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 right-hand 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 well-behaved, 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 left-handed limits of 67 f...
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This document was uploaded on 01/12/2014.
- Winter '14