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-rw-r--r-- | Fourier Series.page | 8 |
1 files changed, 6 insertions, 2 deletions
diff --git a/Fourier Series.page b/Fourier Series.page index e4d4e6f..48f1bdb 100644 --- a/Fourier Series.page +++ b/Fourier Series.page @@ -110,7 +110,7 @@ inner product, (f,g) & = & \int_0^{2\pi} f \,\bar g \,dx\\ \end{array} $$ -*Note: These are purely definitions, and we are now definining the inner product to ensure that inner product of f and f is a real number.* +*Note: These are purely definitions, and we are defining the inner product to ensure that the inner product of f and f is a real number.* ##Basis Vectors of the Hilbert Space The basis vectors of this Hilbert space are taken as follows: @@ -120,7 +120,11 @@ Any basis vectors could conceivable have been assumed on the condition that the In order to prove orthonormality of the basis vectors: ---> orthonormal proof goes here +$$ +\begin{array}{ccl} +(f_n,f_m) = \int_0^{2\pi} \, \frac{1}{\sqrt{2\pi}} \, e^{inx} \, \bar {\frac{1}{\sqrt{2\pi}} \, e^{inx}} \, dx\\ +\end{array} +$$ ##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: |