FOURIER SERIES ON THE CIRCLE 3

This will fit an initial temperature profile/(x) if and only if/ is expressed as a finite

trigonometric sum

(1.1.5) fix) = y^(Aw cos nx + Bn sin nx).

The coefficients An, Bn can be found by using the orthogonality relations

(1.1.6)

J—71

sin mx sin nxdx = 0 m ^ n,

(1.1.7)

(1.1.8)

/ "

J — 7

cos mx cos nxdx = 0 m^n,

f

J—71

sin mx cos

JIX

dx = 0 all m, n,

together with the norms: f*n

sin2

nxdx = n = f*n

cos2

nxdx,n= 1,2,... Thus

(1.1.9) A0 = ^ j

K

f(x)dx,

1 /•*

(1.1.10) An = — I f{x) cos nxdx n = l , 2 , ... ,

*

J.*

I

/•*

(1.1.11) Bn — —\ f{x)sinmxdx n=l,2,—

Fourier's thesis is that (1.1.5) will also remain true for N = oo, if the coeffi-

cients A„, #n are defined by these formulas. This is most easily done in case the series

E^o(l

A

«l +

\Bn\)

converges.

1.1.2 Absolutely Convergent Trigonometric Series

We begin the mathematics by considering functions defined by

(1.1.12) f{0) = ]P(A

n

cos nO + Bn sin nO)

n=0

where X^od

A

«l + l^nl) °°- The values of/ are determined on any interval of length

2TV. A standard choice is the interval T = (—n, 7r], where we identify 2JT-periodic

functions on R with functions on T. A function on T is considered continuous (resp.

differentiable) if the corresponding periodic function on R is continuous (resp. differen-

tiable). In concrete terms this means that/ is continuous (resp. differentiable) on (—n, n]

with/(;r - 0) =f{-jv + 0) (resp. f'{iz - 0) = / ' ( - * + 0)).