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