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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2
votes
1answer
30 views

Space between $L^1$ and $BV$?

I am looking for a function space $X_s$ such that this space has following properties: $X_s$ is a Banach space, and has lower semi-continuous properties with respect to $L^p$ strong convergence. I ...
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 ...
1
vote
1answer
29 views

The intersection of $BV$ space.

Given $\Omega\subset \mathbb R^N$ open bounded smooth boundary. We define $$ TV(u,\alpha):=\sup\left\{\int_\Omega u\operatorname{div}v\,dx,\, v\in C_c^\infty(\Omega;\,\mathbb ...
1
vote
0answers
17 views

The dual function regarding to bounded deformation.

I am studying this short paper and I met some difficulties in understanding Theorem 3.1. There is no need to look at this paper. My question is pretty general and I will state all needed ...
-1
votes
2answers
35 views

Prove that $x^\frac{3}{2}\sin (\frac{1}{x})$ is a function of bounded variation for $x\in(0,1]$. [closed]

I am studying for a test in measure theory. Please help with the following question: Prove that $x^\frac{3}{2}\sin (\frac{1}{x})$ is a function of bounded variation for $x\in(0,1]$.
0
votes
2answers
37 views

Prove that $ \frac {1}{f}$ is a function of bounded variation on $[a,b]$.

I am studying for a test in measure theory. Please help with the following question: Let $f:[a,b]\to R$ a continuous function of bounded variation, when $f(x)\ne 0$ for every $x \in [a,b]$. Prove ...
-2
votes
0answers
28 views

How do I prove that $f(s)=\sum a_{n}{n}^{− s}$ converge for $Re(s)>0$if the partial sum of $a_{n}$ are bounded?? [duplicate]

let $f(s)$ be a power series defined as follow :$$f(s)=\sum a_{n}{n}^{− s}$$ Assume the partial sum of $a_{n}$ are bounded .My question here is : How do I prove that $f(s)$ converge for ...
2
votes
1answer
76 views

Is the space of continuous functions with bounded variation separable?

Let $$BVC([0,1]) = \{f:\mathbb [0,1] \to \mathbb R,\, f\in BV([0,1])\cap C([0,1]),\, f(0)=0\},$$ and $$\|f\|_{BVC}=\mathrm{Var}(f).$$ Is $BVC([0,1])$ separable?
3
votes
1answer
125 views

Total variation measure vs. total variation function

Let $a, b \in \mathbb{R}$ with $a < b$ and define the compact interval $I := [a, b]$. Let $g, h : \mathbb{R} \rightarrow \mathbb{R}$ be non-decreasing and right-continuous on $I$ and constant on ...
0
votes
1answer
59 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
21 views

Bounded Variation imply L^1

If a function $f:\mathbb{R}\to\mathbb{C}$ is of bounded variation, is it true that $f\in L^1(\mathbb{R})$? We need to ask for $f$ to be continuous?
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 ...
1
vote
1answer
22 views

Absolute continuity and sample paths of Brownian motion

An offhand remark in Morters and Peres' book on Brownian motion says that Brownian motion is a.s. absolutely continuous on compact intervals (see page 147, immediately preceding the statement of ...
1
vote
1answer
121 views

Function f is bounded on $[0, \infty)$

Let $f : [0, \infty) \to R$ be continuous such that $lim_{x \to +\infty} f(x) = 0$. How can I Prove that f is bounded on $[0, \infty)$. I know that condition for a function to be bounded is - ...
1
vote
1answer
24 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 ...
0
votes
1answer
29 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 ...
2
votes
1answer
56 views

a question about proving a normed space is complete

Let [a,b] be an interval in R,and denote by E the Vector space of functions f:[a,b]->R such that f is of bounded variation over [a,b] and f(a)=0.Prove that by setting $||f||=Var|_{a}^{b}(f)$ for each ...
0
votes
1answer
60 views

Understanding bounded variation

In my analysis course we are covering the topic of bounded variation fuctions and I am really having a very hard time trying to get the concept. My main problem is that I don't get how can a function ...
1
vote
2answers
27 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} ...
4
votes
1answer
120 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 ...
1
vote
0answers
68 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 ...
0
votes
1answer
48 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 ...
2
votes
1answer
28 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 $g$, ...
1
vote
1answer
177 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 ...
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
52 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
1answer
41 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., ...
1
vote
0answers
39 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 ...
0
votes
2answers
71 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?
9
votes
1answer
106 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 ...
3
votes
1answer
54 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 ...
1
vote
1answer
36 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 ...
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 ...
0
votes
2answers
86 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} ...
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 ...
0
votes
0answers
46 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
1answer
30 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 ...
0
votes
1answer
63 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 ...
0
votes
1answer
33 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 ...
0
votes
1answer
64 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
votes
0answers
123 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 ...
13
votes
4answers
235 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
votes
1answer
104 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
votes
1answer
85 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
votes
1answer
87 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.
1
vote
1answer
37 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
votes
1answer
42 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\{ ...
1
vote
1answer
57 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
votes
1answer
110 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 ...
1
vote
1answer
36 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 $$ \lim_{r\to 0}\frac{\mathcal{L}^N(B(x,r)\cap \{|u-l|>\epsilon\})}{r^N} =0 $$ ...