Problem 6, Fall 2018: Let Ω = {(x, y) : y ≥ 0, x ∈ ℝ}. Let f ∈ C_{c}^{1}(ℝ^{2}) (space of continuous functions with compact support and with continuous first derivatives). Show the following

∫ℝ |f(x, 0)|^{2}dx ≤ 2 (∫_{Ω} |f(x, y)|^{2}dxdy + ∫_{Ω} |\frac{∂f}{∂y} (x, y)|^{2} dxdy) .

Problem 4, Fall 2017: Let [a, b] ⊆ ℝ be a closed interval and let

∥f∥_{∞} = sup_{x∈[a,b]} |f(x)| ∥f∥_{2} = \sqrt{∫_{a}^{b} |f(x)|^{2}dx}

denote the L^{∞} and L^{2} norms. If f ∈ C^{1}([a, b]), prove that

∥f∥_{∞}^{2} ≤ \frac{∥f∥_{2}^{2}}{b − a} + 2∥f∥_{2}∥f′∥_{2}.

Fall 2014 Problem 5: Suppose that f : ℝ → ℝ is a smooth (C^{∞}) function with compact support. Prove that

lim_{n→∞} \{\sqrt{\frac{8}{π}} ∫_{0}^{∞} n sin(n^{2}x^{2})f(x)dx\} = f(0).

Hint. You can use the fact that

lim_{a→∞} \{∫_{0}^{a} sin(t^{2})dt\} = \sqrt{\frac{π}{8}} .

Fall 2016 Problem 6: Let D denote the closed unit disk in ℂ, and consider the complex Hilbert space

H = {f : D → ℂ |f(z) = ∑_{k=0}^{∞} a_{k}z^{k} and ∥f∥H^{2} = ∑_{k=0}^{∞}(1 + k^{2})|a_{k}|^{2} < ∞} .

Prove that the linear functional L : H → ℂ defined by L(f) = f(1) is bounded, and find an element g ∈ H such that L(f) = ⟨g, f⟩. (In other words, so that g represents L as in the Riesz representation theorem.)

Fall 2014 Problem 1: Suppose f : ℝ → ℝ is twice continuously differentiable. Suppose |f(x)| ≤ 1 and |f′′(x)| ≤ 1 for all x ∈ ℝ Prove or disprove that |f′(x)| ≤ 2 for all x ∈ ℝ.