Unformatted text preview: many applications is:
Theorem 1. If f is a continuous periodic function of period T , with condinuous derivative f , then f can be written uniquely as a superposition of sines
and cosines, in accordance with (3.9), where the ak ’s and bk ’s are constants.
Moreover, the inﬁnite series on the righthand side of (3.9) converges to f (t) for
every choice of t.
However, often one wants to apply the theory of Fourier series to functions
which are not quite so wellbehaved, in fact to functions that are not even be
continuous, such as the function in our previous example. A weaker suﬃcient
condition for f to possess a Fourier series is that it be piecewise smooth.
The technical deﬁnition goes like this: A function f (t) which is periodic of
period T is said to be piecewise smooth if it is continuous and has a continuous
derivative f (t) except at ﬁnitely many points of discontinuity within the interval
[0, T ], and at each point t0 of discontinuity, the right and lefthanded limits of 67 f...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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