Homework 9, Real Analysis 2

Due April 27

Problem 1

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

  1. \displaystyle (f*k)(x) = \int_{\R^d} f(x-y)k(y) dy converges for a.e. x.
  2. ||f*k||_{L^2} \le ||f||_{L^2} ||k||_{L^1}.
  3. \widehat{(f*k)}(\xi) = \hat k(\xi) \hat f(\xi) for a.e. \xi.
  4. The operator Tf = f*k is a Fourier multiplier operator with multiplier m(\xi) = \hat k(\xi).

Problem 2

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

  1. \mathscr H is a closed subspace of L^2(\Omega).
  2. If \{\phi_k\} is an orthonormal basis of \mathscr H, then \displaystyle \sum_k |\phi_k(z)|^2 \le \frac{1}{\pi d(z,\C\setminus\Omega)} for z\in\Omega, where d(z,\C\setminus\Omega) is the distance from z to the complement of \Omega.
  3. The sum B(z,w) = \sum_k \phi_k(z)\overline{\phi_k(w)} converges absolutely for z,w\in\Omega, and is independent of the orthonormal basis \{\phi_k\}.
    The function B(z,w) is called the Bergman kernel.
  4. If T is the linear transformation on L^2(\Omega) defined by \displaystyle Tf(z) = \int_\Omega B(z,w)f(w) dudv, w = u + iv, then T is the orthogonal projection of L^2(\Omega) onto \mathscr H.

Problem 3

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

  1. if and only if \displaystyle \sum_{n=0}^\infty \frac{|a_n|^2}{n+1} < \infty.
  2. The sequence \displaystyle \Big\{ \frac{z^n(n+1)}{\pi^{1/2}}\Big\}_{n=0}^\infty is an orthonormal basis for \mathscr H.
  3. \displaystyle B(z,w) = \frac{1}{\pi(1 - z\bar w)}.
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