diff options
| -rw-r--r-- | Fourier Series.page | 7 |
1 files changed, 4 insertions, 3 deletions
diff --git a/Fourier Series.page b/Fourier Series.page index df8441d..6e780a7 100644 --- a/Fourier Series.page +++ b/Fourier Series.page @@ -79,7 +79,7 @@ Summing these two functions we get the following: <center></center> #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 : @@ -90,7 +90,7 @@ f & = & \Sigma e^{inx}\\ \end{array} $$ - +##Definition of Inner Product of Functions We begin proving this hypothesis by considering that any function on the right-hand side of our hypothesis is uniquely defined by the co-efficients of the terms a_0 through a_n and b_1 through b_n. This can be taken to mean that every function is really a vector in an n-dimensional Hilbert space. 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 in Hilbert space: @@ -98,10 +98,11 @@ We now proceed to define certain operations on these functions in Hilbert space. ---> define inner product here This is the definition for a function of real numbers. For a function on complex numbers, the above definition must be altered as follows: - +--> altered function here *Note: These are purely definitions, and we are now definining the inner product to ensure that inner product of f and f is a real number.* + #Why is Fourier series useful? </b> Applications will be covered on Monday July 5, 2010. See you all soon!
\ No newline at end of file |