2 INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS
f(x, 0) =
sin3x
= (3sinx sin3je)/4, whereas
sin2x
cannot be so expressed. In
order to work with these trigonometric sums, we note the property of orthogonality,
expressed as
r
1 sin mx sin nx dx = 0,
Jo
m 7^ n.
If a function has the form f{x) = Ylk=\
aksinfac, m e n w e
must have
JQ
fix) sin nx dx = 0 for n N.
Exercise 1.1.1. Show that ifN is odd,
sinN
x can be written as a finite sum of the
form Ylk=\
a
k
s m
kx.
Exercise 1.1.2. Suppose that we have a convergent series expansion of sin x =
X^fcli
ak
sin be on the interval 0 x n. Prove that a^ is nonzero for infinitely
many values ofk.
Hint: Assume a finite expansion and use the orthogonality relation (1.1.3) to obtain a
contradiction.
Exercise 1.1.3. Generalize Exercise 1.1.2 to any even power of sin x, showing
the impossibility of an expansion sin"* = J2k=\
a
k sin be/or 0 x n where
n = 4, 6,
Any multiple harmonic motion (1.1.2) is a 2TC -periodic function of time:/(jc, t +
2n) = f(x, t) for all oc JC, oo, oo t oo. It also is a 2n-periodic function
of x and is odd with respect to x = 0 and x = n, meaning that/(—JC) = —fix) and
fin +x) = —fin x) for all x.
Exercise 1.1.4. Suppose that fix), —oo x oo is given. Show that any two of
the following properties imply the third: (i)fix + 2n) = fix), Vx; (ii)f(-x) =
—fix), Wx; (iii)fin x) =—fin-\-x), VJC.
/. /. 1.2 Heat flow in solids
The vibrating string suggests the use of sine series, since the ends of the string are fixed.
More general trigonometric series are suggested by the study of heat flow in a circular
ring, assumed to have circumference 2n. In this model it is natural to assume that the
temperature uix, t) is a 2n-periodic function of x (but not periodic in time). Fourier
(1822) formulated the heat equation ut = uxx to describe the time evolution of the
temperature. It is satisfied by any function of the form (An cos nx + Bn sin
nx)e~n t
where
n = 0, 1, 2, ... , t 0 and n x n. Taking linear combinations of these, we arrive
at a "general solution"
TV
(1.1.4) uix, t) = Y^(A„cosn;c +
Bnsinnx)e~n r
.
n=0
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