querent

Spam yourself with math PhD qualifying-exam questions.

Stanford Spring 2014, 5 Real analysis, tempered distributions, distributions, distributional derivatives

Consider $𝑆 (ℝ)$ , the space of Schwartz functions, and its dual $𝑆^{'} (ℝ)$ , the space of tempered distributions.

(a) Suppose $𝑢 \in 𝑆^{'} (ℝ)$ and $𝑥^{𝑘} 𝑢 = 0$ for some $𝑘$ (i.e. $𝑢 (𝑥^{𝑘} 𝜑) = 0$ for all $𝜑 \in 𝑆 (ℝ)$ ). Show that $𝑢$ is a finite sum of derivatives of the delta distribution at $0$ , i.e., there exists $𝑘 \geq 1$ and $𝑎_{𝑗} \in ℂ$ such that $𝑢 (𝜑) = \sum_{𝑗 = 0}^{𝑘 - 1} 𝑎_{𝑗} 𝜑^{(𝑗)} (0)$ for all $𝜑 \in 𝑆 (ℝ)$ .

(b) One says that $𝑥_{0} \notin supp 𝑢$ if $𝑥_{0}$ has a neighborhood $𝑈$ such that for all $𝜑 \in 𝑆 (ℝ)$ with $supp 𝜑 \subseteq 𝑈$ , $𝑢 (𝜑) = 0$ . Suppose now that $𝑢 (𝜑) = 0$ if $0 \notin supp 𝜑$ , so $supp 𝑢 \subseteq {0}$ . Show that $𝑢$ is again a finite sum of derivatives of the delta distribution.

Hint: Prove that if $𝜑 \in 𝑆 (ℝ)$ and $𝜑^{(𝑗)} (0) = 0$ for $𝑗 \leq 𝑘$ then there are $𝜑_{𝑛} \in 𝑆 (ℝ)$ , $𝑛 \geq 1$ , such that $0 \notin supp 𝜑_{𝑛}$ , $𝜑_{𝑛 (𝑥)} = 𝜑 (𝑥)$ for $| 𝑥 | \geq 1$ and $\sum_{𝑗 \leq 𝑘} {sup}_{𝑥 \in ℝ} | 𝜑_{𝑛}^{(𝑗)} (𝑥) - 𝜑^{(𝑗)} (𝑥) | \to 0$ as $𝑛 \to \infty$ .

Stanford Spring 2015, 1 Real analysis, continuity of translation, Lebesgue measure

Suppose $𝐴, 𝐵 \subseteq ℝ / ℤ$ are measurable of positive Lebesgue measure: $𝑚 (𝐴), 𝑚 (𝐵) > 0$ .

(a) Show that there exists $𝑦 \in ℝ / ℤ$ such that $𝑚 ((𝐴 + 𝑦) \cap 𝐵) > 0$ .

(b) Show that in fact there exists $𝑦 \in ℝ / ℤ$ such that $𝑚 ((𝐴 + 𝑦) \cap 𝐵) \geq 𝑚 (𝐴) 𝑚 (𝐵)$ .

Stanford Spring 2015, 1 Real analysis, Lebesgue measure, monotone functions, density

Suppose that $𝑓 \in 𝐿^{1} ([0, 1])$ . Prove that there are nondecreasing sequences of continuous functions, ${𝜑_{𝑘}}_{𝑘 = 1}^{\infty}$ and ${𝜓_{𝑘}}_{𝑘 = 1}^{\infty}$ , on $[0, 1]$ such that for a.e. $𝑥 \in [0, 1]$ (with respect to Lebesgue measure), both $𝜑_{𝑘 (𝑥)}$ and $𝜓_{𝑘 (𝑥)}$ are bounded sequences, and moreover,

𝑓 (𝑥) = lim_{𝑘 \to \infty} 𝜑_{𝑘 (𝑥)} - lim_{𝑘 \to \infty} 𝜓_{𝑘 (𝑥)} .

Stanford Spring 2015, 2 Real analysis, compact operators, finite rank operators, Hilbert space geometry

Let $𝑋, 𝑌$ be Banach spaces.

(a) Show that if $𝑇_{𝑛} \in ℒ︀ (𝑋, 𝑌)$ are compact, and $𝑇_{𝑛} \to 𝑇 \in ℒ︀ (𝑋, 𝑌)$ in norm, then $𝑇$ is compact.

(b) Show that if $𝑋, 𝑌$ are separable Hilbert spaces then every compact operator $𝑇 \in ℒ︀ (𝑋, 𝑌)$ is the norm limit of finite rank operators.

Stanford Spring 2015, 2 Real analysis, weak topology, dual spaces

Let $𝑋$ be a vector space over $ℂ$ , $ℱ︀$ a vector space of linear maps $𝑋 \to ℂ$ , and equip $𝑋$ with the weakest topology in which all members of $ℱ︀$ are continuous. Show that the only continuous linear maps $𝑋 \to ℂ$ are those in $ℱ︀$ .

Stanford Spring 2015, 3 Real analysis, Fourier series, Fourier transform, Schwartz functions

Let $𝑆 (ℝ)$ be the set of Schwartz functions on $ℝ$ , i.e. the set of $𝐶^{\infty}$ functions $𝜑$ on $ℝ$ with bounded for all $𝛼, 𝛽 \in ℕ$ .

(a) With the Fourier transform given by $(ℱ︀ 𝜑) (𝜉) = \int_{ℝ} 𝑒^{- 𝑖 𝑥 𝜉} 𝜑 (𝑥) d 𝑥$ , show the Poisson summation formula for $𝜑 \in 𝑆 (ℝ)$ :

2 𝜋 \sum_{𝑛 \in ℤ} 𝜑 (𝑥 + 2 𝑛 𝜋) = \sum_{𝑛 \in ℤ} (ℱ︀ 𝜑) (𝑛) 𝑒^{𝑖 𝑛 𝑥}, 𝑥 \in ℝ .

(b) Show that for any $𝑡 > 0$ we have

\sum_{𝑛 \in ℤ} exp (- 𝑡 {(2 𝜋 𝑛 + 𝜋)}^{2} / 2) = \frac{1}{\sqrt{2 𝜋 𝑡}} \sum_{𝑘 \in ℤ} {(- 1)}^{𝑘} 𝑒^{- 𝑘^{2} / (2 𝑡)} .

Stanford Spring 2015, 3 Real analysis, Fourier series, uniform boundedness, Baire category

Consider the partial sums $(𝑆_{𝑁} 𝑓) (𝑥) = \sum_{| 𝑛 | \leq 𝑁} 𝑐_{𝑛} 𝑒^{𝑖 𝑛 𝑥}$ of the Fourier series of continuous functions $𝑓$ on $⊤ = ℝ / (2 𝜋 ℤ)$ , where $𝑐_{𝑛}$ are the Fourier coefficients of $𝑓$ , and recall that $𝑆_{𝑁} 𝑓$ is given by the convolution of the Dirichlet kernel $𝐷_{𝑁}$ with $𝑓$ .

(a) Show that ${‖ 𝐷_{𝑁} ‖}_{𝐿^{1}} \to \infty$ as $𝑁 \to \infty$ .

(b) Show that there exists a continuous function $𝑓$ on $⊤ = ℝ / (2 𝜋 ℤ)$ such that the Fourier series of $𝑓$ does not converge uniformly to $𝑓$ , i.e. $𝑆_{𝑁} 𝑓$ does not converge uniformly to $𝑓$ as $𝑁 \to \infty$ .

Stanford Spring 2015, 4 Real analysis, Sobolev spaces, trace theorem, distributions

Let $𝑆 (ℝ^{𝑛})$ denote the set of Schwartz functions, and $𝑆^{'} (ℝ^{𝑛})$ the dual space of tempered distributions. For $𝑢 \in 𝑆^{'} (ℝ^{𝑛})$ let $𝐷_{𝑗} 𝑢$ denote the distributional derivative of $𝑢$ in the $𝑗$ th coordinate. Let $𝐻 = {𝑢 \in 𝐿^{2} (ℝ^{2}) : 𝐷_{2} 𝑢 \in 𝐿^{2} (ℝ^{2})}$ , equipped with the norm ${‖ 𝑢 ‖}_{𝐻}^{2} = {‖ 𝑢 ‖}_{𝐿^{2}}^{2} + {‖ 𝐷_{2} 𝑢 ‖}_{𝐿^{2}}^{2}$ .

(a) Show that $𝐻$ is a Hilbert space (with the norm being induced by the inner product), and $𝑆 (ℝ^{2})$ is dense in $𝐻$ .

(b) Show that the restriction map $𝑅 : 𝑆 (ℝ^{2}) \to 𝑆 (ℝ)$ , $(𝑅 𝜑) (𝑥_{1}) = 𝜑 (𝑥_{1}, 0)$ , to $𝑥_{2} = 0$ , has a unique continuous extension to a map $𝐻 \to 𝐿^{2} (ℝ)$ .

Stanford Spring 2015, 4 Real analysis, spectral theory, ergodic theorem, Hilbert space geometry

Suppose $𝐻$ is a Hilbert space. Recall that $𝑈 \in ℒ︀ (𝐻)$ is unitary if $𝑈 𝑈^{*} = 𝐼 = 𝑈^{*} 𝑈$ .

(a) Show that if $𝑈$ is unitary then (orthogonal direct sum).

(b) Let $𝑃$ be orthogonal projection to $Ker (𝐼 - 𝑈)$ . Let $𝑆_{𝑛} = \frac{1}{𝑛} \sum_{𝑗 = 0}^{𝑛 - 1} 𝑈^{𝑗}$ . Show that $𝑆_{𝑛} \to 𝑃$ in the strong operator topology (i.e. $𝑆_{𝑛} 𝑓 \to 𝑃 𝑓$ in $𝐻$ for all $𝑓 \in 𝐻$ ). (This is the von Neumann, or mean ergodic theorem.)

Stanford Spring 2015, 5 Real analysis, Banach spaces, compact embedding, closed range

Suppose that $𝑋$ is a Banach space, and let $𝐵 = {𝑥 \in 𝑋 : {‖ 𝑥 ‖}_{𝑋} \leq 1}$ be the unit ball in $𝑋$ .

(a) Suppose $𝑍$ is a finite dimensional subspace of $𝑋$ . Show that there exists a closed subspace $𝑊$ of $𝑋$ such that $𝑋 = 𝑍 \oplus 𝑊$ (direct sum).

(b) If $𝑋$ is infinite dimensional, show that $𝐵$ is not compact in the norm topology.

(c) Suppose that $𝑋, 𝑌$ are Banach spaces, $𝑋 \subseteq 𝑌$ with the inclusion map $𝜄 : 𝑋 \to 𝑌$ continuous and compact. Let $𝑇 \in ℒ︀ (𝑋, 𝑌)$ , and suppose that for all $𝑥 \in 𝑋$ , ${‖ 𝑥 ‖}_{𝑋} \leq 𝐶 ({‖ 𝑇 𝑥 ‖}_{𝑌} + {‖ 𝑥 ‖}_{𝑌})$ . Show that $Ker 𝑇$ is finite dimensional, $Ran 𝑇$ is closed, and the induced map $𝑋 / Ker 𝑇 \to Ran 𝑇$ is invertible as a bounded linear map.

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