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
46 views

Need guidance on a problem about oscillation from Spivak's Calculus on Manifolds

I've been stuck on this particular problem for a while now: Let $f: [a,b] \rightarrow \mathbb{R} $ be an increasing function. If $x_1, ... ,x_n \in [a,b]$ are distinct, show that $\sum\limits_{i=1}^n ...
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2answers
38 views

Find limits of value/derivatives defining a polynomial at 2 points to bound it in between

The following properties define the polynomial $p(x)$ uniquely: $\text{deg}(p(x))=7\\p(-1)=y_1,\ p'(-1)=d_{1,1},\ p''(-1)=d_{2,1},\ p'''(-1)=d_{3,1}\\ p(1)=y_2,\ \ \ \ p'(1)=d_{1,2},\ \ \ \ ...
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2answers
44 views

About tight rational bounds

Suppose we have rational functions $f$, $g$, and $h$ defined for all natural numbers $n$ such that $f \leq g \leq h$ for all $n \in \mathbb{N}$. How can we prove that there is no rational functions ...
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1answer
81 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 | ...
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1answer
23 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
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1answer
53 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 $0 < \alpha < 1$ ...
2
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1answer
219 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: ...
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1answer
41 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 ?
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1answer
52 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 ...
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1answer
53 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 ...
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1answer
37 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: $7^n$, $8^n$ ...
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1answer
31 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 $\delta>0$, ...
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1answer
23 views

Definition of Bounded Variation Function with vectorial arguments

I try to found a definition of a function $$f(x)\colon\mathbb R^m\to \mathbb R^n$$ that use the norm. Is the formula below correct? $$TV=\sup\sum_{i=1}^k \|f(x_i)-f(x_{i-1})\|.$$ with k any finite ...
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1answer
35 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., ...
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1answer
76 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 ...
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1answer
88 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 $a,b,c_i>0$
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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
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0answers
44 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 ...
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0answers
81 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
104 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
168 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 ...
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0answers
496 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 ...
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0answers
180 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
101 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
111 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 ...
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0answers
23 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 ...
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0answers
20 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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0answers
21 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 ...
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0answers
28 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 ...
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0answers
24 views

Evaluate the bound of the following complex integral

Let $q$ be a complex analytic function such that $\left|q\right|\neq0$ everywhere in the domain, except at the boundaries ($\left|q\right|\rightarrow0$ as $z\rightarrow\pm\infty$). Is it possible ...
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0answers
40 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 ...
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0answers
64 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. ...
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0answers
131 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 ...
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0answers
87 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 ...
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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
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0answers
251 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: ...
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0answers
79 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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0answers
123 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 ...
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0answers
16 views

Inverse transformation of continous transformation is bounded

I am having a continous transformation: $f: \mathbb C \to \mathbb C $ with $B \subseteq \mathbb C $ bounded. Now I want to proove that $ A = f^{-1} (B)$ is bounded! How can I proove that this ...
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0answers
13 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 ...
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0answers
23 views

Bounded/Unbounded sets. [Mandelbrot set]

This is the last question from my assignment. For Part a I have: $z_{n+1}=z_n^2+c$ $\Rightarrow c =z_{n+1}-z_n^2$ $\Rightarrow |c|=|z_{n+1}-z_n^2|=|z_{n+1}-z_n^2||-1|=|z_n^2-z_{n+1}|$ ...
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0answers
16 views

Why is the weakest bound only possible at these specified mins

Given the following: $$0 < g_0 \leq g_1$$ $$0 \leq B_0 \leq B_1$$ $$D_0 \leq D_1$$ (note: these are the only two variables that could be negative) $$0 \leq c_0$$ $$0 \leq c_1$$ $$0 < f_0$$ ...
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0answers
38 views

Question about deformation

Why if $H\setminus X$ has no crititical point then $X$ is a deformation retract of $H$? $H$ is a Hilbert space $X$ is a subspace of $H$ ($X=\varphi^b=\lbrace x\in H, \varphi(x)\leq b\rbrace$ , ...
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0answers
33 views

Gaussian independent, Mean, expectation, variance

Let $X$ and $Y$ be two independent Gaussian variables with zero mean and variance $\sigma^2$. Define:$$Z = |X-Y|.$$(a) Show that $\operatorname{E}[Z]= 2 \sigma / \sqrt{\pi}$. (b) Show that ...
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0answers
33 views

How to represent?

You are a well-known hedge fund manager in Wall Street circles. One of your wealthy clients has $\$1$ million dollars to invest in XYZ stocks. Currently, XYZ stocks are trading at $\$2$ per share. You ...
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0answers
88 views

Upper bound for logarithmic integral

If i'm not mistaken $li(x)=O(x/logx)$ So we can write $li(x)≤ cx/logx$ . For x>1 tell me any c>0.5 ... I really need this...
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0answers
18 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 ...
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0answers
76 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 ...