that asks for the first order variation and the quadratic variation of a continuous function. For example: $f(x) = ...
3
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1answer
121 views

The question regrading to density argument in analysis.

I know the density argument in $L^p$ space, in Sobolev spaces, and even in BV are really sweet in many many cases. However, some times author just work on nice functions and comment by "the rest can ...
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1answer
35 views

Prove that the sum of a trigonometric series has bounded variation

Let $u$: $(0,1)\to \mathbb R$ be defined as $$ u(x):=\sum_{n=1}^\infty \frac{1}{2^n} \cos(2^n\pi x)$$ I am trying to prove that $\operatorname{Var}(u)<\infty$ It is clear that ...
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1answer
236 views

The Relation between Holder continuous, absolutely continuous, $W^{1,1}$, and $BV$ functions

I am trying to find out the relation between those spaces. Take $I\subset R$ on the real line. $I$ can be unbounded. Then I have: We first assume $I$ is bounded. If $u\in C^{0,\alpha}(I)$, for ...
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0answers
20 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
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1answer
47 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
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1answer
52 views

The approximation of BV functions

From Evans & Gariepy 's book, I learned that generally for any $u\in BV(R^n)$, we can find $u_n\in BV(R^n)\cap C^\infty (R^n)$ such that $$ \lim_{n\to\infty} \|u_n-u\|_{L^1(R^n)} = 0$$ and ...
0
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1answer
89 views

How to show a given path is NOT rectifiable

We've been given a function and asked to find whether it is a rectifiable path or not. Let $\gamma : [0,1]\to \mathbb{C} $ defined as $\gamma (t)= t + \iota t \sin(\frac{1}{t})$ and $\gamma (0)=0$ ...
0
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1answer
30 views

The convergence in Bounded Variation functions

Given a sequence $u_n\in BV(\Omega)$ and $u\in BV(\Omega)$, where $\Omega\in R^n$ is open. We assume $u_n\to u$in $L^1_{loc}(\Omega)$ and we also assume that $$ \lim_{n\to\infty}|D ...
1
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1answer
34 views

About the definition of variation.

If $f$ is a function of bounded variable on $[a,b]$.Then: $$V_a^b(f)=\sup_{\pi}\sum_{i=0}^n|f(x_{i+1})-f(x_i)|$$ where $\pi$ is a partition of $[a,b]$. I wonder if$$V_a^b(f)=\lim_{\|\pi\|\to ...