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)).
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