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Unformatted text preview: sics of Fourier analysis in its simplest context. A
function f : R → R is said to be periodic of period T if it satisﬁes the relation
f (t + T ) = f (t), for all t ∈ R . Thus f (t) = sin t is periodic of period 2π .
Given an arbitrary period T , it is easy to construct examples of functions
which are periodic of period T —indeed, the function f (t) = sin( 2T ) is periodic
of period T because
sin( 2π (t + T )
) = sin(
+ 2π ) = sin(
T More generally, if k is any positive integer, the functions
T and sin( 2πkt
T are also periodic functions of period T .
The main theorem from the theory of Fourier series states that any “wellbehaved” periodic function of period T can be expressed as a superposition of
sines and cosines:
f (t) = a0
+ a1 cos(
) + a2 cos(
) + . . . + b1 sin(
) + b2 sin(
) + ... .
(3.1) In this formula, the ak ’s and bk ’s are called the Fourier coeﬃcients of f , and
the inﬁnite series on the right-hand side is called the Fourier series of f .
Our ﬁrst goal is to determine how to calculate thes...
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This document was uploaded on 01/12/2014.
- Winter '14