All Questions
Tagged with lp-spaces integration
371
questions
0
votes
1
answer
53
views
$ L^p ( X ) \cap L^{\infty}( X) $ is a Banach space with respect only to the $p$-norm $\| \cdot \|_p$, $p<\infty$?
The space $πΏ^π(π) \cap πΏ^\infty(π)$, $p<\infty$, with the norm $||π||_{πΏ^π \cap πΏ^\infty}=||π||_π+||π||_\infty$ is a Banach space. I imagine that if we remove the norm $||π||_\infty$ ...
- 3,004
3
votes
1
answer
67
views
Prove that $\int _Xf_ngd\mu \overset{n\to\infty}{\to}\int _Xfgd\mu$ ,$\forall g\in \mathcal{L}^\infty (\mu )$ if it's true $\forall g\in C_b(X)$
Let $X$ be a Polish space and $\mu :\mathfrak{B}_X\to\overline{\mathbb{R}}$ a finite measure on the Borel subsets of $X$. Suppose $(f_n)_{n\in\mathbb{N}}$ is a sequence of $\mathcal{L}^1(\mu )$ and $f\...
- 1,209
4
votes
1
answer
61
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Finding a function which is $L^1$ not $L^2$ and the integral is bounded by the square root.
So I have been trying to solve the following this past exam problem:
Find $f\in L^1(\mathbb{R})$, not $L^2(\mathbb{R})$ with the property:
$$
\int_{A}|f(x)|dm\leq \sqrt{m(A)}\quad\text{ for all } A\...
- 394
1
vote
0
answers
18
views
Computing $L^2$ norm and quadrature for time-varying solution to PDE?
Setup. On domain $\Omega \times (0,T]$, the parabolic problem
\begin{align}
u_t + \Delta u + u = f
\end{align}
with some appropriate initial and boundary conditions has solution $u(x,t)$, which I ...
- 2,459
0
votes
0
answers
43
views
Is it possible to define $L^p$ spaces using a non-sigma-finite measure space and a Banach space?
Most often (at least in probability), one defines the $L^p$ space as
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $p\geq 1$ be a real number. Then
$$
L^p(\Omega, \mathcal{F},...
- 5,247
0
votes
1
answer
74
views
$f \in C^1([a;b])$, prove that $ \forall x,y \in [a;b]$ we have $|f(x)-f(y)| < || f' ||_2 \sqrt{|x-y|} $
Question:
Let $f \in C^1([a;b])$, prove that $ \forall x,y \in [a;b]$ we have $|f(x)-f(y)| < || f' ||_2 \sqrt{|x-y|} $
Answer:
1- $f \in C^1([a;b]) \Rightarrow f'(\chi) $ exists and is defined: $ |...
- 719
1
vote
2
answers
180
views
Does $\sin (nx)$ converge in $L^2$?
I was just introduced the concept that if $(f_n)$ converges in $L^2$ topology to $g(x)\in L^2([0,2\pi])$ then $\lim_{n\to\infty}\int^{2\pi}_0|f_n(x)-g(x)|^2dx=0$. I would appreciate any hint to how to ...
- 191
1
vote
1
answer
55
views
Integration of discontinuous functions
In order to evaluate its Fourier transform, I want to determine whether $f(x)=\arctan(\frac{1}{x})$ belongs in $L^1(\mathbb{R})$, $L^2(\mathbb{R})$ or both. Therefore, we have to check the continuity ...
- 165
5
votes
1
answer
78
views
Is the function $x \mapsto \mu(A+x)$ continuous, where $\mu$ is a finite Borel measure on $\mathbb R^n$ and $A \in \mathcal B(\mathbb R^n)$
Let $\mu$ be a finite regular Borel measure on $\mathbb R^n$ and $A$ is a Borel set. I am trying to prove that $x \mapsto \mu(A+x)$ is continuous. Here $\mu$ is regular means it satisfies assumptions ...
2
votes
1
answer
110
views
Let $(X, \mathcal M, \mu)$ be a finite measure space. If $f_n, f$ are $L^1$ functions, $f_n$ unif. integrable, $f_n \to f$, then $f_n \to f$ in $L^1$
Let $(X, \mathcal M, \mu)$ be a measure space with $\mu(X)$ finite, and $f,f_1,f_2,\dots$ be $L^1$ functions. Show that if $\{ f_n \}$ is uniformly integrable and $f_n \to f$ for a.e. $x \in X$, then $...
- 3,800
1
vote
1
answer
60
views
Example of a function such that $f\in C(\Bbb R)$, $f\in L^2\setminus L^1$ and $f(n)=n$ for all $n\ge 1$, $n\in \Bbb N$
I'm looking for an example of a function with the following properties:
$f$ is continuous over $\Bbb R$;
$f\in L^2(\Bbb R)\setminus L^1(\Bbb R)$;
$f(n)=n$ for all $n\ge1$, $n\in \Bbb N$.
My attempt
...
- 5,840
0
votes
1
answer
31
views
Proof of unitairity of regular representation
Let $G$ be a compact group and $\varphi:G\to L^2(G), \ \sigma\mapsto L_\sigma$ the regular representation.
In the proof of unitairity of $\varphi$ i dont understand one step.
First, the haar measure $\...
4
votes
1
answer
184
views
Counting measure and integrability condition
Let $\mathcal{A}$ be the $\sigma$-field on $[0,1]$ that consists of all subsets of $[0,1]$ that are (at most) countable or whose complement is (at most) countable. Let $\mu$ be the counting measure on ...
- 103
0
votes
1
answer
37
views
$\frac{f}{|x|^4+1}$ lies in $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ for $f \in L^2(\mathbb{R}^n)$ and $n > 6$
I'm trying to solve the following problem:
Show that $\frac{f}{|x|^4+1}$ lies in $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ for $f \in L^2(\mathbb{R}^n)$ and $n \le 7$, where $|x|$ denotes the ...
- 55
2
votes
1
answer
115
views
$\|f \otimes g\|_{L^p} \leq \|f\|_{L^p} \|g\|_{L^p}$ for $1 \leq p \leq \infty$
I am looking to prove the inequality
$$\|f \otimes g\|_{L^p} \leq \|f\|_{L^p} \|g\|_{L^p}$$
for $1 \leq p \leq \infty$. Here $f$ takes values in a Banach space $X_1$ and $g$ takes values in a Banach ...
- 6,275