Fall 2018 Problem 3: Show that for every f ∈ C(𝕋) and ϵ > 0 there is an initial condition g ∈ C(𝕋) for which there is a solution u(x, t) to the heat equation on a ring with u(x, 0) = g(x) and |u(x, 1) − f(x)| < ϵ for every x ∈ 𝕋.

Fall 2014 Problem 6:

(a) By choosing a suitable even, periodic extension for f, calculate the Fourier cosine series for f = sin x, x ∈ [0, π].

(b) Deduce that

∑_{n=1}^{∞} \frac{1}{4n^{2} − 1} = 1/2.

Spring 2016 Problem 4: Suppose f is a function in the Schwartz space S(ℝ) which satisfies the normalizing condition ∫_{−∞}^{∞} |f(x)|^{2}dx = 1. Let f̂ denote the Fourier transform of f. Show that

(∫_{−∞}^{∞} x^{2}|f(x)|^{2}dx) (∫_{−∞}^{∞} ω^{2}|f̂(ω)|^{2}dω) ≥ \frac{1}{16π^{2}} .

Fall 2018 Problem 4: Consider the functions f_{N}(x) = (2π)^{−1} ∑_{n=−N}^{N} e^{inx}. Show that if g ∈ L^{2}(𝕋) then {f_{N} ∗g} converges in ∥·∥_{L^{2}}-norm to g (here, ∗ denotes convolution).

Fall 2016 Problem 2: Let f : 𝕋 → ℂ be a C^{1} function such that ∫_{−π}^{π} f(x)dx = 0. Show that

∫_{−π}^{π} |f(x)|^{2}dx ≤ ∫_{−π}^{π} |f′(x)|^{2}dx.