Ricardo A. Sáenz

Escuela de Análisis matemático Análisis de Fourier y operadores de multiplicación Si $latex f\in L^p(\mathbb R^d)$ y $latex g\in L^1(\mathbb R^d)$, entonces $latex f*g\in L^p(\mathbb R^d)$ y $latex ||f*g||_{L^p} \le ||f||_{L^p} ||g||_{L^1}.$ Si $latex f\in L^1(\mathbb R^d)\cap L^2(\mathbb R^d)$, entonces $latex ||\hat f||_{L^2} = ||f||_{L^2}.$ ( Sugerencia: Considera $latex h=f*g$, con $latex g(x) = \overline{f(-x)}$, y nota que $latex h(0) = \int \hat h$.) El operador $latex f\mapsto \hat f$ es sobreyectivo en $latex L^2(\mathbb R^d)$. ( Sugerencia: Si no, existiría $latex g\in L^2$ tal que $latex \int \hat f g = 0$ para toda $latex f\in L^2$. Utiliza el pasito del sombrero.) Sea $latex h(x) = \begin{cases} 0 & x<0\\1 & x\ge 0\end{cases}$ la función de Heaviside. Como distribuciones, $latex h' = \delta,$ la función delta de Dirac. Sea $latex \mathbf 1$ la función constante igual a 1. Como distribuciones, $latex \hat{\mathbf 1} = \delt...

Escuela de Análisis matemático Análisis de Fourier y operadores de multiplicación Si $latex T:V\to V$ es simétrico y $latex \lambda_i\not=\lambda_j$ son eigenvalores reales distintos de $latex T$, con eigenvectores $latex u_i, u_j$ correspondientes, entonces $latex u_i\perp u_j$. Sea $latex T:\mathscr H\to\mathscr H$ un operador acotado diagonalizado con sucesión multiplicadora $latex \lambda_k$. T es unitario si, y solo si, $latex |\lambda_k|=1$ para todo k . T es una proyección ortogonal si, y solo si, todo $latex \lambda_k = 0\text{ o } 1$. T es un operador compacto si, y solo si, $latex \lambda_k \to 0$. Si $latex \sum a_n$ es una serie convergente, entonces es Cesàro-sumable. Si $latex \sum a_n$ es Cesàro-sumable, entonces es Abel-sumable.

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis Prove by induction, for the interval case, that $latex \min\{ \mathscr E_m(u): u|_{\{0,1\}}=v\} = \mathscr E_0(v) = (v(0)-v(1))^2,$ with the minimizer satisfying $latex \displaystyle u\Big(\frac{2k+1}{2^m}\Big) = \frac{1}{2}\Big(u\Big(\frac{k}{2^{m-1}}\Big) + u\Big(\frac{k+1}{2^{m-1}}\Big)\Big).$ The minimum of $latex f(x,y,z) = (a-x)^2 + (x-y)^2 + (y-a)^2 + (x-b)^2 + (b-z)^2 + (z-x)^2 + (y-z)^2 + (z-c)^2 + (c-y)^2$ is attained at $latex \displaystyle x^* = \frac{2a+2b+c}{5},\; y^* = \frac{2a+b+2c}{5},\; z^* = \frac{a+2b+2c}{5},$ with $latex f(x^*, y^*, z^*) = \dfrac{3}{5}\big((a-b)^2 + (b-c)^2 + (c-a)^2\big).$ Prove that one can obtain the values $latex u(F_2(q_1)) = x, u(F_3(q_1)) = y$ of a harmonic function in terms of the values $latex a,b,c$ at the points $latex p_2, q_1, p_3$, respectively  (as in the figure below). Use the previous problem to show that, if $latex u$ is a ha...

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis Show that $latex ||\cdot||_{H^1}$ is a norm, by showing it is induced by an inner product. Consider, for a connected domain $latex \Omega$, the energy form $latex \mathscr E(u,v) = \int_\Omega \nabla u\cdot \nabla v$. $latex \mathscr E(u,v)$ is an inner product on $latex H^1$ modulo constants. $latex \mathscr E(u,v)$ is an inner product on $latex H_0^1$. Show the equivalences of the Dirichlet principle.