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August 14

Convergence Question

Consider the following complex-valued two-way sequences: $S=...,a_{-2},a_{-1},a_0,a_1,a_2,...$ with the property that $\sum _{n\in \mathbb{Z}}|a_n{|}^2$ converges. Suppose you have a sequence of such sequences $S_1,S_2,S_3,...$ which and converges to $...,b_{-2},b_{-1},b_0,b_1,b_2...$ Show that $\sum _{n\in \mathbb{Z}}|b_n{|}^2$ converges too. Widener (talk) 12:34, 14 August 2012 (UTC)

How are you defining convergence in the space of sequences? Rckrone (talk) 21:48, 14 August 2012 (UTC)

They're convergent sequences, so presumably you just identify them with their limits. Any other definition would seen strange to me. --Tango (talk) 22:49, 14 August 2012 (UTC)

I think it means pointwise convergence. So if $S_i=...,a_{-2}^i,a_{-1}^i,a_0^i,a_1^i,a_2^i,...$, then

$b_n=\underset{i}{\lim }{a}_n^i$. Of course, in this interpretation, the stated result is false. Define $a_n^i=1$ if $|n|<i$ and $=0$ otherwise.--121.73.35.181

Sorry. It does not mean pointwise convergence; it means convergence under the metric $d(S_i,S_j)=\sum _{k\in \mathbb{Z}}|a_k^i-a_k^j{|}^2$. --Widener (talk) 00:06, 15 August 2012 (UTC)(talk) 23:54, 14 August 2012 (UTC)

You should think of using the triangle inequality with $b_n=a_n^i-(a_n^i-b_n)$. Sławomir Biały (talk) 01:40, 15 August 2012 (UTC)

If it helps: a sequence $a_n$ such that $\sum |a_n{|}^2$ converges must converge to ____? Sławomir Biały (talk) 22:41, 14 August 2012 (UTC)

I misunderstood your question. This observation will not help. Sławomir Biały (talk) 01:37, 15 August 2012 (UTC)

So you are dealing with a convergent sequence $S_i$ in the Hilbert space ${\ell }^2(\mathbb{Z})$. A convergent sequence is bounded, and the norm is continuous. So $\Vert S{\Vert }_2=\underset{i\to \mathrm{\infty }}{\lim }\Vert S_i{\Vert }_2<+\mathrm{\infty }$ (which you can prove directly by the triangle inequality as suggested above). --pma 09:39, 15 August 2012 (UTC)

Thanks ! 00:28, 16 August 2012 (UTC)Widener (talk)

Example of Limitation of Riemann integration

How do You show That $\sum _{k=2}^{\mathrm{\infty }}\frac{1}{\log (k)}\sin (kx)$ is Not a Fourier series Of a Riemann integrable Function? Widener (talk) 14:36, 14 August 2012 (UTC)

Why don't you start by trying to explain why

$\frac{{\pi }^2}{3}+\sum _{k=1}^{\mathrm{\infty }}\frac{4(-1{)}^k}{k^2}\cos (kx)$

is a Fourier series of a Riemann integrable function? — Fly by Night (talk) 17:17, 14 August 2012 (UTC)

Is it because the sequence of Fourier coefficients is not in $l_2(\mathbb{Z})$? Widener (talk) 23:06, 14 August 2012 (UTC)

Yes. Sławomir Biały (talk) 00:19, 15 August 2012 (UTC)