# John Lindsay Orr

$\newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\RR}{\mathbb{R}} \newcommand{\TT}{\mathbb{T}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\H}{\mathcal{H}} \newcommand{\e}{\epsilon} \newcommand{\x}{\mathbf{x}} \newcommand{\y}{\mathbf{y}}$

$\newcommand{\F}{\mathcal{F}}$

# Fourier Notes

## Fourier Series

Definition 1. Let $f\in L^1(0,1)$ be a Lebesgue integrable function on $[0, 1]$ and define the Fourier coefficients by

Theorem 2. Suppose that $f$ is continuous on $\TT$ and $\hat{f}(n)$ is absolutely summable. Then where the series converges uniformly on $[0,1]$.

Theorem 3. (Parseval's Theorem) Let $f \in L^2(0,1)$. Then its Fourier coefficients are in $\ell^2(\ZZ)$ and Thus the map $f' \in L^2(0,1) \rightarrow {f}\in\ell^2(\ZZ)$ is unitary.

## Fourier Transforms

Definition 4. Let $f\in L^1(\RR)$ and define the Fourier transform Likewise for integrable $g(\xi)$ define the inverse Fourier transform

Theorem 5. (Fourier Inversion Theorem) If $f \in L^1(\RR)$ and also $\F f \in L^1(\RR)$ then $(\F^{-1}\F f)(x) = f(x)$ for almost every $x\in\RR$.

Corollary 6. If, in addition, $f$ is continuous, then $(\F^{-1}\F f)(x) = f(x)$ for all $x\in\RR$.

Theorem 7. (Plancherel Theorem) If $f\in L^1(\RR)\cap L^2(\RR)$ then $\F f\in L^2(\RR)$ and

Thus the map $f\rightarrow\F f$ is an isometry from $f\in L^1(\RR)\cap L^2(\RR)$ into $L^2(\RR)$, which extends to a unitary $L^2(\RR)\rightarrow L^2(\RR)$. This unitary is called the Plancherel transform.