Can anybody tell me why $\sin(nx)$ converges weakly in $L^2(-\pi,\pi)$. I can't see how $\sin(nx)$ can converge?
Explanation with any other example will be nice as well.
Why $\sin(nx)$ converges weakly in $L^2(-\pi,\pi)$?
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1This is probably an overkill, but from [Parseval's identity](http://en.wikipedia.org/wiki/Parseval%27s_identity) we know that for each $f\in L_2$ we have \sum (c_k^2+d_k^2)<\infty, where $c_k=\int_{-\pi}^\pi f(x) \sin kx$ and $d_k$ is the same thing with cosines. This implies that $c_k\to 0$. Since this is true for each $f$, this is the weak convergence of $\sin nx$. – 2012-05-28
3 Answers
(I assume you mean $L^2$ on a bounded interval since $\sin(nx)$ has to be an element of $L^2$.) A sequence converges weakly in a Hilbert space if its image under any bounded linear functional converges. What are the bounded linear functionals on $L^2$? Then use the Riemann Lebesgue lemma.
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2@Norbert: I assumed $L^2$ on a bounded interval. Otherwise $\sin(nx)$ would not be an element in $L^2$. – 2012-05-28
$\newcommand{\norm}[1]{\lVert{#1}\rVert}$Up to a constant, the functions $\sin nx$ are elements of the standard orthonormal basis in the Hilbert space $L_2(-\pi,\pi)$. (The constant is the same for all $n$'s.)
If $\{e_n; n\in\mathbb N\}$ is an orthonormal basis in some Hilbert space $X$, then for any $f\in X$ we have $f=\sum_{n\in\mathbb N} \langle f,e_n \rangle e_n$ and $\norm{f}^2 = \sum_{n\in\mathbb N} |\langle f,e_n \rangle|^2.$
This formula is called Parseval's identity.
The fact that the series $\sum\limits_{n\in\mathbb N} |\langle f,e_n \rangle|^2$ converges implies that $\langle f,e_n \rangle \to 0$. This is true for each $f$, which means that $e_n$ converges weakly to $0$.
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0I like your argument. It is quite neat. However, it doesn't work if you replace $L^2$ by $L^p$ for a general $p \in [1,\infty)$ because then you don't have an inner product. Alex's works for $p \in [1,\infty)$. – 2012-06-26
Edit: My previous proof was incorrect, showing that $\sin(x/n)\to 0$ weakly rather than $\sin(nx)$.
Note that $\sin(nx)\in L^2([a,b])$ for any $a,b\in\mathbb R$ but $\sin(nx)\notin L^2(\mathbb R)$, so I assume you are working over some finite interval $[a,b]$. To see that $\sin(nx)\to 0$ weakly, note that the linear functionals on $L^2([a,b])$ are of the form $\int_a^b f(x)\cdot dx$ where $f\in L^2([a,b])$. For any $f\in L^2([a,b])$ and $\epsilon>0$, we have some step function $f'\in L^2([a,b])$ such that $\|f-f'\|_2<\epsilon$. Since $f'$ is a step function, we have (by definition) constants $c_1,\ldots,c_n$ such that $a=c_1<\cdots
This technique is fairly standard for proving uniform convergence of $L^2$ functions: you prove convergence for any linear functional defined by a step function and then use the density of step functions in $L^2$ to complete the proof.
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0@MartinSleziak Whoops. Totally reversed that. Editing to fix. – 2012-05-28