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Spam yourself with math PhD qualifying-exam questions.

Stanford Practice, 39 Applied mathematics, Fourier series, divergence of Fourier series, Dirichlet kernel

(i) Let $𝑎_{𝑘}$ be the Fourier coefficients of the function $𝑢 (𝑥) = 1 / 2 - 𝑥$ , $0 < 𝑥 < 1$ , extended periodically:

𝑎_{𝑘} = \int_{0}^{1} (\frac{1}{2} - 𝑥) 𝑒^{- 2 𝜋 𝑖 𝑘 𝑥} d 𝑥 .

Show that there exists a constant $𝑀 > 0$ so that for all $𝑥 \in [0, 1]$ and all $𝑁 \in ℤ$ the partial sums of the Fourier series satisfy

| \sum_{𝑘 = - 𝑁}^{𝑁} 𝑎_{𝑘} 𝑒^{2 𝜋 𝑖 𝑘 𝑥} | \leq 𝑀 .

(ii) Let two integer sequences $𝑁_{𝑘} \geq 1$ and $𝑚_{𝑘} \geq 1$ be such that

𝑁_{𝑘 + 1} - 𝑚_{𝑘 + 1} > 𝑁_{𝑘} + 𝑚_{𝑘},

and define

𝑓 (𝑥) = \sum_{𝑘 = 1}^{\infty} 𝑒^{2 𝜋 𝑖 𝑁_{𝑘} 𝑥} \frac{𝐵_{𝑘} (𝑥)}{𝑘^{2}},

with

𝐵_{𝑘} (𝑥) = \sum_{𝑗 = - 𝑚_{𝑘}}^{𝑚_{𝑘}} 𝑎_{𝑗} 𝑒^{2 𝜋 𝑖 𝑗 𝑥} .

Show that the function $𝑓 (𝑥)$ is continuous and find its Fourier coefficients in terms of $𝑎_{𝑘}$ .

(iii) Show that the partial Fourier sums $𝑆_{𝑁_{𝑘}} 𝑓 (0)$ satisfy a lower bound

𝑆_{𝑁_{𝑘}} 𝑓 (0) \geq \frac{𝐶 log 𝑚_{𝑘}}{𝑘^{2}} .

(iv) Find a choice of $𝑚_{𝑘}$ and $𝑁_{𝑘}$ so that the Fourier series of $𝑓 (𝑥)$ at $𝑥 = 0$ diverges.

Stanford Practice, 4 Applied mathematics, elliptic PDE, subharmonic functions, mean value inequality

Let $𝑎 (𝑥)$ be a function such that $𝜆^{- 1} \leq 𝑎 (𝑥) \leq 𝜆$ for all $𝑥 \in ℝ^{𝑛}$ , with some fixed $𝜆 > 0$ , and $𝑣 (𝑥)$ , $𝑥 \in ℝ^{𝑛}$ , be a smooth function such that

- \nabla \cdot (𝑎 (𝑥) \nabla 𝑣) \geq 0

in a ball $𝐵 (0, 𝑟)$ . Show that there exists a constant $𝐶 (𝜆, 𝑛)$ such that

𝑣 (0) \geq \frac{𝐶 (𝜆, 𝑛)}{𝑟^{𝑛}} \int_{𝐵 (0, 𝑟)} 𝑣 (𝑥) d 𝑥 .

Stanford Practice, 40 Applied mathematics, Fourier transform, convolution, L^1 spaces

(i) Let $𝑔_{𝑛} = 𝜒_{[- 𝑛, 𝑛]}$ , compute $ℎ_{𝑛} = 𝑔_{𝑛} ⋆ 𝑔_{1}$ and show that $ℎ_{𝑛}$ is a Fourier transform of a multiple of the function

𝑓_{𝑛} (𝑥) = \frac{sin 𝑥 sin (𝑛 𝑥)}{𝑥^{2}} .

(ii) Use (i) to show that the Fourier transform maps $𝐿^{1}$ into a proper subset of $𝐶_{0} (ℝ)$ and not onto $𝐶_{0} (ℝ)$ .

(iii) Show that the image of the Fourier transform is dense in $𝐶_{0} (ℝ)$ .

Stanford Practice, 41 Applied mathematics, Fourier coefficients, modulus of continuity, analytic functions

(i) Let $𝑓 \in 𝐶 (𝑆^{1})$ have a modulus of continuity $𝜔 (𝛿) = {sup}_{| 𝑥 - 𝑦 | \leq 𝛿} | 𝑓 (𝑥) - 𝑓 (𝑦) |$ . Show that $| \hat{𝑓} (𝑛) | \leq 𝐶 𝜔 (1 / (2 𝑛))$ .

(ii) Assume that $𝑓$ is absolutely continuous, show that $\hat{𝑓} (𝑛) = 𝑜 (1 / 𝑛)$ as $𝑛 \to + \infty$ .

(iii) Show that $𝑓 \in 𝐿^{1} (𝑆^{1})$ is equal to an analytic function almost everywhere on $𝑆^{1}$ if and only if there exist $𝑐 > 0$ and $𝐴 > 0$ so that $| \hat{𝑓} (𝑛) | \leq 𝐴 𝑒^{- 𝑐 𝑛}$ .

Stanford Practice, 42 Applied mathematics, Fourier series, Dini kernel, Gibbs phenomenon

Consider the function $𝑓 (𝑥)$ defined as

𝑓 (𝑥) = \frac{1}{2} - 𝑥, for 0 < 𝑥 < 1,

and extended periodically to all of $ℝ$ (the resulting function has a jump discontinuity at all integers $𝑥 \in ℤ$ ).

(i) Show that ${\hat{𝑓}}_{𝑛} = 1 / (2 𝜋 𝑖 𝑛)$ for $𝑛 \neq 0$ and ${\hat{𝑓}}_{0} = 0$ .

(ii) Show that

{(𝑆_{𝑁} 𝑓)}^{'} (𝑥) = 𝐷_{𝑁} (𝑥) - 1,

where $𝐷_{𝑁} (𝑥) = sin ((2 𝑁 + 1) 𝜋 𝑥) / sin (𝜋 𝑥)$ is the Dini kernel, and that

𝑆_{𝑁} 𝑓 (𝑥) = - 𝑥 + \int_{0}^{𝑥} \frac{sin ((2 𝑁 + 1) 𝜋 𝑡)}{sin (𝜋 𝑡)} d 𝑡 .

(iii) Let

{\tilde{𝑆}}_{𝑁} 𝑓 (𝑥) = - 𝑥 + \int_{0}^{𝑥} \frac{sin ((2 𝑁 + 1) 𝜋 𝑡)}{𝜋 𝑡} d 𝑡

and show that there exists some constant $𝐶 > 0$ so that

| 𝑆_{𝑁} 𝑓 (𝑥) - {\tilde{𝑆}}_{𝑁} 𝑓 (𝑥) | \leq 𝐶 / 𝑁, for all 𝑥 \in [0, 1 / 4] .

(iv) Let $𝑥_{𝑁} = 1 / (2 𝑁)$ and show that

(v) Show that (use a computer if you need).

Stanford Practice, 43 Applied mathematics, wave equation, leapfrog scheme, CFL condition

Consider the 1D wave equation

𝑢_{𝑡 𝑡} = 𝑢_{𝑥 𝑥} .

(i) Describe the leapfrog scheme.

(ii) Let $𝑘$ and $ℎ$ be the temporal and spatial steps, respectively. Use the Fourier analysis and multistep analysis to derive the stable condition (in terms of $ℎ$ and $𝑘$ ).

Stanford Practice, 44 Applied mathematics, discrete Fourier transform, fast Fourier transform

Consider the discrete Fourier transform.

(i) Concisely describe the $𝑂 (𝑁 log 𝑁)$ fast Fourier transform of size $𝑁 = 2^{𝑛}$ for $𝑛$ integer.

(ii) Give an $𝑂 (𝑁 log 𝑁)$ fast Fourier transform that works for any integer $𝑁$ (including primes).

Stanford Practice, 45 Applied mathematics, orthogonal polynomials, Gaussian quadrature

Consider a weight function $𝑤 (𝑥)$ on $(- 1, 1)$ with $𝑤 (𝑥) > 0$ .

(i) Describe how to construct the orthogonal polynomial family on $(- 1, 1)$ with respect to $𝑤 (𝑥)$ .

(ii) Describe how to construct the $𝑛$ -point Gaussian quadrature weights with respect to $𝑤 (𝑥)$ .

(iii) What is the order of this $𝑛$ -point Gaussian quadrature?

Stanford Practice, 5 Applied mathematics, Haar wavelets, Brownian motion, orthonormal basis, L^2 convergence

Set

𝜓 (𝑥) = {\begin{cases} 1 & if 0 \leq 𝑥 < 1 / 2 \\ - 1 & if 1 / 2 \leq 𝑥 < 1 \\ 0 & otherwise \end{cases}

and

𝜓_{𝑗 𝑘} (𝑥) = 2^{𝑗 / 2} 𝜓 (2^{𝑗} 𝑥 - 𝑘), 𝑗, 𝑘 \in ℤ .

(i) Show that the functions $𝜑_{𝑛} (𝑥) = 𝜓_{𝑗 𝑘} (𝑥)$ for $𝑛 = 2^{𝑗} + 𝑘$ , $0 \leq 𝑘 \leq 2^{𝑗} - 1$ , and $𝜑_{0} (𝑥) = 1$ form an orthonormal basis for $𝐿^{2} [0, 1]$ . You may use this result in the exercises below.

(ii) Let $𝑍_{𝑛} (𝜔)$ , $𝑛 \geq 0$ , be a collection of independent Gaussian random variables of mean zero and variance one, that is,

ℙ (𝑍_{𝑛} (𝜔) < 𝑦) = \int_{- \infty}^{𝑦} 𝑒^{- 𝑦^{2}} d 𝑦 / \sqrt{2 𝜋} .

Consider the processes $𝑋_{𝑚} (𝑡; 𝜔)$ defined by

𝑋_{𝑚} (𝑡; 𝜔) = \sum_{𝑛 = 0}^{𝑚} 𝑍_{𝑛} (𝜔) \int_{0}^{𝑡} 𝜑_{𝑛} (𝑠) d 𝑠,

and the process

𝑋 (𝑡; 𝜔) = \sum_{𝑛 = 0}^{\infty} 𝑍_{𝑛} (𝜔) \int_{0}^{𝑡} 𝜑_{𝑛} (𝑠) d 𝑠 .

Show that the series defining $𝑋 (𝑡; 𝜔)$ converges in $𝐿^{2} (Ω)$ for any fixed $𝑡 \in [0, 1]$ .

(iii) Show that $𝑋 (𝑡; 𝜔)$ is a Brownian motion.

Stanford Practice, 6 Applied mathematics, heat kernel, fundamental solution, Laplace equation

Let $𝑢 (𝑡, 𝑥)$ satisfy the heat equation

\frac{𝜕 𝑢}{𝜕 𝑡} - Δ 𝑢 = 0, 𝑡 > 0, 𝑥 \in ℝ^{𝑛}

in dimension $𝑛 \geq 3$ , with the initial condition $𝑢 (0, 𝑥) = 𝑓 (𝑥)$ , where $𝑓 (𝑥)$ is a continuous function that vanishes outside the ball ${| 𝑥 | \leq 1}$ .

(i) Define

𝑣 (𝑥) = \int_{0}^{\infty} 𝑢 (𝑡, 𝑥) d 𝑡,

and show that this function is well-defined: the integral above converges.

(ii) Show that $𝑣 (𝑥)$ is the unique bounded solution to the Poisson equation

- Δ 𝑣 = 𝑓 (𝑥), 𝑥 \in ℝ^{3} .

(iii) Use this to obtain the expression

𝐻 (𝑥) = 𝑐_{𝑛} / {| 𝑥 |}^{𝑛 - 2}

for the fundamental solution of the Laplace equation, starting with the heat kernel

𝐺 (𝑡, 𝑥) = \frac{𝑒^{- {| 𝑥 |}^{2} / (4 𝑡)}}{{(4 𝜋 𝑡)}^{𝑛 / 2}} .

(iv) In $𝑛 = 2$ the function $𝑣 (𝑥)$ is not well-defined. How can one mitigate this to use a similar process to obtain the expression for the fundamental solution of the Laplace equation in dimension $𝑛 = 2$ ?

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