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## 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. $\displaystyle f(z) = \sum_{n=0}^\infty a_n z^n \in\mathscr H$ 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. $\displaystyle B(z,w) = \frac{1}{\pi(1 - z\bar w)}$.