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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 < ℓ . 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 = ℓ . For the application, we only care about x ’s between 0 and ℓ . But we are free to extend the definition of f ( x ) to x < 0 and x > ℓ , for computational purposes. We define F ( x ) to be a periodic extension of f ( x ) if i ) F ( x ) = f ( x ) for 0 < x < ℓ ii ) F ( x ) is periodic of period 2 ℓ There are many different periodic extensions of f ( x ). Most of them are pretty useless. For example define f ( x ) = 1 for 0 < x < π . We are leaving f ( x ) undefined for x ≥ π or x ≤ 0. x f ( x ) − π π 2 π 3 π 4 π Then the function F ( x ) graphed in x F ( x ) − π π 2 π 3 π...
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- Spring '10
- Fourier Series, 1 K, Bk, 4 k, 0 4 k, 1 4 k