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Unformatted text preview: Periodic Extensions We know that every (sufficiently smooth) periodic function has a Fourier series expansion. It is fairly common for functions arising from certain applications to be defined only on a finite interval < x < . This is the case if, for example, f ( x ) is the vertical displacement of a string from the x axis at position x and if the string only runs from x = 0 to x = . For the application, we only care about x s between 0 and . But we are free to extend the definition of f ( x ) to x < 0 and x > , for computational purposes. We define F ( x ) to be a periodic extension of f ( x ) if i ) F ( x ) = f ( x ) for 0 < x < ii ) F ( x ) is periodic of period 2 There are many different periodic extensions of f ( x ). Most of them are pretty useless. For example define f ( x ) = 1 for 0 < x < . We are leaving f ( x ) undefined for x or x 0. x f ( x ) 2 3 4 Then the function F ( x ) graphed in x F ( x ) 2 3...
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- Spring '10
- Fourier Series