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| -rw-r--r-- | Fourier Series.page | 9 |
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></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 |