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Diffstat (limited to 'Fourier Series.page')
| -rw-r--r-- | Fourier Series.page | 9 |
1 files changed, 8 insertions, 1 deletions
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 $f$ |