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-rw-r--r--Fourier Series.page9
1 files changed, 7 insertions, 2 deletions
diff --git a/Fourier Series.page b/Fourier Series.page
index 67b9027..430d536 100644
--- a/Fourier Series.page
+++ b/Fourier Series.page
@@ -78,7 +78,7 @@ Summing these two functions we get the following:
<center>![$\cos^{2n}(x) + cos^{2n+1}(x)$](/cos10x-cos11x.gif)</center>
-#<b>What is the Fourier Series actually</b>
+#<b>What is the Fourier Series actually?</b>
##Initial Hypothesis
Now, to prove that the Fourier series is indeed true, we begin with the following hypothesis:
Let $f : \mathbb I \rightarrow \mathbb C$ be a continuous, periodic function where $I$ is some time interval(period of the function). Then it can be expressed as :
@@ -95,7 +95,12 @@ We begin proving this hypothesis by considering that any function on the right-h
We now proceed to define certain operations on these functions in Hilbert space. One operation that will be particularly useful is that of the inner product of two functions:
----> define inner product here
+$$
+\begin{array}{ccl}
+inner product, (f,g) & = & \int_0^{2\pi} f g dx\\
+\mid f \mid ^2 & = & (f, f) & = & \int_0^{2\pi} f^2 dx\\
+\end{array}
+$$
This is the inner product of 2 real-number functions. For a function on complex numbers, the above definition must be altered as follows:
--> altered function here