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Spam yourself with math PhD qualifying-exam questions.

UC Irvine Fall 2023, 6 Real analysis, absolute convergence, almost-everywhere series

Fix $𝑓 \in 𝐿^{1} (ℝ)$ . Show that the series

\sum_{𝑘 = 1}^{\infty} \frac{1}{\sqrt{𝑘}} 𝑓 (𝑥 + \sqrt{𝑘})

converges absolutely for almost every $𝑥 \in ℝ$ .

Stanford Fall 2024, 1 Real analysis, Taylor's theorem, smooth functions

Let $𝑓 : ℝ^{2} \to ℝ$ be an infinitely differentiable function. Suppose $𝑓 (𝑥, 𝑦) = 0$ whenever $𝑥 𝑦 = 0$ . Prove that there exists $𝐶 > 0$ such that

| 𝑓 (𝑥, 𝑦) | \leq 𝐶 | 𝑥 | | 𝑦 |, for all (𝑥, 𝑦) \in 𝐵 (0, 1) .

Stanford Fall 2024, 1 Real analysis, Fourier transform, convolution

Suppose $𝑓 \in 𝐿^{1} (ℝ; ℝ)$ and $ℱ︀ 𝑓$ has compact support, where $ℱ︀$ denotes the Fourier transform. Find a function $𝑔 \in 𝐿^{1} (ℝ; ℝ) \cap 𝐶 (ℝ; ℝ)$ such that $𝑓 * 𝑔 = 0$ (where $*$ denotes the convolution) but $𝑔$ is non-zero Lebesgue almost everywhere.

UC Irvine Fall 2024, 1 Real analysis, harmonic series, logarithm bounds

Show that for every $𝑁 \in ℕ$ ,

\sum_{𝑘 = 1}^{𝑁} \frac{1}{𝑘} \geq log (𝑁 + 1) .

UC Irvine Fall 2024, 1 Real analysis, L^p norms, limit as p to negative infinity, essential infimum

Let $𝑓$ be a positive continuous function on $[0, 1]$ . Prove that

lim_{𝑟 \to - \infty} {(\int_{0}^{1} 𝑓^{𝑟})}^{1 / 𝑟} = min_{[0, 1]} 𝑓 .

Stanford Fall 2024, 2 Real analysis, sequence spaces, Hahn-Banach, dual spaces

Prove that there exists $Λ \in {(ℓ^{\infty})}^{*}$ such that the following all hold:

(a) ${‖ Λ ‖}_{{(ℓ^{\infty})}^{*}} = 1$ .

(b) If $𝑎 = {(𝑎_{𝑖})}_{𝑖 = 1}^{\infty}$ with $𝑎_{𝑖} = 1$ for all $𝑖$ , then $Λ (𝑎) = 1$ .

(c) If $𝑏 = {(𝑏_{𝑖})}_{𝑖 = 1}^{\infty} \in ℓ^{\infty}$ and $𝑐 = {(𝑐_{𝑖})}_{𝑖 = 1}^{\infty}$ is given by $𝑐_{𝑖} = 𝑏_{𝑖 + 1} - 𝑏_{𝑖}$ for all $𝑖$ , then $Λ (𝑐) = 0$ .

Stanford Fall 2024, 2 Real analysis, Fourier series, bounded operators, closed graph theorem

Suppose $𝜑 : ℤ \to ℂ$ satisfies the property that for every $𝑓 \in 𝐿^{1} (ℝ / (2 𝜋 ℤ))$ , there exists $𝑔 \in 𝐿^{1} (ℝ / (2 𝜋 ℤ))$ such that

ℱ︀ 𝑔 (𝑛) = 𝜑 (𝑛) ℱ︀ 𝑓 (𝑛), \forall 𝑛 \in ℤ, (1)

where $ℱ︀ 𝑓 (𝑛) = \int_{⊤} 𝑒^{- 𝑖 𝑛 𝑥} 𝑓 (𝑥) d 𝑥$ . Define $𝑇 : 𝐿^{1} (ℝ / (2 𝜋 ℤ)) \to 𝐿^{1} (ℝ / (2 𝜋 ℤ))$ by $𝑇 𝑓 = 𝑔$ , where $𝑔$ is given by (1). Prove that $𝑇$ is a bounded linear operator on $𝐿^{1} (ℝ / (2 𝜋 ℤ))$ .

UC Irvine Fall 2024, 2 Real analysis, moment conditions, continuous functions, vanishing integrals

Let $𝑓$ be continuous on $[- 1, 1]$ and suppose that

\int_{- 1}^{1} 𝑓 (𝑥) 𝑥^{2 𝑛} d 𝑥 = 0

for all integers $𝑛 \geq 0$ . Show that $𝑓 (0) = 0$ .

UC Irvine Fall 2024, 2 Real analysis, L^1 functions, pointwise decay

Suppose that $𝑓 \in 𝐿^{1} (ℝ)$ . Show that

lim_{𝑛 \to \infty} 𝑓 (𝑥 + 𝑛) = 0 for a.e. 𝑥 \in ℝ .

Here $𝑛 = 1, 2, 3, \dots$ .

Stanford Fall 2024, 3 Real analysis, Lp spaces, Chebyshev inequality

Let ${𝑓_{𝑛}}_{𝑛 = 1}^{\infty} \subset 𝐿^{2} ([0, 1])$ . Suppose ${‖ 𝑓_{𝑛} ‖}_{𝐿^{2}} \leq 2024 {‖ 𝑓_{𝑛} ‖}_{𝐿^{1}}$ for all $𝑛 \in ℤ_{> 0}$ . If ${lim}_{𝑛 \to \infty} {‖ 𝑓_{𝑛} ‖}_{𝐿^{1}} = \infty$ , prove that there exists a sequence $𝑦_{𝑛} \to \infty$ such that

inf_{𝑛} | {𝑥 : | 𝑓_{𝑛 (𝑥)} | \geq 𝑦_{𝑛}} | > 0 .

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