Ricardo A. Sáenz

Due March 29 Problem 1 $latex A\subset\mathbb R^d$ is measurable if and only if, for all $latex B\subset\mathbb R^d$,  $latex |B|_* = |B\cap A|_* + |B\setminus A|_*$. Problem 2 Let $latex A\subset\mathbb R^d$. The following are equivalent. $latex A$ is measurable. $latex A = P\setminus M$, where $latex P$ is a $latex G_\delta$ set and $latex |M|=0$. $latex A = Q\cup N$, where $latex Q$ is an $latex F_\sigma$ set and $latex |N|=0$. Problem 3 Let $latex A\subset\mathbb R^d$ be a measurable set. For $latex \delta > 0$, let $latex \delta A = \{ \delta x: x\in A\}$. Then $latex \delta A$ ls measurable and $latex |\delta A| = \delta^d |A|$. For a $latex d$-tuple $latex \bar{\delta} = (\delta_1, \ldots, \delta_d)$ with each $latex \delta_j>0$, $latex j=1,\ldots,d$, define $latex \bar\delta A = \{(\delta_1 x_1, \ldots, \delta_d x_d): (x_1, \ldots, x_d)\in A\}$. Then $latex \bar\delta A$ is measurable and $latex |\bar\delta A| = \delta_1\cdots\delta_d |A|.$

Notes on the Poisson kernel for the upper half-space:  Chapter07.pdf Recall that you can find the bibliography in the link:  Bibliography.pdf

Due March 22 Problem 1 For any dimension $latex d\ge 1$, $latex \displaystyle \int_{\mathbb R^d} \frac{dx}{(|x|^2 + 1)^{(d+1)/2}} = \frac{\pi^{(d+1)/2}}{\Gamma((d+1)/2)},$ and verify that $latex \displaystyle \int_{\mathbb R^d} P_t(x) dx = 1.$ ( Hint:  Use spherical coordinates and the identity $latex \displaystyle \int_0^\infty t^\alpha e^{-ts} \frac{dt}{t} = \frac{\Gamma(\alpha)}{s^\alpha}$ for any $latex \alpha, s>0$. Problem 2 Let $latex f$ be integrable on $latex \mathbb R^d$. For any $latex h\in\mathbb R^d$, $latex \displaystyle \int_{\mathbb R^d} f(x) dx = \int_{\mathbb R^d} f(x - h) dx.$ For any $latex r>0$, $latex \displaystyle \int_{\mathbb R^d} f(x) dx = \frac{1}{r^d} \int_{\mathbb R^d} f\big( \frac{x}{r}\big) dx.$ Problem 3 If $latex f\in C_0(\mathbb R^d)$, then $latex u(x,t) \to f(x)$ as $latex t\to 0$, uniformly in $latex x\in\mathbb R^d$. Problem 4 Let $latex f\in C_c(\mathbb R^d)$. Then $latex \displaystyle \int_{\mathbb R^d} |f(x -

Notes on spherical harmonics:  Chapter06.pdf These notes don't include the bibliography, as the previous chapters had. Rather, I have prepared a link for the bibliography of the course here:  Bibliography.pdf . This link is included in the course webpage , listed as Extended bibliography .

Due March 15 Problem 1 A polynomial $latex p\in\mathcal P_k$ if and only if $latex \displaystyle \sum_{j=1}^d x_j \frac{\partial p}{\partial x_j} = k p.$ In fact, if $latex f$ is a differentiable function on $latex \mathbb R^d$ that is homogeneous of degree $latex \kappa\in\mathbb R$, i. e. , $latex f(tx) = t^\kappa f(x)$ for any $latex t>0$ and $latex x\in\mathbb R^d$, then $latex \displaystyle \sum_{j=1}^d x_j \frac{\partial f}{\partial x_j} = \kappa f.$ Problem 2 If $latex p\in\mathcal H_k$, then $latex \dfrac{p(x)}{|x|^{2k+d-2}}$ is harmonic in $latex \mathbb R^d\setminus\{0\}.$ Problem 3 If $latex R$ is any rotation on $latex \mathbb R^d$, then $latex Z_k^{R\zeta}(R\xi) = Z_k^\zeta(\xi).$ Problem 4 Use Parseval's identity to verify $latex ||Z_k^\zeta||^2 = d_k/\omega_d$. Use Cauchy-Schwarz inequality to verify $latex ||Z_k^\zeta||_u = d_k/\omega_d$. Problem 5 The Gegenbauer polynomials $latex C_k^{\alpha}(t)$ are defined by the generating fu

As stated in the syllabus, 50% of the midterms grades consist of a written essay of a short research project. The projects may be worked by a student alone or in pair and, for the first midterm, the essay must be turn in by March 29. Each enrolled student or pair of student must choose a different project. Unenrolled students sitting in the course may work on a project, and are free to choose independently of other students, but cannot work with enrolled students. The following is the list of projects to choose from. This list is not yet complete, and new projects will be added in the following days, depending on the student choices. Littlewood's Theorem Karla Paulette Flores Silva and Jaime Daniel Hernández Palacios Sharpness of Bernstein's Theorem Yair Antonio Castillo Castillo and Rafael Morales Jiménez Uniform convergence of Fourier series of Hölder continuous functions Bernardo Ameneyro Rodríguez and José Gabriel Rosales Castañeda

Notes on Fourier series and square means convergence:  Chapter04.pdf

As noted in class, we gave Fejér's original proof of his theorem on the Cesàro summability of a Fourier series of a continuous function (except for the fact the he assumed the function to be even and, thus, its Fourier series contains only cosines). Fejér's theorem was published in Comptes Rendus Hebdomadaries, Seances de l’Academie de Sciences, Paris 131 (1900), and you can find the paper here:   Sur les fonctions bornées et intégrables .

Due March 8 Problem 1 If $latex \sum a_n$ converges to $latex s$, then it is Abel-summable to $latex s$. If $latex \sum a_n$ is Cesàro-summable to $latex s$, then is it Abel-summable to $latex s$. Problem 2 (Tauber's theorem) If $latex \sum a_n$ is Abel-summable to $latex s$ and $latex na_n \to 0$, then $latex \sum a_n$ coverges to $latex s$. Problem 3 Suppose $latex f$ has left and right limits at $latex \theta_0$, say $latex \displaystyle \lim_{\theta\to\theta_0^-}f(\theta) = f(\theta_0-) \qquad \text{and}\qquad \lim_{\theta\to\theta_0^+}f(\theta) = f(\theta_0+),$ then  $latex \displaystyle \sigma_N(\theta_0) \to \frac{f(\theta_0-) + f(\theta_0+)}{2},$ where $latex \sigma_N(\theta)$ are the Cesàro sums of its Fourier series. Problem 4 (Bernstein's theorem) If $latex f\in C^\alpha(\mathbb S)$, for some $latex \alpha > 1/2$, then $latex \sum |\hat{f}(n)| < \infty$. $latex f\in C^\alpha(\mathbb S)$ means that $latex f$ is Holder continuous