Math 121A: Homework 6-

1. Consider the function f(x) = x(π - x) defined on 0 ≤ x ≤ π.

(a) Calculate the Fourier sine series f_s and Fourier cosine series f_c of f.

(b) For both f_s and f_c, 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 f_s(π/2) and f_c(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^-∞ c_ne^inx. Suppose that f'(x) = ∞Σ-∞ d_ne^inx and f(x - l) = _∞Σ^-∞ q_ne^inx. Express the coefficients d_n and q_n in terms of c_n.

3. Let f and g be periodic functions on the interval -π ≤ x < π, with complex Fourier series

f(x) = _∞Σ^n=-∞c_ne^inx,           g(x) = _∞Σ^n=-∞ d_ne^inx.

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 c_n and d_n.

4. (a) Calculate the Fourier series of the function defined on -π ≤ x < π as

(b) Let f_λ(x) be a filtered version of f, in which the higher frequency components are damped: if f has Fourier coefficients a_n and b_n, then let f_λ have coefficients λⁿa_n, λⁿb_n. Plot f, and f_λ for the cases of λ = 0.7, 0.8, 0.9.

(c) By making use of the result from question 3, find a function K_λ(x) such that f_λ = K_λ ∗ f for any function f. Plot K_λ for λ = 0.7, 0.8, 0.9.