0
votes
0answers
15 views

Showing equivalence of weak convergence on closed and open intervals

Quick question. Let $I$ be an open bounded subset of $\mathbb{R}^{n}$. If I am given that $u_{m},u \in W^{1,\infty}(I)$ and I want to show that $u_{m} \rightharpoonup^{*} u$ in $L^{\infty}(I)$. Then I ...
1
vote
1answer
36 views

The Gaussian Integral

Hi I am trying to calculate the expected value of $$ \mathbb{E}\big[x_i x_j...x_N\big]=\int_{-\infty}^\infty x_ix_jx_k...x_N \exp\bigg({-\sum_{i,j=1}^N\frac{1}{2}x^\top_i A_{ij}x_j}-\sum_{i=1}^Nh_i ...
2
votes
1answer
19 views

What assumptions are needed to get two integrals close to each other?

I have functions $A,B,C$, where $\int_{\mathbb{R}} |A\cdot B - C| <\varepsilon$, and want to be able to say that $\int_{\mathbb{R}} A \approx \int_{\mathbb{R}} \frac{C}{B}$. What extra assumptions ...
0
votes
0answers
19 views

Mean value of a function over the n-sphere superficie.

We know that we can use the bloch sphere to represent an unitary vectors $v$ in $\mathbb{C}^{2}$, due to the fact $su(2) \approx so(3)$. Then, if we have the function $f:\mathbb{C}^{2} \rightarrow ...
4
votes
2answers
74 views

On the convexity of a particular form of integrals

EDIT: I made some critical corrections below. Let $\mathcal{H}\colon\mathbf{w}\cdot\mathbf{x}+c=0$ be a hypeplane in $\mathbb{R}^n$. Also, let $g\colon\mathbb{R}^n\to\mathbb{R}_+$, be a non-negative, ...
3
votes
0answers
57 views

Rising Sun Inequality (Dunford-Schwartz maximal inequality)

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := ...
11
votes
2answers
149 views

Integration in Banach spaces - interesting, nice and non-trivial examples needed

I am interested in $\textbf{Integration in Banach spaces}$. Here is a little motivation for my question: Let $\left(X,\|\cdot\| \right)$ be a Banach space, $a,b \in \mathbb{R}$ with $a<b$ and $f ...
2
votes
0answers
53 views

Is this Neumann series solution unique?

I have a Fredholm integral equation of the second kind given as $$f(x)=g(x)+\lambda\int_{-\infty}^\infty K(x,y)f(y)dy, $$ where $\lambda\in(0,1)$, the kernel $K(x,y)=\phi(x-y)$ is a Gaussian ...
0
votes
0answers
23 views

proof of an relation

Let $\Omega = \mathbb{R}^2_+=\{(x,y)\in \mathbb{R}^2; y>0\}$ et soit $v \in H^1_0(\Omega)$ and let $h \neq 0$. Let $$D_h v = \dfrac{v(x+h,y) - v(x,y)}{h}$$ The questions are: 1- Prouve that ...
0
votes
0answers
29 views

Bound of integral involving theta function

I need to prove that the following bound is true. I thought this might follow from the inversion property of the theta function, as the infinite sum in the integrand is precisely ...
1
vote
1answer
30 views

spectral measure and integral query

I have proved the 'resolution of the identity' for a normal operator, namely that there is a unique spectral measure E such that $\int_{{\sigma}(T)} {\lambda}\,dE=T$ If (${\lambda}_{n}$) is the ...
2
votes
0answers
34 views

How is this the definition of equi-integrable?

Let $Q=(0,T)\times\Omega.$ I am completely lost with this: No definition of equi-integrability I have seen looks anything like this. Can someone help me please? Presumably it is a fact that a ...
1
vote
0answers
71 views

Question about integration [duplicate]

I have this, and I don't understand how to do the change of variable. Please help me Thank you.
1
vote
1answer
53 views

Getting the bound $\frac{1}{h}\int_0^{T-h}\int_t^{t+h}\int_\Omega |\nabla u(\tau)| |\nabla u(t+h) - \nabla u(t)|\;dxd\tau dt \leq C$

Let $u \in L^2(0,T;H^1(\Omega)) \cap L^\infty(0,T;L^2(\Omega)).$ Is it possible to find the following bound: $$\frac{1}{h}\int_0^{T-h}\int_t^{t+h}\int_\Omega |\nabla u(\tau,x)| |\nabla u(t+h,x) - ...
2
votes
1answer
89 views

An equality that holds with $v_t \in L^2(0,T;L^2(\Omega))$ but its proof requires $v_t \in L^2(0,T;H^1(\Omega))$

Let $Q=(0,T)\times \Omega$. For all $\varphi \in C_c^\infty(Q)$ such that $0 \leq \varphi \leq 1$, the following holds $$\int_Q \varphi^2 (\Delta v)v_t = \int_Q |\nabla v|^2 \varphi \varphi_t - ...
1
vote
0answers
29 views

Integration on associated vector bundle

Let $G$ be a compact lie group and $\mathfrak{g}$ be its Lie algebra then we can construct the integral on $G\times \mathfrak{g}$ by $$\int_G\int_{\mathfrak{g}}f(x,Y)dxdY$$ Where $x\in G$ and $Y\in ...
-1
votes
1answer
175 views

Why?$\int_0^1\int_{u(t)}^{u(t)+w(t)} f(t,v(t)) dv dt = \int_0^1 f(t,u(t)+\theta w(t))w(t) dt; ~~\theta\in[0,1] $

why: $$\int_0^1\int_{u(t)}^{u(t)+w(t)} f(t,v(t)) dv dt = \int_0^1 f(t,u(t)+\theta w(t))w(t) dt; ~~\theta\in[0,1] $$ how to get this ? Please help me Thank you.
1
vote
1answer
87 views

Support of a function and the struggle…

The concept of support is very confusing to me, I'm just getting used to it. Lets consider $\Omega\subseteq\Bbb R^n$ an open set, $C^1_c(\Omega):=\{f\in C^1(\Omega)\mid\operatorname{supp}(f)\text{ is ...
0
votes
0answers
72 views

Convolution of piecewise function

I would like to compute the convolution of piece wise function Following is the piecewise function $$ C_a(t) = \begin{cases}0& t\leq t_d\\ ...
1
vote
3answers
35 views

Notation of this set in a set?

I am currently struggeling with the following notation: For $\epsilon \in (0,1)$ and $p \in (0,\infty)$, consider the following subset of $L ^p$: $M(p,\epsilon)=\{f \in L^p:m \{x:|f(x)| \ge \epsilon ...
1
vote
0answers
31 views

Comparing Methods For Solving Double Integral

For my assignment I'm asked to compute a given double integral using these methods: ...
1
vote
2answers
78 views

How to prove that $\int_{-1}^{1}\exp\left(\frac{1}{x^2-1}\right) \ dx=1$?

I have some trouble to prove that $$\int_{-1}^{1}\exp\left(\frac{1}{x^2-1}\right) \ dx=1\ ? $$
0
votes
2answers
79 views

Two definitions of integral on boundary $\int_{\partial\Omega}f$?

I have seen two definitions of an integral of a function $f:\partial\Omega \to \mathbb{R}$ from the boundary of an open set $\Omega \subset \mathbb{R}^n$ where the domain is Lipschitz. 1) ...
0
votes
1answer
79 views

Partial derivative w.r.t an integration

For example, I have a functional $$J(f)=\int \frac{f(x)}{1+x^2}dx.$$ How to calculate $\frac{\partial J}{\partial f(x)}$? Does it equal to $\int \frac{1}{1+x^2}dx$? It seems that the question is ...
1
vote
0answers
72 views

Variants of the bump function.

The title of this question isn't really clear because of the 150 char limit. What I actually want to ask is this: If I would have a bump function for $-1 < x < 1$ and I would have some ...
4
votes
1answer
38 views

For which $L^p[0,1]$, $1\le p < \infty$ does the function $n^\alpha*\chi_{[0,1/n]}$ converge weakly to 0?

My hypothesis is that this function converges to 0 weakly iff $ p < 1/\alpha$, but I am not sure how to prove this. We are working in the space $[0,1]$ with the Borel sets and Lebesgue measure. I ...
3
votes
1answer
50 views

Convergence in $L^1$ of a sequence of functions

I have to see if the following sequence of functions is convergent in the space $L^1[(0,\infty)]$ $$f_n(x)= n\frac{\exp\left(-\frac{n}{2x^2}\right)}{x^3}$$ By definition, $f_n(x)$ is convergent in ...
6
votes
1answer
123 views

Positivity of the Coulomb energy in 2d

Let $$D(f,g):=\int_{\mathbb{R}^3\times\mathbb{R}^3}\frac{1}{|x-y|}\overline{f(x)}g(y)~dxdy$$ with $f,g$ real valued and sufficiently integrable be the usual Coulomb energy. Under the assumption ...
0
votes
1answer
47 views

convolution of two functions and relations with their p-norms

Let $f\in L^p(\mathbb{R})$, $g\in L^1(\mathbb{R})$, $1\leq p< \infty$. Then I have proved the convolution $f\ast g\in L^p(\mathbb{R})$ and $||f\ast g||_p\leq ||f||_p||g||_1$. Does $f\ast g$ ...
2
votes
1answer
39 views

Prove lower bound of integral

I have a continuous function $h:[a,b]\rightarrow\Bbb C$. Let $$M=\sup_{x\in [a,b]}|h(x)|$$ I need to find function $f\in L^2[a,b]$ with: $${||f||}^2=\int_{a}^b|f(x)|^2dx=1$$ such that: $$ ...
2
votes
0answers
32 views

Riemann integrability in not sequentially complete LCS?

For $E$ any Hausdorff locally convex space, I have been wondering whether Riemann integrability of all continuous functions $f:[\,0,1\,]\to E$ implies that $E$ be sequentially complete. For example, ...
0
votes
1answer
61 views

Prove x(t) is bounded given a integral inequality

I want to answer the following question: $x=x(t)$ is defined and continuous on $[0,T)$ and satisfies an integral inequality $$1 \leq x(t) \leq A_1 + A_2\int_0^t x(s)\big(1+\log x(s)\big) ds$$ for ...
2
votes
1answer
40 views

When can we interchange Fourier transform and countable sum?

When does $\mathcal{F}\left ( \sum_{n=1}^{\infty} f_n (x)\right ) = \sum_{n=1}^{\infty} \mathcal{F}(f_n(x))$ where $\mathcal{F}$ the Fourier transform operator.
0
votes
1answer
23 views

Inequality with partial integration in one dimension

Is it possible to prove $ \| u \|_{L^2(0,1)} \leq \| u' \|_{L^2(0,1)} $ for $u \in C^1([0,1])$ with $u(0)=0$ by using partial integration?
0
votes
2answers
169 views

Dirchlet Riemann Integrable in certain interval

Considering the Dirichlet function f : f (x) = { 1 if x is rational 0 if x is irrational } I want to know if this function can be Riemann Integrable in ...
1
vote
1answer
120 views

On why the Vitali Covering Lemma does not apply when the covering collection contains degenerate closed intervals

I believe I have a fundamental misunderstanding of the concept of the Vitali Covering Lemma. Definition - A closed bounded interval $[c, d]$ is said to be nondegenerate provided $c < d$. ...
5
votes
1answer
133 views

Proving a few things about $ L^{p} $-spaces

I am new to $ L^{p} $-spaces and am trying to prove a few things about them. Therefore, I would like to ask you whether I have gotten the following right. Prove that $ {L^{\infty}}(I) \subseteq ...
3
votes
1answer
26 views

Integral constraints for positive function

Let $C={\cal C}([0,1],(0,\infty))$ denote the set of all continuous maps $[0,1]\to (0,\infty)$. Let $g_1,g_2 \in C$ ; one can then define $$ \begin{array}{rcl} \Phi &: C& \to (0,\infty)^2 \\ ...
0
votes
0answers
30 views

How to prove a duality of $L^p$ spaces? [duplicate]

Let $(\Omega,\Sigma,\mu)$ be a finite measure space and $f:\Omega\longrightarrow \mathbb{R}$ be a measuable function. Let $1\leq p< \infty$ and $1/p+1/q=1$. Prove that the following are equivalent: ...
0
votes
1answer
80 views

Integrals Regulated functions

stuck on an example for this question, Give an example of a regulated function $f \colon [a,b] \to \mathbb{R}$ with the properties that $\forall x \in [a,b] f(x) \ge 0 , \int_a^b f = 0$ and there is ...
2
votes
1answer
71 views

A variance-mixture model

So I've tried to make a probability distribution which has a tunable degree of kurtosis and which becomes Gaussian if the control-parameter goes to 0. Now Levy-distributions are out of the question, ...
1
vote
1answer
32 views

Changing domain of integration with measure

Let $\Omega$ a bounded open set in $\mathbb{R}^n$. Let $\mu$ be a measure on $\Omega.$ Suppose $\Gamma$ is a bounded open set in $\mathbb{R}^n$ and that $F:\Omega to \Gamma$ is a diffeomorphism. Let ...
2
votes
1answer
170 views

Integration by parts on the $N$-dimensional torus

My problem. Consider the $N$-dimensional torus $$\mathbb{T}^N = \mathbb{R}^N /(2\pi n \mathbb{Z})^N \simeq [-\pi n, \pi n)^N$$ and consider two function $v,w \in H^2(\mathbb{T}^N)$. I want to ...
1
vote
1answer
101 views

Steklov Average in time and spatial gradient can be interchanged?

I'm trying to understand a proof on Steklov average and weak derivatives. Let $\Omega$ be a bounded domain of $\mathbb{R}^n$, $T>0$, $h\neq 0$ and $u\in L^2(0,T;H^1(\Omega))$ and extend $u$ by ...
1
vote
1answer
43 views

maximum of derivatives of lipschitz functions

Say $f$, and $g$ are nondecreasing functions on [a,b], differentiable with derivatives bounded by 1. can one infer that $$\int_a^b \min\left(\frac{d}{dx} f(x),\frac{d}{dx} g(x)\right)\,dx \leq ...
3
votes
2answers
112 views

Banach space integral via defining it in $X^{**}$ and then proving it's in $X$

Vector-valued integration is something I generally try not to think about very much. I have the impression that it can be a sort of "rabbit hole" of a subtlety if one allows it to be. So, I tend to ...
1
vote
1answer
66 views

Null sets on Daniell integral

Can someone help me clarify the definition of null sets related to Daniell integrals. The book I'm using is Introduction to analysis and integration theory by Philips. The definition that I'm provided ...
6
votes
4answers
440 views

For $f$ continuous, show $\lim_{n\to\infty} n\int_0^1 f(x)x^n\,dx = f(1).$

Suppose $f:[0,1]\to \mathbb{R}$ is continuous. Show thatĀ  $$\lim_{n\to\infty} n\int_0^1 f(x)x^n\,dx = f(1).$$ My answer so far: First I want to assume that $f\in C^1$. ThenĀ  $$n\int_0^1f(x)x^n\,dx ...
7
votes
1answer
146 views

Does this inner product on $L^1([0,1])$ have a name?

Math people: For $f, g \in L^1([0,1])$, define $$\langle f,g \rangle = \int_0^1 \int_0^1 f(t)g(t')\exp(-|t-t'|)dt'\,dt.$$ Although we don't normally think of $L^1([0,1])$ as an inner product space, ...
4
votes
2answers
85 views

(From Lang $SL_2$) Orthonormal bases for $L^2 (X \times Y)$

Lang $SL_2$ p. 13 :Let $\{\phi_i\}$, $\{\psi_i\}$ be orthonormal bases for $L^2(X)$ and $L^2(Y)$ respectively. Let $$\theta_{ij}(x,y) = \phi_i(x)\psi_i(y).$$ Then $\{\theta_{ij}\}$ is an ...