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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