Fourier Series

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. PLACE THIS ORDER OR A SIMILAR ORDER WITH US TODAY AND GET AN AMAZING DISCOUNT :)