Math 121A: Homework 6-
1. Consider the function f(x) = x(π - x) defined on 0 ≤ x ≤ π.
(a) Calculate the Fourier sine series fs and Fourier cosine series fc of f.
(b) For both fs and fc, plot the sum of the first four non-zero terms over the range -π ≤ x < π and compare with the exact solution. Which series is more accurate and why?
(c) By evaluating fs(π/2) and fc(π/2), find two expressions for π3 and π2 as infinite series of fractions.
(d) By using Parseval's theorem find expressions for π6 and π4 as infinite series of fractions.
2. Let f(x) = ∞Σ-∞ cneinx. Suppose that f'(x) = ∞Σ-∞ dneinx and f(x - l) = ∞Σ-∞ qneinx. Express the coefficients dn and qn in terms of cn.
3. Let f and g be periodic functions on the interval -π ≤ x < π, with complex Fourier series
f(x) = ∞Σn=-∞cneinx, g(x) = ∞Σn=-∞ dneinx.
Let f ∗ g be the convolution of f and g, defined as
(f ∗ g)(x) = -π∫π f(y)g(x - y)dy.
Calculate the complex Fourier series coefficients of f ∗ g in terms of cn and dn.
4. (a) Calculate the Fourier series of the function defined on -π ≤ x < π as
(b) Let fλ(x) be a filtered version of f, in which the higher frequency components are damped: if f has Fourier coefficients an and bn, then let fλ have coefficients λnan, λnbn. Plot f, and fλ for the cases of λ = 0.7, 0.8, 0.9.
(c) By making use of the result from question 3, find a function Kλ(x) such that fλ = Kλ ∗ f for any function f. Plot Kλ for λ = 0.7, 0.8, 0.9.