querent

Spam yourself with math PhD qualifying-exam questions.

Stanford Spring 2023, 4 Real analysis, Fourier transform, analytic continuation, boundary values of analytic functions

Let $ℝ^{+} = [0, \infty)$ and let $𝐶_{𝑐}^{\infty} (ℝ^{+})$ be the set of functions which are restrictions of $𝐶_{𝑐}^{\infty} (ℝ)$ functions to $ℝ^{+}$ . For $𝑓 \in 𝐶_{𝑐}^{\infty} (ℝ^{+})$ , define

𝑇 𝑓 (𝑧) = \int_{0}^{\infty} 𝑒^{- 𝑖 𝑥 𝑧} 𝑓 (𝑥) d 𝑥, 𝑧 \in ℂ .

Define also $𝑇_{𝜂} 𝑓 (𝜉) = 𝑇 𝑓 (𝜉 + 𝑖 𝜂)$ for $𝜉, 𝜂 \in ℝ$ .

(a) For $𝜂 \leq 0$ , show that $𝑇_{𝜂}$ , a priori defined on $𝐶_{𝑐}^{\infty} (ℝ^{+})$ , extends to a bounded map ${\hat{𝑇}}_{𝜂} : 𝐿^{2} (ℝ^{+}) \to 𝐿^{2} (ℝ)$ and satisfies

\int_{- \infty}^{\infty} {| {\hat{𝑇}}_{𝜂} 𝑓 |}^{2} (𝜉) d 𝜉 \leq 2 𝜋 \int_{0}^{\infty} {| 𝑓 |}^{2} (𝑥) d 𝑥, \forall 𝜂 \leq 0 .

(b) Let $𝑎 > 0$ and define the space $𝐿^{2} (ℝ^{+}, 𝑒^{2 𝑎 𝑥} d 𝑥)$ to be the closure of $𝐶_{𝑐}^{\infty} (ℝ^{+})$ under the norm $\int_{0}^{\infty} {| 𝑓 |}^{2} 𝑒^{2 𝑎 𝑥} d 𝑥$ . Prove that for any $𝑓 \in 𝐿^{2} (ℝ^{+}, 𝑒^{2 𝑎 𝑥} d 𝑥)$ , the function $𝑇 𝑓 (𝑧)$ extends to be holomorphic in the half-plane ${𝑧 \in ℂ : Im (𝑧) < 𝑎}$ .

Stanford Spring 2023, 4 Real analysis, tempered distributions, distributions, Borel measures

For $𝜀 > 0$ , define $𝑢_{\pm, 𝜀} : 𝒮︀ (ℝ) \to ℂ$ by $𝑢_{\pm, 𝜀} (𝜑) = \int_{ℝ} {(𝑥 \pm 𝑖 𝜀)}^{- 1} 𝜑 (𝑥) d 𝑥$ , $𝜑 \in 𝒮︀ (ℝ)$ .

(a) Show that for all $𝜀 > 0$ , $𝑢_{\pm, 𝜀} \in {𝒮︀}^{'} (ℝ)$ , and that there exist $𝑢_{\pm} \in {𝒮︀}^{'} (ℝ)$ such that for all $𝜑 \in 𝒮︀ (ℝ)$ , $𝑢_{\pm, 𝜀} (𝜑) \to 𝑢_{\pm} (𝜑)$ as $𝜀 \to 0$ .

(b) Compute $𝑢_{+} - 𝑢_{-}$ , and show that it can be represented by a locally finite Borel measure, i.e., there exists a locally finite Borel measure $𝜇$ such that $𝑢_{+} (𝜑) - 𝑢_{-} (𝜑) = \int_{ℝ} 𝜑 d 𝜇$ , $\forall 𝜑 \in 𝒮︀ (ℝ)$ .

UC Irvine Spring 2023, 4 Real analysis, intermediate value theorem, continuous functions

Let $𝑓, 𝑔 : [0, 1] \to [0, \infty)$ be continuous functions satisfying

sup_{0 \leq 𝑥 \leq 1} 𝑓 (𝑥) = sup_{0 \leq 𝑥 \leq 1} 𝑔 (𝑥) .

Prove that there exists $𝑥_{0} \in [0, 1]$ such that

𝑓 (𝑥_{0}) = 𝑔 (𝑥_{0}) .

Stanford Spring 2023, 5 Real analysis, Fourier transform, uncertainty principle, Schwartz functions

(a) Prove that the following inequality holds for any $𝑓 \in 𝒮︀ (ℝ)$ :

{‖ 𝑓 ‖}_{𝐿^{2}}^{2} \leq 2 {‖ 𝑥 𝑓 ‖}_{𝐿^{2}} {‖ 𝑓^{'} ‖}_{𝐿^{2}} .

(Hint: apply the Cauchy–Schwarz inequality to the function $𝑥 𝑓 𝑓^{'}$ .)

(b) Show that equality holds in the inequality in part (a) if and only if $𝑓 (𝑥) = 𝑎 𝑒^{- 𝑏 𝑥^{2} / 2}$ for some $𝑏 \in ℂ$ with positive real part and some $𝑎 \in ℂ$ .

(c) Given $𝑓 \in 𝒮︀ (ℝ)$ , denote its Fourier transform by $\hat{𝑓}$ . Prove that there exists $𝐶 > 0$ such that the following inequality holds for any $𝑐, 𝑑 \in ℝ$ and for any $𝑓 \in 𝒮︀ (ℝ)$ ,

{(\int_{- \infty}^{\infty} {| 𝑓 |}^{2} (𝑥) d 𝑥)}^{2} \leq 𝐶 (\int_{- \infty}^{\infty} {(𝑥 - 𝑐)}^{2} {| 𝑓 (𝑥) |}^{2} d 𝑥) (\int_{- \infty}^{\infty} {(𝜉 - 𝑑)}^{2} {| \hat{𝑓} (𝜉) |}^{2} d 𝜉) .

Stanford Spring 2023, 5 Real analysis, Sobolev spaces, weak convergence, Hilbert space geometry

Let $⊤^{2} = ℝ^{2} / {(2 𝜋 ℤ)}^{2}$ with coordinates $(𝑥_{1}, 𝑥_{2})$ . Define the Hilbert spaces $𝐻_{1}$ and $𝐻_{2}$ to be the completions of $𝐶^{\infty} (⊤^{2})$ under the respective inner products

angle.l f, g angle.r_(H_1) = angle.l f, g angle.r_(L^2(TT^2)) +
angle.l diff_(x_1) f, diff_(x_1) g angle.r_(L^2(TT^2)), quad
angle.l f, g angle.r_(H_2) = angle.l f, g angle.r_(L^2(TT^2)) +
angle.l diff_(x_2) f, diff_(x_2) g angle.r_(L^2(TT^2)).

(a) Prove that if $𝑓 \in 𝐻_{1}$ and $𝑔 \in 𝐻_{2}$ , then $𝑓 𝑔 \in 𝐿^{2}$ .

(Hint: first show that $𝐻_{1}$ embeds continuously into the $𝐿_{𝑥_{2}}^{2} 𝐿_{𝑥_{1}}^{\infty}$ space, i.e., the normed space with norm ${(\int_{0}^{2 𝜋} {esssup}_{𝑥_{1}} {| 𝑓 |}^{2} (𝑥_{1}, 𝑥_{2}) d 𝑥_{2})}^{1 / 2}$ .)

(b) Suppose ${‖ 𝑓_{𝑛} ‖}_{𝐻_{1}} \leq 1$ , ${‖ 𝑔_{𝑛} ‖}_{𝐻_{2}} \leq 1$ , $𝑓_{𝑛} \to 0$ weakly in $𝐻_{1}$ and $𝑔_{𝑛} \to 0$ weakly in $𝐻_{2}$ . Prove that .

UC Irvine Spring 2023, 5 Real analysis, Kronecker's lemma, series convergence

Let ${𝑏_{𝑛}}$ be a monotonic increasing sequence of positive numbers and suppose

lim_{𝑛 \to \infty} 𝑏_{𝑛} = + \infty .

Show that, if

\sum_{𝑛 = 1}^{\infty} 𝑎_{𝑛}

converges, then

lim_{𝑛 \to \infty} \frac{𝑎_{1} 𝑏_{1} + \dots + 𝑎_{𝑛} 𝑏_{𝑛}}{𝑏_{𝑛}} = 0 .

UC Irvine Spring 2023, 6 Real analysis, mean value theorem, Riemann sums, error bounds

Let $𝑓$ be a real-valued continuous function on $[0, 1]$ that is differentiable on $(0, 1)$ . Assume

sup_{0 < 𝑥 < 1} | 𝑓^{'} (𝑥) | = 𝑀 < \infty .

Show that for all $𝑛 \in ℕ$ ,

| \frac{1}{𝑛} \sum_{𝑗 = 0}^{𝑛 - 1} 𝑓 (𝑗 / 𝑛) - \int_{0}^{1} 𝑓 (𝑥) d 𝑥 | \leq \frac{𝑀}{𝑛} .

UC Irvine Spring 2023, 7 Real analysis, difference quotients, derivatives at a point

Suppose $𝑓 : ℝ \to ℝ$ is continuous at $𝑥 = 0$ and satisfies

lim_{𝑥 \to 0} \frac{𝑓 (2 𝑥) - 𝑓 (𝑥)}{𝑥} = 𝑚 .

Show that

𝑓^{'} (0) = 𝑚 .

UC Irvine Spring 2023, 8 Real analysis, implicit function theorem, smooth solutions

Let $𝑓, 𝑔 : ℝ \to ℝ$ be smooth functions with

𝑓 (0) = 0

and

𝑓^{'} (0) \neq 0 .

Consider the equation

𝑓 (𝑥) = 𝑡 𝑔 (𝑥), 𝑡 \in ℝ .

Show that there exists $𝜀 > 0$ such that on the interval $| 𝑡 | < 𝜀$ , there is a unique smooth solution $𝑥 (𝑡)$ to the above equation with $𝑥 (0) = 0$ . Then calculate $𝑥^{'} (0)$ and $𝑥^{″} (0)$ .

UC Irvine Spring 2023, 9 Real analysis, volume of balls, multivariable integrals

Compute the volume of the balls

𝐵_{𝑛} (𝑟) = {𝑥 \in ℝ^{𝑛} : 𝑥_{1}^{2} + 𝑥_{2}^{2} + \dots + 𝑥_{𝑛}^{2} \leq 𝑟^{2}},

for $𝑛 = 3$ and $𝑛 = 4$ .

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