From c4a2bf1158888668f27faa7c0de0a7d8264a2057 Mon Sep 17 00:00:00 2001 From: siveshs Date: Sat, 3 Jul 2010 04:49:59 +0000 Subject: section 3 editing --- Fourier Series.page | 9 ++++++++- 1 file changed, 8 insertions(+), 1 deletion(-) diff --git a/Fourier Series.page b/Fourier Series.page index e339854..c764b6a 100644 --- a/Fourier Series.page +++ b/Fourier Series.page @@ -159,7 +159,14 @@ $$ Extending this principle to the case of an n-dimensional vector: -Let $f$ be the periodic function expressed as $f= \Sigma a_n \frac{1}{\sqrt{2\pi}} \, e^{inx} = \Sigma a_n \, f_n$ where $a_n \in \mathbb C$ +Let $f$ be the periodic function expressed as $f= \Sigma a_n \frac{1}{\sqrt{2\pi}} \, e^{inx} = \Sigma a_n \, f_n$ where $a_n \in \mathbb C$ and $f_n$ are the basis vectors. + +Inner product of the vector (in this case the function $f$) with the some basis vector $f_m$ is: +$$ +\begin{array}{ccl} +(f, f_m) & = & \left( \Sigma a_n\,f_n , f_m \right)\\ +& = & \Sigma a_n\,\left(f_n , f_m \right)\\ + ##Proving that this function is does indeed completely represent $f$ -- cgit v1.2.3