b2 f sin 2t use of the inner product makes

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Unformatted text preview: and f , lim (f (t)), t→t0 + lim (f (t)), lim (f (t)), t→t0 − t→t0 + lim (f (t)), t→t0 − all exist. The following theorem, proven in more advanced books,3 ensures that a Fourier decomposition can be found for any function which is piecewise smooth: Theorem 2. If f is any piecewise smooth periodic function of period T , f can be expressed as a Fourier series, f (t) = a0 + 2 ∞ ak cos( k=1 2πkt )+ T ∞ bk sin( k=1 2πkt ) T where the ak ’s and bk ’s are constants. Here equality means that the inﬁnite sum on the right converges to f (t) for each t at which f is continuous. If f is discontinuous at t0 , its Fourier series at t0 will converge to the average of the right and left hand limits of f as t → t0 . Exercises: 3.1.1. The function f (t) = cos2 t can be regarded as either periodic of period π or periodic of period 2π . Choose one of the two periods and ﬁnd the Fourier series of f (t). (Hint: This problem is very easy if you use trigonometric identities instead of trying to integrate directly.) 3.1.2. T...
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This document was uploaded on 01/12/2014.

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