# Lecture 16: Fourier Transform
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$\newcommand{\F}{\mathcal{F}}$
$\newcommand{\R}{\mathbb{R}}$
So far in the course, we studied differential operators on bounded intervals. We began with the Laplacian $-d^2/dx^2$ on $[a,b]$, which we "solved" using Fourier series, and then generalized this to Schrodinger operators of type $L=-d^2/dx^2+q(x)$ on $[a,b]$. The generalization involved developing the theory of compact operators, and then observing that the Green's function $(z-L)^{-1}$ (which is really just the resolvent at some point not in $\sigma(L)$) is compact, so $L$ inherits the (complete) set of eigenvectors of the compact case.

In the next three lectures we will generalize this to differential operators on $\R$. The analogue of Fourier series is the Fourier transform, which will give us a spectral representation of the momentum and Laplacian operators on $\R$. We will study Schrodinger operators on $\R$ by observing that $(z-L)^{-1}$ is *bounded* for a judicious choice of $z$.

In this lecture we develop the basic theory of Schwartz spaces and the Fourier transform, covering the following topics. The treatment is very similar to Dimock 1.1.4 so I will not (re)write notes.

Schwartz Space
Fourier Transform
Multiplication and Differentiation
Inversion Theorem, Plancharel
Extension to a unitary operator on L^2

The punch line of this lecture was a spectral representation for the momentum operator $A=-id/dx$ on the Schwartz space $S(\R)$:
$$ \F A = M_x \F\quad on \quad S(\R)$$
where $\F$ is the $L^2$ Fourier transform. In particular, this means that
$$ \F A \F^{-1} = M_x \quad on \quad S(\R)\quad (*).$$
where $M_x$ is multiplication by $x$.

For the purposes of spectral theory, however $S(\R)$ is not a convenient subspace of $L^2$ to work with; the right one is the "maximal domain"
$$D(M_x):=\{f\in L^2(\R): M_xf\in L^2(\R)\}.$$
In order to make sense of the identiy $(*)$ on this extended domain, however, we need to extend $A$ to $\F^{-1}D(M_x)$, and this is done by *defining* $\hat{A}:\F^{-1}D(M_x)\rightarrow L^2(\R)$ by $\hat{A} f = \F^{-1} M_x \F f$. The domain of $\hat{A}$ thus obtained is called a *Sobolev space* and the resulting notion of derivative is called a *weak derivative*.

In the next lecture we will define the spectral theory of such operators and show how to do this for any self-adjoint unbounded operator (which we will also define).