Convergence of Fourier Series The period 2L function f (t) is called piecewise smooth if there are a only ﬁnite number of points 0 ≤ t1 < t2 < . . . < tn ≤ 2L where f (t) is not differentiable, and if at each of these points the left and right-hand limits lim f � (t) and lim f � (t) exist (although they might not be equal). t→ti− t→ti+ Recall that when we ﬁrst introduced Fourier series we wrote a0 f (t) ∼ + a1 cos(t) + a2 cos(2t) + a3 cos(3t) + . . . 2 +b1 sin(t) + b2 sin(2t) + b3 sin(3t) + . . . = ∞ a0 + ∑ an cos(nt) + bn sin(nt), 2 n =1 where we used ’∼’ instead of an equal sign. The following theorem shows that our subsequent use of an equal sign, while not technically correct, is close enough to be warranted. Theorem: If f (t) is piecewise smooth and periodic then the Fourier series for f 1. converges to f (t) at values of t where f is continuous 2. converges to the average of f (t− ) and f (t+ ) where it has a jump discon tinuity. Example. Square wave. No matter what the endpoint behavior of f (t) the Fourier series converges to: • ◦ • ◦ • ◦ ◦ ◦ ◦ ◦ ◦ ◦ � • ◦ • ◦ • Original f (t) • ◦ • • • • • ◦ ◦ ◦ ◦ ◦ Fourier series for f (t) Example. Continuous sawtooth: Fourier series converges to f (t). Example. • ◦ • ◦ • ◦ � Original f (t) Fourier series ◦