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Homework 9, Real Analysis 2

Due April 27

Problem 1

Let $latex f\in L^2(\R^d), k\in L^1(\R^d)$.

  1. $latex \displaystyle (f*k)(x) = \int_{\R^d} f(x-y)k(y) dy$ converges for a.e. x.

  2. $latex ||f*k||_{L^2} \le ||f||_{L^2} ||k||_{L^1}$.

  3. $latex \widehat{(f*k)}(\xi) = \hat k(\xi) \hat f(\xi)$ for a.e. $latex \xi$.

  4. The operator $latex Tf = f*k$ is a Fourier multiplier operator with multiplier $latex m(\xi) = \hat k(\xi)$.

Problem 2

Let $latex \Omega\subset\C$ be open, and $latex \mathscr H\subset L^2(\Omega)$ be the subspace of holomorphic functions on $latex \Omega$.

  1. $latex \mathscr H$ is a closed subspace of $latex L^2(\Omega)$.

  2. If $latex \{\phi_k\}$ is an orthonormal basis of $latex \mathscr H$, then $latex \displaystyle \sum_k |\phi_k(z)|^2 \le \frac{1}{\pi d(z,\C\setminus\Omega)}$ for $latex z\in\Omega$, where $latex d(z,\C\setminus\Omega)$ is the distance from z to the complement of $latex \Omega$.

  3. The sum $latex B(z,w) = \sum_k \phi_k(z)\overline{\phi_k(w)}$ converges absolutely for $latex z,w\in\Omega$, and is independent of the orthonormal basis $latex \{\phi_k\}$.
    The function $latex B(z,w)$ is called the Bergman kernel.

  4. If T is the linear transformation on $latex L^2(\Omega)$ defined by $latex \displaystyle Tf(z) = \int_\Omega B(z,w)f(w) dudv$, $latex w = u + iv$, then T is the orthogonal projection of $latex L^2(\Omega)$ onto $latex \mathscr H$.

Problem 3

Consider the definitions of the previous problem with $latex \Omega$ the unit disc.

  1. $latex \displaystyle  f(z) = \sum_{n=0}^\infty a_n z^n \in\mathscr H$ if and only if $latex \displaystyle \sum_{n=0}^\infty \frac{|a_n|^2}{n+1} < \infty.$

  2. The sequence $latex \displaystyle \Big\{ \frac{z^n(n+1)}{\pi^{1/2}}\Big\}_{n=0}^\infty$ is an orthonormal basis for $latex \mathscr H$.

  3. $latex \displaystyle B(z,w) = \frac{1}{\pi(1 - z\bar w)}$.