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diff --git a/Fourier Series.page b/Fourier Series.page index 5704fd1..4d025a9 100644 --- a/Fourier Series.page +++ b/Fourier Series.page @@ -91,18 +91,21 @@ f & = & \Sigma e^{inx}\\ $$ ##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 $2n+1$-dimensional 'Hilbert space'.(*perhaps someone can clarify this?*) +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 a $2n+1$-dimensional 'Hilbert space'.(*perhaps someone can clarify this?*) 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 -This is the definition for a function of real numbers. For a function on complex numbers, the above definition must be altered as follows: +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 *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.* - +#Basis Vectors of the Hilbert Space +The basis vectors of this Hilbert space are taken as follows: +basis vectors, $f_n = \frac{1}{\sqrt{2\pi}}e^{inx}$ + #Why is Fourier series useful? </b> Applications will be covered on Monday July 5, 2010. See you all soon!
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