From ea7f1a48a7443539420fa50bf04b0d356f560703 Mon Sep 17 00:00:00 2001 From: luccul Date: Sun, 11 Jul 2010 00:56:04 +0000 Subject: A bit about L2 convergence --- ClassJuly5.page | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/ClassJuly5.page b/ClassJuly5.page index 7e19aff..f6b879d 100644 --- a/ClassJuly5.page +++ b/ClassJuly5.page @@ -73,6 +73,8 @@ In fact, every function of the kind described above does have a Fourier sine exp How do we know that the Fourier series of a square wave or sawtooth function converges? -The answer to this question depends greatly on the type of convergence desired. Aside from the convergence we already proved, the next easiest type of convergence is $L^2$ or root-mean-square convergence. The formal statement is that +The answer to this question depends greatly on the type of convergence desired. Aside from the convergence we already proved, the next easiest type of convergence is $L^2$ convergence. The formal statement is that -$$ \lim_{N \to \infty} \sqrt{\int_0^{2\pi} \left| \sum_{n = - N}^N c_n e^{in\theta} - f(\theta) \right|^2} = 0 $$ \ No newline at end of file +$$ \lim_{N \to \infty} \sqrt{\int_0^{2\pi} \left| f(\theta) - \sum_{n = - N}^N c_n e^{in\theta} \right|^2} = 0 $$ + +In other words, as we let $N$ go to infinity, the root mean square error of our approximation gets arbitrarily small. \ No newline at end of file -- cgit v1.2.3