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
4
votes
1answer
55 views
Total variation of a Fourier series
Let $f(x) = f(x+2\pi)$ be a bounded real function given by the Fourier series of the form
$$
f(x) = \sum_{k=1}^N a_k \sin(kx + \phi_k).
$$
What is the total variation $V(f)$ of this function over one ...
3
votes
1answer
55 views
Bounded variation implies Borel measurable
Suppose that $f\colon[a, b] \to \mathbb{R}$ is a function of bounded variation. Show that $f$ is Borel measurable.
I was wondering if I could get a hint.
2
votes
1answer
59 views
Construct a sequence of functions that does not converge in $B[a, b]$
Construct an example of a sequence of functions $(f_n)$ in $BV[0, 1]$ such that $f_n \to f$ uniformly on $[0, 1]$ for some function $f \in BV[0, 1]$, whereas $(f_n)$ does not converge to $f$ in the ...
1
vote
1answer
108 views
$f(0)=0,\;\;f(x)=e^{-2/x}\sin\left(e^{1/x}\right)$, is $f$ bounded variation on [0,1]?
Let $$f(0)=0,\;\;f(x)=e^{-2/x}\sin\left(e^{1/x}\right),$$ is $f$ bounded variation on $[0,1]$?
Here is my thinking:
Since $f$ is differentiable on $(0,1]$ and continuous on $[0,1]$
If $f^\prime$ ...
1
vote
1answer
101 views
looking for a Poincare-type lemma for BV functions
Given a smooth bounded open subset $\Omega$ of $\mathbb{R}^n$, does there exist $A >0$ such that if $f\in BV(\Omega)$ with zero trace on $\partial \Omega$, and $\int_\Omega |Df| = 1$, then ...
0
votes
1answer
35 views
Using Chernoff bound to analysis the Lazyselect algorithm
It's my homework of the course of randomized algorithm. In the textbook (Randomized Altorithm by Rajeev Motwani et.al.), the author analyzed this algorithm using Chebyshev bound, but are there any ...
0
votes
1answer
49 views
Can I make a BV function right-continuous this way?
Math people:
This question is related to how can you "fix" one of the definitions of a BV function of one variable? . Suppose $f \in BV([0,1])$. I really have two-three questions. The ...
4
votes
0answers
82 views
minimizing the total variation of BV function with given trace on the boundary of the domain
In 1-2 papers, Sternberg, Williams, and Ziemer proved the following result: if $\Omega$ is a bounded connected open set in $\mathbb{R}^n$ whose boundary is smooth and has positive mean curvature ...
4
votes
0answers
478 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
90 views
Trace of BV function
Let $\Omega$ be a bounded open set in ${\mathbb R}^n$ with smooth boundary. Let $t > 0$ be small enough so that for every $x \in \partial \Omega$, there exists a unique $y \in \Omega$ with $|x-y| ...
2
votes
0answers
111 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
162 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:
...
2
votes
0answers
47 views
when is variation continuous?
Reading my lecture notes, I was faced with this question:
If $M$ is a continuous martingale of finite variation, why is its variation continuous? The variation is a number, how we see this as a ...
2
votes
0answers
79 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 ...
1
vote
0answers
19 views
Positive solutions for third order semipositone boundary value problems
please what is "semipositone"
http://www.sciencedirect.com/science/article/pii/S0893965909000500
thak you
1
vote
0answers
120 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
64 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 ...
1
vote
0answers
55 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?
1
vote
0answers
198 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 ...
1
vote
0answers
200 views
upper bound of exponential function
I am looking for a tight upper bound of exponential function (or sum of exponential functions):
$e^x<f(x)$ when $x<0$
or
$\sum_{i=1}^n e^{x_i} < g(x_1,...,x_n)$ when $x_i<0$
Thanks a ...
1
vote
0answers
66 views
Riemann integrals with respect to bounded variation.
Let $\alpha$ be of bounded variation on $[a,b]$. let $V(x)$ be the total variation of $\alpha$ on $[a,x]$ and let $V(a)=0$. If $f \in R(\alpha)$, prove that $f \in R(V)$?
Can anyone help me with the ...
1
vote
0answers
71 views
Non-periodic BV function
I want to know the definition of non-periodic bounded variation function.
I know the definition for periodic function of bounded variation, which is,
Let $f:[a,b]\to \mathcal c$ and ...
0
votes
0answers
16 views
Lemma 3.3 from “Positive solutions for third order semipositone boundary value problems”
I have this lemma :
Assume that : $w(t)$ is nondecreasing and $w(t)>0$ on $(q,1]$ , and let $M(t)$ such that $M\in L(0,1)$; $M(t)>0 $ on $(0,1)$ and $f(t,x+\gamma(t))\geq -M(t)$ for $(t,x)\in ...
0
votes
0answers
18 views
Lemma 3.2 from “Positive solutions for third order semipositone boundary value problems”
How de prove this lemma please :
Assume that: $w(t)$ is nondercreasing and $w(t)>0$ on $(q,1]$ , $\frac12<p<q<1$ hods .
Let $z\in C^2[0,1]\cap C^3(0,1)$ satisfy $z'''(t)\geq 0$ 0n ...
0
votes
0answers
34 views
which condition says that $f$ is necessarily bounded variation
Which of the following condition below imply that the $f:[0,1]\to\mathbb{R}$ is necessarily Bounded Variation?
monotone;
continuous and monotone;
has derivative on $(0,1)$;
bounded derivative on ...
0
votes
0answers
28 views
Composition of bounded variation functions
Assume $f\in BV[a,b]$ and $g : [c,d]\rightarrow[a,b]$ is increasing, continuous, and onto. Prove that $F:=f\circ g\in BV[c,d]$ and $V^b_a f=V^d_c F$
0
votes
0answers
46 views
what are some applications of functions of bounded variation?
Math people:
I have been studying functions of bounded variation for several years. While I find it a beautiful subject intrinsically, I would like to know if they have any applications in science, ...
0
votes
0answers
67 views
Functions of bounded variation and differentiablity
Let $f$ be of bounded variation on $[a,b]$, and define $\nu(x)=TV(f_{[a,x]})\ \forall x \in [a,b]$.
Show that $|f'| \leq \nu'\ a.e\ $on $[a,b]$, and infer from this that $\int_a^b |f'| \leq TV(f)$.
...
0
votes
0answers
56 views
About regularity of the maximal operator of Hardy-Littlewood
What difficulties arise when you consider the centered maximal operator, such that you can't prove that it maps BV into BV?
Does someone have some reference which that maps BV into BV?
PS: The ...
