18.03 Class 21, April 3 Fun with Fourier series [1] If f(t) is any decent ...

pi |---/ t = (pi/L)x| /|| / ||/ |-------------- x| Lso we have g(x) = f(t)= (4/pi) (sin(t) + (1/3) sin(3t) + ... ))= (4/pi) (sin(pi x / L) + (1/3) sin(3 pi x / L) + ...Then general appearance of the Fourier series for a function of period2L isg(x) = a0/2 + a1 cos(pi x / L) + a2 cos(2 pi x / L) + ...+ b2 sin(pi x / L) + b2 sin(2 pi x / L) + ...The integral formulas for an and bn can of course be translatedinto the variable x:an = (1/L) integral_{-L}^L g(x) cos(pi n x / L) dxbn = (1/L) integral_{-L}^L g(x) sin(pi n x / L) dx[5] Gibbs effect. What happens at points where f(t) isdiscontinuous?Definition: A function f(t) is "piecewise continuous" if it iscontinuous except at some sequence of points $c_n$, and for each $n$both one-sided limits exist at $c_n$. (If the two limits are equal thenthe function is continuous there.)Notation: lim_{t-->c from above} f(x) = f(c+)lim_{t-->c from below} f(x) = f(c-)To find out, I got Matlab to sum the first 10 nonzero terms of theFourier series for (pi/4) sq(t) ; that is,sin(t) + (1/3) sin(3t) + ... + (1/19) sin(19t)One thing's clear: the value of the Fourier series at 0 is 0 .This is a general fact:If f(t) isn't continuous at t = a , then the sum at t = aconverges to(f(a+) + f(a-))/2 .More surprising is that the Fourier approximation has visibleoscillation about the constant values of sq(t),