querent

Spam yourself with math PhD qualifying-exam questions.

Stanford Fall 2015, 3 Real analysis, Fourier transform, positive-definite functions

Show that there exists a compactly supported $𝐶^{\infty}$ function $𝜑$ on $ℝ$ such that $𝜑 \geq 0$ , $𝜑 (0) > 0$ , and the Fourier transform $ℱ︀ 𝜑$ of $𝜑$ is non-negative, where $(ℱ︀ 𝜑) (𝜉) = \int_{ℝ} 𝑒^{- 𝑖 𝑥 𝜉} 𝜑 (𝑥) d 𝑥$ .

Hint: When is the Fourier transform of a function real?

Stanford Fall 2015, 3 Real analysis, strong operator topology, weak operator topology, bounded operators, adjoint operators

(a) Give an example (with proof) of a sequence ${𝐴_{𝑛}}_{𝑛 = 1}^{\infty}$ of bounded linear operators on $ℓ^{2}$ such that (i) $𝐴_{𝑛} \to 0$ in the strong operator topology, but not in norm, and (ii) $𝐴_{𝑛} \to 0$ in the weak operator topology, but not in the strong operator topology.

(b) Show that the map $𝐴 \mapsto 𝐴^{*}$ is continuous in the norm topology and in the weak operator topology, but not in the strong operator topology.

Stanford Fall 2015, 4 Real analysis, distributions, distributional derivatives

(a) Suppose $𝑓 \in 𝐶^{2} (ℝ^{2})$ . Show that the mixed partials (partial derivatives are defined as limits of difference quotients) are equal: .

(b) Give an example of $𝑓 \in 𝐶^{1} (ℝ^{2})$ such that there is a point $𝑥_{0} \in ℝ^{2}$ at which the mixed partials , exist, but are not equal.

Stanford Fall 2015, 4 Real analysis, Fredholm operators, compact operators, adjoint operators

Suppose that $𝐻_{1}$ and $𝐻_{2}$ are separable Hilbert spaces and $𝐴 : 𝐻_{1} \to 𝐻_{2}$ is a bounded linear operator. Suppose that there exist $𝐵 \in ℒ︀ (𝐻_{2}, 𝐻_{1})$ and compact operators $𝐸_{𝑗}$ on $𝐻_{𝑗}$ , $𝑗 = 1, 2$ , such that $𝐵 𝐴 = 𝐼_{1} - 𝐸_{1}$ , $𝐴 𝐵 = 𝐼_{2} - 𝐸_{2}$ , where $𝐼_{𝑗}$ is the identity operator on $𝐻_{𝑗}$ . Show that the nullspace of $𝐴$ is finite dimensional, the range of $𝐴$ is closed in $𝐻_{2}$ , and its orthocomplement is finite dimensional.

Stanford Fall 2015, 5 Real analysis, stationary phase, oscillatory integrals

Let $𝑓 (𝑥) = 𝑥^{4} - 2 𝑥^{2} + 1$ , $𝑥 \in ℝ$ . Prove that there exists an $𝛼 \in ℝ$ (and find its value) such that

𝐶^{- 1} 𝜆^{- 𝛼} \leq | \int_{- 𝜋}^{𝜋} 𝑒^{𝑖 𝜆 𝑓 (𝑥)} {sin}^{4} 𝑥 d 𝑥 | \leq 𝐶 𝜆^{- 𝛼}

as $𝜆 \to + \infty$ , for some $𝐶 > 0$ .

Stanford Fall 2015, 5 Real analysis, Cantor set, weak-star convergence of measures, Borel measures

Let $𝐴 \subseteq [0, 1]$ be the middle third Cantor set: $𝐴 = \cap_{𝑛} 𝐴_{𝑛}$ , where $𝐴_{0} = [0, 1]$ and $𝐴_{𝑛}$ is obtained from $𝐴_{𝑛 - 1}$ by removing the middle third of each component interval. Show that if $𝑓 \in 𝐶 ([0, 1])$ , then ${lim}_{𝑛 \to \infty} {𝑚 (𝐴_{𝑛})}^{- 1} \int_{𝐴_{𝑛}} 𝑓$ exists, where $𝑚 (𝐴_{𝑛})$ is the Lebesgue measure. Show moreover that there is a Borel measure $𝜇$ on $[0, 1]$ such that ${lim}_{𝑛 \to \infty} {𝑚 (𝐴_{𝑛})}^{- 1} \int_{𝐴_{𝑛}} 𝑓 = \int_{[0, 1]} 𝑓 d 𝜇$ .

Stanford Fall 2016, 1 Real analysis, spectral theory, bounded operators

Suppose $𝑋$ is a Banach space and $𝐴 \in ℒ︀ (𝑋)$ is a bounded linear operator on it. Show that the spectrum of $𝐴$ is a closed, bounded subset of $ℂ$ .

Stanford Fall 2016, 1 Real analysis, absolute continuity

Show that if $𝜇$ is a $𝜎$ -finite measure on a measurable space $(𝑋, 𝒜︀)$ (i.e. on $𝒜︀$ , where $𝒜︀$ is a $𝜎$ -algebra of subsets of $𝑋$ ), then there is a finite measure $𝜈$ on $(𝑋, 𝒜︀)$ with $𝜈 ≪ 𝜇$ and $𝜇 ≪ 𝜈$ .

Stanford Fall 2016, 2 Real analysis, tempered distributions, distributions

Recall that $𝑆 (ℝ)$ is the space of Schwartz functions on $ℝ$ , and $𝑆^{'} (ℝ)$ the space of tempered distributions. Show that there exists no $𝑢 \in 𝑆^{'} (ℝ)$ such that for $𝜑 \in 𝑆 (ℝ)$ with $supp 𝜑 \subseteq (0, \infty)$ , $𝑢 (𝜑) = \int_{ℝ} 𝑒^{1 / 𝑥} 𝜑 (𝑥) d 𝑥$ .

Hint: If such a distribution $𝑢$ existed, it would satisfy an estimate!

Stanford Fall 2016, 2 Real analysis, compact operators, integral operators

Suppose $𝐾 \in 𝐿^{2} (ℝ^{2 𝑛})$ , and define $(𝑇 𝑓) (𝑥) = \int_{ℝ^{𝑛}} 𝐾 (𝑥, 𝑦) 𝑓 (𝑦) d 𝑦$ . Show that $𝑇 \in ℒ︀ (𝐿^{2} (ℝ^{𝑛}))$ , and $𝑇$ is compact.

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