3
votes
0answers
26 views

Difference between almost everywhere convergence of whole Fourier series and a subseries of $L^2$ functions

Let $f \in L^2(\mathbb{R})$ and $F_{N}f$ denote the $N$-th partial sum of its Fourier series. Then $||F_{N}f -f||_{L^{2}} \rightarrow 0$ as $N \rightarrow \infty$. But this implies there exists a ...
1
vote
0answers
23 views

A useful criterion in dispersive PDE.

I would like to prove the following theorem: Consider two Banach spaces $X\hookrightarrow Y$ and $1<p,q\leq\infty$. Let $(f_n)_{n\geq 0}$ be a bounded sequence in $L^q(I,Y)$ and let $f:I\mapsto Y$ ...
2
votes
1answer
73 views

Convergent sequence in Lp has a subsequence bounded by another Lp function

For $E$ a measurable set and $1\leq p<\infty $, assume $f_n\to f$ in $L^p(E)$. Show that there is a subsequence $\{f_{n_k}\}$ and a function $g\in L^p(E)$ such that $|f_{n_k}|\leq g$ almost ...
2
votes
1answer
46 views

Integral convergence involving Lp and Sobolev spaces

Quick question, any contribution or hint would be appreciated: How does it follow that: $$\lim\limits_{k \rightarrow \infty}\int_{\Omega}a(u_{k})\frac{\partial u_{k}}{\partial x}\frac{\partial ...
0
votes
1answer
31 views

Convergence of a sequence in two different $L_p$ spaces

If I have a sequence $\{f_n\}_{n\in \mathbb{N}} \subset L_p\cap L_q$ that is Cauchy under both norms, I am wondering if $f_n \to f$ in $L_p$ implies that $f_n\to f$ in $L_q$. I have been working on ...
1
vote
1answer
35 views

Implications of Weak convergence in Sobolev Spaces

A quick question regarding weak convergence in Sobolev Spaces. If $u_{k} \rightharpoonup u$ in $W^{1,p}(\Omega)$ for bounded $\Omega$ then can we show that $\nabla u_{k} \rightharpoonup \nabla u$ in ...
1
vote
0answers
38 views

Convergence of $L^1$ functions

Given that $\Omega$ is bounded and $a_{ij}(u_{k}) \rightarrow a_{ij}(u)$ in $L^{1}(\Omega)$, $a_{i0}(u_{k}) \rightarrow a_{i0}(u)$ in $L^{1}(\Omega)$, $\frac{\partial u_{k}}{\partial x_{j}} ...
1
vote
1answer
30 views

Relation between convergences in $L^{p}$ for probability spaces.

I have read that for a probability space $(\Omega,\Sigma,P)$ it is true that $f \in L^{p}(\Omega,\Sigma,P)$ implies $f \in L^{q}(\Omega,\Sigma,P)$ if $p>q$, and hence $L^{2} \subset L^{1}$. I'm ...
2
votes
1answer
37 views

Integral convergence and weak convergence

Given that $\Omega \subset \mathbb{R}^{n}$ is a connected bounded Lipshitz domain and $u_{k} \rightharpoonup u$ in $W^{1,p}(\Omega)$. We denote $\Gamma$ as the boundary of the domain. We have the ...
2
votes
1answer
62 views

How Lp spaces relate regarding convergence

I know that for a bounded $\Gamma$ it follows that $L^{q}(\Gamma) \subset L^{p}(\Gamma)$ if $q > p$. I have a few questions regarding how $L^{p}$ spaces relate with regard to convergence. Consider ...
-1
votes
1answer
29 views

The convergence in $L^{p_1}$ and $L^{p_2}$

Suppose $f_k$ is a sequence of $M-$measurable function. Let $p_1$ and $p_2\in[1,\infty)$,and $f_k\in L^{p_1}\cap L^{p_2}$.Also suppose $\exists g\in L^{p_1}$ and $h\in L^{p_2}$ s.t. $f_k\rightarrow g$ ...
1
vote
0answers
20 views

Show $\lim_{k \rightarrow \infty} (f_{n_k}, g) = (f,g)$ for a subsequence $f_{n_k}$ of $f_n$ and $g$ in $L^2([0,1])$

I would like to show that for any sequence $f_n(x) \in L^2([0,1])$ with $||f_n||_2=1$ for all $n$, there exists an $f\in L^2([0,1])$ and a subsequence $(f_{n_k})$ such that for every $g\in L^2([0,1])$ ...
5
votes
1answer
102 views

Subsequence convergence in $L^p$

I recall a fact that for functions $f_1,f_2,\ldots\in L^1$ such that $\|f_n-f\|_1\rightarrow 0$ as $n\rightarrow\infty$, there exists a subsequence $f_{n_i}$ that converges to $f$ almost everywhere. ...
3
votes
1answer
77 views

$L^2$ norm of series with complex exponentials

Suppose that $f\in L^2(\mathbb{R}/2\pi\mathbb{Z})$ takes the form $$f(\theta)=\sum_{n=1}^\infty a_ne^{in\theta}.$$ The function $$f_r(\theta)=\sum_{n=1}^\infty r^na_ne^{in\theta}$$ is a harmonic ...
0
votes
1answer
63 views

Constructing Sequences in Lp

Consider Banach Space $L^{p}(U)$ where $U$ is bounded open subset in $\mathbb{R}^{n}$. Take a bounded sequence $\{u_{m}\}_{m}^{\infty}$ in $L^{p}(U)$. Consider a subsequence ...
3
votes
1answer
68 views

Bounding for convolution convergence

Suppose $f\in L^p(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Define $$K_t(x)=\dfrac{1}{t}K\left(\dfrac{x}{t}\right)$$ I'm trying to prove that $\lim_{t\rightarrow ...
0
votes
1answer
79 views

Uniform convergence with Lp functions

I have a convergence question: Say we have a sequence of functions $\{u_{m}\}_{m=1}^{\infty}$ where $\{u_{m}\}_{1}^{\infty} \subset L^{p}(U)$ and where $U$ is bounded. Consider $u^{\epsilon} := ...
0
votes
1answer
39 views

Implication of $L^p$ convergence

Take $U$ as open subset of $\mathbb{R}^{n}$. If $u_{m} \rightarrow u$ in $L^{p}(U)$ then does it follow that $||u_{m}||_{L^{p}(U)} \rightarrow ||u||_{L^{p}(U)}$?
1
vote
1answer
401 views

Does $L^p$-convergence imply pointwise convergence for $C_0^\infty$ functions?

It is stated in my professor's notes that, given a sequence $\{f_j\}$ of $C_0^\infty(\Omega)$ functions (infinitely differentiable with compact support), and a function $g\in C_0^\infty(\Omega)$, all ...
2
votes
0answers
43 views

Lebesgue's convergence for $H(u_n)\nabla u_n$ where $H$ is not everywhere defined

Consider the Heaviside function that is undefined in zero, i.e. $$H(t)=\begin{cases} 1&t>0 \\ 0&t< 0\end{cases}$$ Now consider a sequence of $H^1(\Omega)$-functions $u_n\to u$ in the ...
2
votes
1answer
67 views

$\int_{B} f_n \phi \rightarrow 0$ if the Weak-$L^p$ norm of $f$ tends to zero?

Let $f_n \in L^p(B)$ be a sequence where $B$ is some ball in $\mathbb{R}^n$. Assume that $\|f_n\|_{L^p(B)} \rightarrow 0$ when $n\rightarrow \infty$, then by some $\phi \in C^\infty_0(B)$ applying ...
2
votes
1answer
54 views

Convergence of $\varphi_n(x):=\frac{\varphi(nx)}{n}$ in Schwartz space

I want to find all $\varphi\in\mathcal S(\mathbb R)$ for which the sequence $\varphi_n(x):=\frac{\varphi(nx)}{n}$ converges in $\mathcal S(\mathbb R)$. The first step, I have already managed to do by ...
0
votes
2answers
147 views

Determine if the following sequence converge in the quadratic mean

"For integer $n$ let $f_n(x) = \dfrac{1}{\sqrt{1 + nx^2}}$ say if the sequence $f_n$ converges in quadratic mean." This is what I have concluded so far: $$\lim\limits_{n\rightarrow \infty}f_{n}(0) ...
2
votes
2answers
60 views

Convergence in $L_1$ and Convergence of the Integrals

Am I right with the following argument? (I am a bit confused by all those types of convergence.) Let $f, f_n \in L_1(a,b)$ with $f_n$ converging to $f$ in $L_1$, meaning $$\lVert f_n-f \rVert_1 = ...
1
vote
1answer
77 views

Need help in showing that $F(x)/x^{1/q}$ goes to $0$ as $x$ goes to $0$ and $\infty$.

$1<p<\infty$, $f\in L^{p}(0,\infty)$, $p^{-1}+q^{-1}=1$, define $$F(x)=\int_{0}^{x}f(t)dt,$$ then I need to show that $\frac{F(x)}{x^{\frac{1}{q}}}\rightarrow 0$ as $x\rightarrow 0$ and ...
1
vote
1answer
53 views

Continuity of conditional expectation in $L_p$

I'm looking at a probability space $(\Omega,\mathcal{F},P)$. Let $1\leq p<\infty$, and let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$. I'm then asked to show that, for $X\in L_p(P)$, ...
1
vote
1answer
43 views

$L^p$ convergence proof check

I don't have much experience with measure theory, so I want to make sure that I'm not making any bad mistakes. I also want to be sure that the theorem is true so I can use it. Theorem: Let $\{u_i\}$ ...
1
vote
1answer
25 views

Remainder of a series converges uniformly?

Let $B \subset \Bbb R^{\Bbb N}$ and $p \geq 1$. Suppose $$ \sup_{u\in B}\sum_{n=0}^\infty |u_n|^p \leq 1,\qquad \sup_{u\in B}\sum_{n=0}^\infty |u_{n+1}-u_n|^p\leq 1 $$ Is it true that $$ \sup_{u\in ...
2
votes
2answers
148 views

Weak limit of disjoint normalized sequence in $L^p$

I want to prove that the weak limit of a disjoint normalized (pairwise disjoint supports, elements of norm $1$) sequence $(f_n)$ in $L^p$ for $p >1$ is zero ? I started with the measure of ...
1
vote
0answers
440 views

Rademacher function and weak convergence

The function $r_{n}:[0,1]\rightarrow \{-1,1\}$ be defined by $r_{n}(t)=\operatorname{sgn}(\sin(2^{n}\pi t))$ is known as the $n$-th Rademacher function.a) Show that $r_{n}\xrightarrow{w}0$ in ...
2
votes
1answer
130 views

Uniform convergence in $L^p$-spaces

Let $f\in L^p(0,\infty)$, $p>1$. Show that $\int_0^\infty f(x)\frac{\sin xy}{x} dx$ converges uniformly in $y$ in every finite interval. Show also that $|g(t+y)-g(y)|\leqslant M|t|^{\frac{1}{p}}$. ...
1
vote
2answers
39 views

$\ell_{p}$ space closed to addition

I'm trying to show that $\ell_{p}$ is a vector space for any $1\leqslant p<\infty$ . So given two infinite series $\left(x_{n}\right)_{n=1}^{\infty}$ and $\left(y_{n}\right)_{n=1}^{\infty}$ ...
8
votes
1answer
261 views

Convergence in $L^1$ space

Suppose that $f_{n}$ is a sequence of measurable functions, in a finite measure space, $f_{n}\to f $ in $m$-measure and that there exists $g$ in $L^1$ such that $\vert f_n\vert \le g$. Prove that ...
3
votes
1answer
94 views

$f_k \rightarrow f$ in $L^p$ and that $g_k \rightarrow g$ weakly in $L^q$. Show that $f_k g_k \rightarrow fg$ weakly in $L^1$

I want to solve the following exercise, and I thankfully welcome some hints. Note that this is not homework. Problem: Let $1 < p,q < \infty$ be conjugate exponents. Assume $f_k \rightarrow ...
1
vote
1answer
456 views

Convergence in $\ell^p$ norm provided it weakly converges.

I need some help with the following problem : $1<p < \infty$ , let $x_n$ be a sequence in $\ell^p$ and also $x\in \ell^p$ . I am interested in showing $$\lim_{n\to \infty} \|x_n-x\|_p\to0$$ ...
1
vote
2answers
86 views

$f_n$ $\in$ $L_2(\mu)$, the limit $ f \in L_2(\mu)$

If $f_n \in L_2(\mu)$, $f_n\rightarrow f$ almost everywhere, this is not enough to conclude $f\in L_1(\mu)$. But is it enough to conclude whether $f\in L_2(\mu)$ or $$\lim_{n \to ...
4
votes
3answers
764 views

Convergence in $L^{\infty}$ norm implies convergence in $L^1$ norm

Let $\{f_n\}_{n\in \mathbb{N}}$ be a sequence of measurable functions on a measure space and $f$ measurable. Assume the measure space $X$ has finite measure. If $f_n$ converges to $f$ in ...
11
votes
1answer
1k views

Strong and weak convergence in $\ell^1$

Let $\ell^1$ be the space of absolutely summable real or complex sequences. Let us say that a sequence $(x_1, x_2, \ldots)$ of vectors in $\ell^1$ converges weakly to $x \in \ell^1$ if for every ...
7
votes
3answers
211 views

Convergence of functions in $L^p$

Let $\{f_k\} \subset L^2(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain and suppose that $f_k \to f$ in $L^2(\Omega)$. Now if $a \geq 1$ is some constant, is it possible to say ...
1
vote
0answers
178 views

Convergent in $L^1(0,1)$ but not in $L^2(0,1)$ help understanding a paper from arxiv

http://arxiv.org/pdf/math/0205003v1 In around equation (1.1) the author says "By necessity all authors have been led in one way or another to the natural approximation $$F(n) := \sum_{a=1}^n \mu(a) ...
2
votes
1answer
184 views

Convergence of integrals in $L^p$ and $L^{p/(p-1)}$

Let $X$ be a measure space and let $f_{n}$ be a sequence of functions which converge pointwise to a function $f$ in $L^{p}(X)$ where $p>1$ and suppose $g_{n}$ is a sequence of functions which ...