For questions about $L^p$ spaces, that is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence class of measurable functions with $p$-th power of the absolute value integrable. Question can be about properties of elements of these spaces, or when the ambient space on a ...

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2
votes
1answer
29 views

Question on Inequality from Bartle's Elements of Integration: Riesz Fischer Theorem

I am puzzled how did Bartle get $$|g_k|\leq\sum_{j=k}^\infty |g_{j+1}-g_j|$$ (second last line)? I tried using Triangle Inequality and ended up with one extra term: $$\begin{align*} |g_k|&=|g_k-...
4
votes
2answers
90 views

Supremum of absolute value of the Fourier transform equals $1$, and it is attained exactly at $0$

Suppose that $f \in L^1(\mathbb{R}^n)$, $f \ge 0$, $\|f\|_{L^1} = 1$. How do I see that $\sup_{\xi\in\mathbb{R}^n} |\mathcal{F}(f)(\xi)| = 1$, and it is attained exactly at $0$?
0
votes
1answer
32 views

Writing an operator $T$ defined by $(T f)(t) = \int_{-\pi}^\pi h(t − s)f(s)ds$ as $\sum_{n \in \mathbb Z} \mu_n \langle f, \varphi_n\rangle \varphi_n$

Let $h$ be a continuous function with period $2\pi$. Define $T : L_2[−\pi, \pi] \to L_2[−\pi, \pi]$ by $(T f)(t) = \int \limits _{-\pi}^\pi h(t − s)f(s)ds$. Let $\{\varphi_n(t) =\frac{1}{\sqrt{2\pi}} ...
3
votes
1answer
58 views

Upper and lower bound on $L^1$ norm purely in terms of measure

Suppose $f$ is a measurable almost everywhere finite function on $\mathbb{R}^d$, and let$$E_n = \{x : 2^n \le |f(x)| < 2^{n + 1}\}, \quad n \in \mathbb{Z}.$$What is a non-trivial upper and lower ...
0
votes
1answer
21 views

$L^1$ approximation by a slightly “displaced” copy

Let $f:\Bbb R\to \Bbb R$ be an $L^1$ function and $f_\epsilon(x):=f(x+\epsilon)$, $\mu$ is the Lebesgue measure, prove that $$\lim_{\epsilon\to 0}\int|f_\epsilon-f|\mathrm d\mu=0.$$ I tried to ...
0
votes
1answer
37 views

If $f_n$ converges to $f$ in $L^1(\mathbb{R})$ and $f_n$ converges to $g$, what relation exists between $f$ and $g$?

Take $(f_n)$ to be a sequence in $L^1(\mathbb{R})$ and suppose it is true that $(f_n)$ converges in $L^1(R)$ to a function $f \in L^1(\mathbb{R})$. Let $g$ be a function such that $(f_n)$ converges to ...
0
votes
0answers
34 views

Is there any relationship between $\mathcal{l}^p$ spaces and $\mathcal{L}^P$ space?

Due to the similarity of the names, I guessed that there may be some relationship between the two spaces. Is there such a relationship, or is there nothing more to it other than the fact that they ...
0
votes
1answer
43 views

Suppose $f_n\to f$ in $L^1([0,1],\lambda)$. Prove or disprove: $\exists \{f_{n_j}\}$ such that $f_{n_j}(x)\to f(x)$ for almost every $x\in[0,1]$. [duplicate]

This is part of an old preliminary exam in Analysis I am reviewing to prepare for my own prelim. $\lambda$ is the Lebesgue measure. $f_n\to f$ with respect to the $L^1-$norm. I know that there exists ...
6
votes
0answers
133 views

Inequality for Lévy SDE

Let $X_{s}^{t,x}$ denote the solution at time $s$ of an Ito SDE whose coefficients are Lipschitz continuous with initial condition $X_t=x$. Let $t\leq s\leq T<\infty$. The inequality $$ \mathbb{E}\...
0
votes
0answers
32 views

Bounding $L^p$ norm of a function defined by averaging

Let $\Delta=\{t_0, t_1, ... t_m\}$ be a partition of $[a, b]$ and let $f{\in}L^{p}[a, b]$ for $1\le p\le\infty$. Let $T\Delta$ be the function on $[a, b]$ defined by $T\Delta(f)(a)=0$ and $$T\Delta(f)...
2
votes
1answer
29 views

Non-convergent Cauchy sequence in $\ell^1$ with respect to the $\ell^2$ norm

Let $X = \ell^1$, the set of absolutely convergent real valued sequences and let $d_2(\mathbf{x},\mathbf{y}) = \left(\sum_{k=1}^\infty |x_k - y_k|^2\right)^{1/2}.$ This is the $2$-norm on the $1$ ...
1
vote
0answers
29 views

$f_n\rightharpoonup f$ weakly in $\sigma(L^1,L^{\infty})$ and $\|f_n\|_1\to \|f\|_1$ no implies $\|f_n-f\|_1\nrightarrow 0$ [duplicate]

I'm studying Brezis' book of Functional Analysis. I'm trying to do the exercise $4.19$ and I would like of a little help. Le be $\Omega=(0,1)\in \Bbb{R}.$ Construct $(f_n)$ in $L^1 (\Omega),$ $f_n ...
4
votes
1answer
73 views

Polynomials dense in $L^{2}(\mathbb{R})$?

I was reading something about a week ago and there was a line that said something to the effect of "the space of polynomials is dense in $L^{2}(\mathbb{R})$" and then there was another line that said ...
0
votes
1answer
42 views

If $f \in L^1([a, 1])$ for all $a \in (0, 1)$, is it true that $f \in L^1((0, 1])$?

I'm learning about measure theory and need help with the following question: True or Fasle (justify): If $f \in L^1([a, 1])$ for all $a \in (0, 1)$, then $f \in L^1((0, 1])$. While it is very ...
0
votes
0answers
20 views

Show property of convolution

Proposition: Let $ u, v \in L^2(\mathbb{R}^n) $ then $ \widehat{u \ast v}=\widehat{u} \cdot \widehat{v}$. Proof: We want to show that $\mathcal{F}^{-1} (\widehat{u} \cdot \widehat{v})=u \ast v $. We ...
1
vote
0answers
46 views

Show $L^2$-convergence

Lemma: Let $\psi \in C_C^{\infty}(\mathbb{R}^n), \psi \geq 0, \int \psi(x)dx=1, \psi_{\epsilon}(x)= \epsilon^{-n} \psi{\left( \frac{x}{\epsilon}\right)}, \epsilon>0$. Let $f \in L^2(\mathbb{R}^n)$ ...
-1
votes
1answer
79 views

Show that the following is a bounded linear operator on $L^2(R_+)$ Calculate the adjoint operator.

The following is a question I have been working on for some time with help from my teacher. Unfortunately we have a solution but are not 100% confident with it. Some guidance on if part(s) of our ...
0
votes
1answer
18 views

Uniform convergence of a sequence of functions given as product and convolution.

Suppose we have, for an open bounded set $\Omega \subset \mathbb{R}^n$: A function $u \in L^p(\mathbb{R}^n) \cap C(\mathbb{R}^n)$. A sequence of mollifiers $(\rho_n) \subset C_c^{\infty}(\mathbb{R}^...
2
votes
0answers
49 views

Let $T(f):=\frac{1}{x}\int_{0}^{x}{f(t)\,\mathrm{d}t}$ (the Hardy operator) find the norm of $T$ on $L^p$ [duplicate]

We have the operator $T: L^p(\mathbb{R}^+) \to L^p(\mathbb{R}^+) $ with $p \in (1,+\infty)$, defined by $T(f):=\frac{1}{x}\int_{0}^{x}{f(t)dt}$. We define $\tilde{f}(x)=e^{x/p}f(e^x)$ for all $f \in ...
3
votes
1answer
42 views

If $\{\nabla u_j\}$ is Cauchy in $L^p(\mathbb{R}^n)$ and $\int_{B(0,1)} u_j dx = 0$, does $\{u_j\}$ converge in $L^p_{\text{loc}}(\mathbb{R}^n)$?

Let $1 < p < \infty$. Let $\{u_j\}_{j=1}^\infty$ be a sequence of functions in $W^{1,p}_{\text{loc}}(\mathbb{R}^n)$ such that $\nabla u_j \in L^p(\mathbb{R}^n)$ for all $j$, $\int_{B(0,1)} u_j ...
4
votes
3answers
67 views

Properties of sequences of bounded functions in $L^2$

Let $f_n: [0, 1] \to \mathbb{R}$ be a bounded sequence of functions in $L^2([0, 1])$. This means that there exists $C_0 > 0$ so that$$\|f_n\|_{L^2} \le C_0 \text{ for all }n.$$Assume that $f_n$ ...
0
votes
0answers
37 views

Subspace of $L^1(\Omega)$ closed

Let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space, $\mathcal{A}\subset\mathcal{B}$ two sub-sigma-algebras and $f$ a $\mathcal{B}$-measurable function. I want to show, that the subspace $$\...
4
votes
1answer
44 views

Is it natural how $L^p$ spaces measure local and global sizes the same?

This is a continuation of my question Spaces of functions similar to $L^p$ but with different local and global sizes. I have been bothered by the fact that the $L^p$ norm on $\mathbb R^n$, which is ...
6
votes
1answer
82 views

Spaces of functions similar to $L^p$ but with different local and global sizes

Obviously much of analysis on $\mathbb R^n$ considers $L^p$ spaces or other Banach spaces derived from them. The definition of $L^p(\mathbb R^n)$ looks very natural, but I've been bothered for some ...
1
vote
1answer
43 views

Is it true that $\lim\limits_{r \rightarrow 1}f(r x) = f(x)$ in $L^1$?

Suppose $f \in L^1(\mathbb{R})$ with Lebesgue measure and $r > 0$. Does $f(rx)$ converges to $f$ in $L^1$ as $r \rightarrow 1$ ? Put differently, does $$ \| f(rx) - f(x)\|_1 \rightarrow 0$$ as $r \...
1
vote
2answers
40 views

Minimizing an integral — Hilbert space

Find the real values of $a, b$ which minimize $$\int_1^{\infty} \left| \frac{1}{x^2} - a \frac{1}{x^3} - b\frac{1}{x^4}\right|^2 \; dx.$$ Hint : Work in an appropriate Hilbert space. Here is why I ...
2
votes
1answer
59 views

Open set in $\ell^2$

Let $a=(a_n)_n\subset(0,\infty)$ be a sequence and $S^{(a)}:=\{(x_n)_n\in\ell^2:\lvert x_n\rvert\ <a_n \forall n \}$. I want to prove that $S^{(a)}$ is open in $\ell^2$ iff $\inf_{n\in\mathbb{N}} ...
2
votes
1answer
98 views

Parseval's identity holds

Theorem: If $u \in L^2(\mathbb{R}^n)$ then the Fourier transform $\widehat{u} \in S'(\mathbb{R}^n)$ is a $L^2(\mathbb{R}^n)$ function and the Parseval's identity holds: $||\widehat{u}||_{L^2(\mathbb{R}...
1
vote
2answers
27 views

Let $f\in L^p(0,1)$ and define $f_h$

Let $f\in L^p(0,1)$ ($1\leq p<\infty$) and define $f_h$ as $$f_h(x)=\begin{cases}f(x+h)&\text{ for } x+h\in [0,1]\\ 0 &\text{ for } x+h\not\in[0,1]\end{cases}$$ Prove that for all $\...
1
vote
2answers
52 views

Inclusion of Schwartz space on $L^p$

I'm looking for a proof of $\mathcal{S}(\mathbb{R}) \subset L^p(\mathbb{R})$ for $1 \leq p \leq \infty$. My informal probe follow like this: For any function $f \in L^p(\mathbb{R})$ exists a ...
0
votes
1answer
36 views

Is the sequence of functions $g_n=ng_1(nx)$ a Cauchy sequence?

Given a function $g_1(x) \in \mathcal L^2(\Bbb R)$ that satisfies: $$\int_{-\infty}^{\infty}dx \space g_1(x)=1$$ one can define a sequence of functions $g_n=ng_1(nx)$. Does $g_n(x)$ define a ...
1
vote
2answers
43 views

List of counter-examples to $\mathcal{L}_p(\mathbb{R})$ inclusions.

Given $1 \leq p < q \leq \infty$, it is well-known that $$\ell_p \subseteq \ell_q$$ and that $\mathcal{L}_p(\mu) \supseteq \mathcal{L}_q(\mu)$ whenever $\mu$ is finite. However $\mathcal{L}_p(\...
1
vote
1answer
61 views

The convolution is in $L^1$

According to my notes: The convolution is in $L^1$. Indeed $$\left| \int_{\mathbb{R}^n} \left( \int_{\mathbb{R}^n} f(y) g(x-y) dy\right) dx\right| \leq \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} |f(y) ...
0
votes
1answer
61 views

if $\int{}f$ is finite, then $\int{}f$ exists?

My textbook said, If $\int_E f$ exists then, of course, $-\infty\le\int_E f\le+\infty$. If $\int_E f$ exists and is finite, we say that $f$ is Lebesgue integrable, or simple integrable, on $E$ and ...
-1
votes
1answer
35 views

$L_p(\mu)\subseteq L_q(\mu)$ [closed]

Given a measure space $(\Omega,\mathfrak A,\mu)$ and $1≤q≤p$, how can I show that $$L_p(\mu)\subseteq L_q(\mu)$$ if the measure $\mu$ is finite, that means $\mu(\Omega)<\infty$?
1
vote
1answer
40 views

Inequality of $L^p$ type

If $a\geq 1,$ $b\geq c\geq 1$ and $p>0$ then is it true that $$\frac{a+b}{\left\{\int_0^{2\pi}|e^{i\theta}+b|^pd\theta\right\}^{1/p}}\leq \frac{a+c}{\left\{\int_0^{2\pi}|e^{i\theta}+c|^pd\theta\...
0
votes
1answer
18 views

When summation of two sequences is finite, is one finite?

$|\cdot|$ is Lebesgue measure. Let $w(\alpha) := |\{x:f(x)>\alpha\}|$ Let $f$ be a nonnegative function. Then, the proof uses that $\displaystyle\sum_{k=-\infty}^{\infty} 2^{kp}w(2^k)\lt\infty \...
1
vote
1answer
20 views

For $E \subset \mathbb{R^n}$, is $f^p$ finite almost everywhere in $E$ if $f \in L^p(E)$?

Q1) I didn't learn $L^p(E)$ yet, only learned $L(E)$. In order to solve problems, however, I need to know that the following theorem is correct or not. Let $E \subset \mathbb{R^n}$. Then, if $f \in ...
0
votes
1answer
49 views

Three questions on measurable functions and $L^p$ spaces

I'm learning about measure theory and $L^P$ spaces and need help with the following questions: True or False (justify): $(1)$ Let $f:(-1, 1) \to \mathbb{R}$ measurable on $(-n, n), \; \forall ...
1
vote
1answer
22 views

Easy example of $f\in L_1^*\backslash L_\infty$?

If I'm not mistaken the dual of $L_1$ is $L_\infty$ whenever the measured space is $\sigma$-finite. So I know where not to look for an easy example of $f\in L_1^*\backslash L_\infty$. Does anyone know ...
0
votes
0answers
14 views

Definition of $L^p(C,B)$ where $C$ is a subset of the domain of the function and $B$ is a Banach space.

I am looking for the definition of $L^p(C,B)$ as declared in the title. I know that $C^s(C,B)$ is defined with the norm: $$\frac{ \| f(x) - f(y)\|_B}{|x-y|^s}\le A$$ i.e $f\in C^s(C,B)$ iff the ...
1
vote
1answer
75 views

In Rudin's proof of the completeness of $L^\infty$

This is closely related to a previous question: In a proof of the completeness of $L^\infty$ The following is a proof of completeness of $L^\infty$ by Rudin in his Real and Complex Analysis: Here ...
5
votes
2answers
94 views

In a proof of the completeness of $L^\infty$

The following is a proof of the completeness of $L^\infty$ in a lecture note by Hunter: Here are my questions: Can one (literally) replace $1/m$ in the whole proof with $\epsilon$ and replace $...
2
votes
2answers
41 views

If for $u \in L^2(\mathbb{R}^n)$ , we define $v(t)=u(x+th) $ $v: [0,1] \to \mathbb{R}$ $\Rightarrow^?$ $v \in L^2((0,1))$

I have a function $u(x) \in L^2(\mathbb{R}^n)$ ($n \geq 2$). Suppose we define another function $v$ as $$v:[0,1] \to \mathbb{R} $$ $$\quad \quad \quad \quad \quad \quad \ t \to u(x+th)$$ where $h \in ...
4
votes
1answer
49 views

What's an example of a function in $L^1(0,1)$ but not $L^p(0,1)$ for $p>1$?

What's an example of a function in $L^1(0,1)$ but not $L^p(0,1)$ for $p>1$? I've seen this answer but this is on an infinite domain. I'm interested only in $(0,1)$. I tried playing around with $\...
-2
votes
1answer
24 views

Convergence pw if converges in Lp space

Let $p\in[1,\infty]$ be given. If $f$ and $g$ are non-negative analytic functions such that the following holds: \begin{equation} \int_{\mathbb{R}}|\frac{\partial^ig(t)}{\partial^it}|^pdt=\int_{\...
0
votes
1answer
20 views

Riesz theorem and $L^p$ norm in expectation

I am reading a paper that uses the following fact, which claims to be from the Riesz's theorem: For a continuous stochastic process $\{ X_t \}$, let $u_t$ be its density function at each time ...
1
vote
1answer
29 views

Theorem 2.14 (The dual of $L^p(\Omega)$) in Lieb's Analysis book

The following pictures are Theorem 2.14 (The dual of $L^p(\Omega)$ in Lieb's Analysis book and its proof of the case $1<p<\infty$. My question is how to get the inequility (3) in the red box? ...
3
votes
0answers
29 views

Some sort of generalized Jensen inequality?

Let $(X, \mathcal{A},\mu)$ a measure space such that $\mu(X) > 0$ and let $f, g : X \rightarrow (0,\infty)$ be such that $f, g, f\log(f), f\log(g) \in L^1(\mu)$. Show that $$ \|f\|_1\log \|f\|_1 ...
1
vote
1answer
34 views

Construct an isometric isomorphism between $L_p(\mathbb{R})$ and $L_p[0;1]$

I know that $L_p(\mathbb{R})$ and $L_p[0;1], \; p<+\infty$ are isometrically isomorphic, which means that there is an isomorphism that respects norm. The question is how to construct it?