diff options
| -rw-r--r-- | Fourier Series.page | 19 |
1 files changed, 19 insertions, 0 deletions
diff --git a/Fourier Series.page b/Fourier Series.page index 4d025a9..90a64f1 100644 --- a/Fourier Series.page +++ b/Fourier Series.page @@ -106,6 +106,25 @@ This is the inner product of 2 real-number functions. For a function on complex The basis vectors of this Hilbert space are taken as follows: basis vectors, $f_n = \frac{1}{\sqrt{2\pi}}e^{inx}$ +Any basis vectors could conceivable have been assumed on the condition that the basis vectors are orthonormal. (*Note: These particular basis vectors are chosen to prove that Fourier series exists*) + +In order to prove orthonormality of the basis vectors: + +--> orthonormal proof goes here + +##Determining Coefficients of the Basis vectors +In any vector space, the inner product of a vector and its basis vector gives the coefficient. For example, consider a 2-dimensional vector as shown below: + +--> image of a 2d vector + +--> demo of coefficient being true + +Extending this principle to the case of an n-dimensional vector: +--> compute inner product here and then continue to show what the coefficient formula is + +##Proving that this function is does indeed completely represent $f$ + +--> don't quite remember this part #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 |