Fourier Series

Consider the function f(x) = a: for 0 < :17 < 7r.
(a) [5 marks] Extend f to an even function F(:c) of period 21r.
i. Sketch F(a:) in the range -37r < :17 < 31r.
ii. Find the Fourier Series of F(:c)
(b) [5 marks] Extend f to an odd function G(a:) of period 21r.
i. Sketch G(a:) in the range -37r < 17 < 37r.
ii. Find the Fourier Series of G(x)

(c) (Gibbs Phenomenon) The American mathematician J. W. Gibbs observed that near points of discon-
tinuity of f. the partial sums of the Fourier Series for f may overshoot by approximately 9% of the
jump. regardless of the number of terms. Consider

-1, -7r < (E < O,
f(”)'{ +1, O<a:<1r.
i. [2 marks] Show that the partial sums are given by
4 . 1 . ‘ 2 – 1
f2,,_1(a:) = ; [s1n(x) + § s1n(3a:)
ii. [2 marks] Sketch the partial sums f11,f51 and the original function f.
iii. [6 marks] Assume that for each partial sum the maximum occurs at (E = 1r/(2n). Show that
n1i_1’1;3f2n_1(7r/(2n)) m 1.18.