# Fourier transform

# The Fourier transform on the space of functions of moderate decrease

Before extend the integral of a function on a closed and bounded interval, we should check whether the value exists or not. A useful conditino is as follows.

Definition 5.1.1 A function $f$ is said to be of moderate decrease if $f\in C(\mathbb R)$ and $\exists A>0,\epsilon>0$ so that

$$|f(x)|\leq \dfrac {A}{1+x^{1+\epsilon}},\ \forall x\in \mathbb R$$

We shall denote $\mathcal M(\mathbb R)$ the set of functions of moderate decrease on $\mathbb R$.

Definition 5.1.2 For all $f\in\mathcal M(\mathbb R)$, we may define its integral

$$\int^\infty_{-\infty}f(x)\mathrm dx:=\lim_{N\to\infty}\int^N_{-N}f(x)\mathrm dx$$

We summarize some elementary properties of integration over $\mathbb R$ in a property

Property 5.1.2 The integral of a function of moderate decrease defined above satisfies the following properties, if $f,g\in \mathcal M(\mathbb R)$ and $a,b\in\mathbb C$, then

1. Linearity:

$$\int^\infty_{-\infty}(af(x)+bg(x))\mathrm dx=a\int^\infty_{-\infty}f(x)\mathrm dx+b\int^\infty_{-\infty}g(x)\mathrm dx$$

2. Translation invariance: $\forall h\in\mathbb R$ we have

$$\int^\infty_{-\infty}f(x-h)\mathrm dx=\int^\infty_{-\infty}f(x)\mathrm dx$$

3. Scaling under dilations: if $\delta \neq 0$, then

$$\delta\int^\infty_{-\infty}f(\delta x)\mathrm dx=\mathrm{sgn}(\delta )\cdot \int^\infty_{-\infty}f(x)\mathrm dx$$

4. Continuity:

$$\int ^\infty_{-\infty}|f(x-h)-f(x)|\mathrm dx\to 0,\ \text{ as }h\to 0$$

Definition 5.1.3 If $f\in\mathcal M(\mathbb R)$, we define its Fourier transform for $\xi\in\mathbb R$ by

$$\hat f(\xi)=\int^\infty_{-\infty}f(x)e^{-2\pi i x\xi}\mathrm dx$$

Obviouly the integral makes sense. However, nothing in the definition above guarantees that $\hat f$ is of moderate decrease.

Roughly speaking, the Fourier transform is a continuous version of the Fourier coefficients.

# The Fourier transform on the Schwartz space

Definition 5.1.4 The Schwartz space on $\mathbb R$ consists of the se t of all $f\in C^\infty(\mathbb R)$ so that $f$ and its derivatives $f',...,f^{(l)},...$ are rapidly decreasing, in the sense that

$$\sup_{x\in\mathbb R}|x|^k|f^{(l)}(x)|<\infty,\ \forall k,l\geq 0$$

We denote this space by $\mathcal S=\mathcal S(\mathbb R)$

Definition 5.1.5 The Fourier transform of a function $f\in \mathcal S(\mathbb R)$ is defined by

$$\hat f(\xi)=\int^\infty_{-\infty}f(x)e^{-2\pi i x\xi}\mathrm dx$$

We use the notation

$$f(x)\to \hat f(\xi)$$

to mean that $\hat f(\xi)$ denotes the Fourier transform of $f$.

Property 5.1.6 If $f\in\mathcal S(\mathbb R)$, then

1. Whenever $h\in\mathbb R$

$$f(x+h)\to \hat f(\xi)e^{2\pi i h\xi}$$

2. Whenever $h\in\mathbb R$

$$f(x)e^{-2\pi ix h}\to \hat f(\xi +h)$$

3. Whenever $\delta >0$

$$f(\delta x)\to \delta^{-1}\hat f(\delta^{-1}\xi)$$

4. Derivatives:

$$f'(x)\to 2\pi i\xi\hat f(\xi)$$

$$-2\pi ixf(x)\to \dfrac {\mathrm d}{\mathrm d\xi}\hat f(\xi)$$

Theorem 5.1.7 If $f\in\mathcal S(\mathbb R)$, then $\hat f\in\mathcal S(\mathbb R)$.

Theorem 5.1.8 If $|f|\in\mathcal R(\mathbb R)$, then $\hat f$ is uniformly continuous in $\mathbb R$.

# Fourier inversion

# The Gaussians as good kernels

Theorem 5.2.1 If $f(x)=e^{-\pi x^2}$, then $\hat f(\xi)=f(\xi)$

Corollary 5.2.2 If $\delta>0$ and $K_\delta(x)=\delta^{1/2}e^{-\pi x^2/\delta}$, then $\hat K_\delta(\xi)=e^{-\pi \delta \xi^2}$.

We have now constructed a family of good kernels on the real line.

# Good kernel and convolution

Theorem 5.2.3 The collection $\{K_\delta\}_{\delta>0}$ is a family of good kernels as $\delta \to 0$. With

$$K_\delta=\delta^{-1/2}e^{-\pi x^2/\delta}$$

we have:

1. For every $\delta>0$, we have

$$\displaystyle \int^\infty_{-\infty}K_\delta(x)\mathrm dx=1$$

2. $\forall \delta>0, \exists M>0$, we have

$$\displaystyle \int ^\infty_{-\infty}|K_\delta|\mathrm dx\leq M$$

3. For every $\eta>0$, we have

$$\displaystyle \int_{|x|>\eta}|K_\delta(x)|\mathrm dx\to 0, \text{ as } \delta \to 0$$

Definition 5.2.4 If $f,g\in \mathcal S(\mathbb R)$, their convolution is defined by

$$(f*g)(x)=\int^\infty_{-\infty}f(x-t)g(t)\mathrm dt$$

Corollary 5.2.5 If $f\in \mathcal S(\mathbb R)$, then

$$(f*K_\delta)(x)\rightrightarrows f(x), \text{ as } \delta\to 0$$

# Fourier inversion

Property 5.2.6 If $f,g \in\mathcal S(\mathbb R)$, then we have the multiplication formula

$$\int^\infty_{-\infty}f(x)\hat g(x)\mathrm dx=\int^\infty_{-\infty}\hat f(y)g(y)\mathrm dy$$

Property 5.2.7 If $f,g\in\mathcal S(\mathbb R)$, then

1. $f*g\in \mathcal S(\mathbb R)$
2. $f*g=g*f$
3. $\widehat{f*g}(\xi)=\hat f(\xi)\hat g(\xi)$

Theorem 5.2.7 If $f\in \mathcal S(\mathbb R)$, then

$$f(x)=\int^\infty_{-\infty}\hat f(\xi)e^{2\pi ix\xi}\mathrm d\xi$$