Ricardo A. Sáenz

Due May 3 Problem 1 Let $latex \gamma$ be the lower semicircle of radius $latex N$ around the origin, and $latex \xi > 0.$ Then $latex \displaystyle \int_\gamma f(z) dz = 2\pi i\text{Res}_{z=-it} f(z) = i e^{-2\pi t\xi},$ where $latex f(z)$ is the function defined by $latex f(z) = \dfrac{1}{\pi} \dfrac{z}{z^2+t^2} e^{-2\pi iz\xi}.$ Problem 2 If $latex f\in L^1(\mathbb R)$ and diferentiable at $latex x\in\mathbb R$, then the limit $latex \displaystyle \lim_{t\to 0} \int_{|y|\ge t} \frac{f(x-y)}{y} dy$ exists. ( Hint:  Use the identity, for any $latex \delta_n>0$, $latex \displaystyle \int_{t\le|y|<\delta_n} \frac{f(x-y)}{y} dy = \int_{t\le|y|<\delta_n} \frac{f(x-y) - f(x)}{y} dy + \int_{t\le|y|<\delta_n} \frac{f(x)}{y} dy,$ and take $latex \delta_n\to 0$.) Problem 3 If $latex f_n, g_n$ are sequences in $latex C_c^\infty(\mathbb R)$ that converge in $latex L^1(\mathbb R)$ to $latex f$, then $latex Hf_n$ and $latex Hg_n$ converge in measu...

Notes on pointwise and uniform convergence of Fourier series of continuous functions:  Chapter05.pdf

Notes on the Hardy-Littlewood maximal functions and boundary limits of harmonic functions:  Chapter09.pdf Notes on the Fourier transform, the inversion formula and square means convergence:  Chapter10.pdf

Due April 12 Problem 1 For any $latex \xi\in\mathbb R$, $latex \displaystyle \lim_{N\to\infty} \int_{-N}^N e^{-\pi (x+i\xi)^2}dx = \lim_{N\to\infty} \int_{-N}^N e^{-\pi x^2}dx = 1.$ ( Hint:  Consider the contour integral $latex \int_\gamma e^{-\pi z^2} dz = 0$ over the rectangle $latex \gamma$ with vertices $latex N, N+i\xi, -N+i\xi$ and $latex -N$.) Problem 2 Let $latex u>0$. Then $latex \displaystyle \frac{1}{\pi} \int_{-\infty}^\infty \frac{e^{-2\pi iuv}}{1 + v^2} dv = \frac{1}{\sqrt\pi} \int_0^\infty \frac{1}{\sqrt s} e^{-s} e^{-\pi^2 u^2/s} ds$; $latex \displaystyle \frac{1}{\pi} \int_{-\infty}^\infty \frac{e^{-2\pi iuv}}{1 + v^2} dv = e^{-2\pi u}$. Problem 3 Prove the Riemann-Lebesgue Lemma : If $latex f\in L^1(\mathbb R^d)$, then $latex \hat f(\xi) \to 0$ as $latex |\xi|\to\infty$. Problem 4 If $latex \Phi(x) = e^{-\pi |x|^2}$, the collection $latex \{\Phi_t(x)\}_{t>0}$ if its dilations is a collection of better kernels. Problem 5 ...

Debido a que la hora de oficina para los estudiantes de cálculo tenía un conflicto con otra clase, las horas de oficina cambian de horario: la de Análisis Armónico será a las 4:00pm, cada miércoles.

Due April 5 Problem 1 The set $latex L^1(\mathbb R^d)$ of integrable functions is a complex vector space, and $latex f\mapsto \int f$ is a linear functional. $latex \displaystyle ||f||_1 = \int f$ is a norm on $latex L^1(\mathbb R^d)$, when $latex L^1(\mathbb R^d)$ is seen as a set of equivalence classes of $latex f\sim g$ if and only if $latex f=g$ a.e. Problem 2 Let $latex L^2(\mathbb R^d)$ be the set of measurable functions $latex f$ such that $latex \displaystyle \int |f|^2 < \infty$, seen as a set of equivalence classes of $latex f\sim g$ if and only if $latex f=g$ a.e. $latex L^2(\mathbb R^d)$ is a complex vector space. The bilinear form $latex \displaystyle \langle f, g \rangle = \int f \bar g$ is well defined on $latex L^2(\mathbb R^d)$ and is an inner product. $latex L^2(\mathbb R^d)$ is complete with the norm $latex \displaystyle ||f||_2 = \sqrt{\langle f, f \rangle}.$ ( Hint:  Proceed as in the case of $latex L^1$ seen in class.) Problem 3 If...