Home Analysis Fourier Transform properties and applications

Fourier Transform properties and applications


In this post, we study Fourier transform properties and give some applications. In fact, this transformation helps in converting partial differential equations to ODE. A simple method to solve the heat equation is the Fourier transform technique.

The Riemann-Lebesgue Lemma

We denote by $L^1(\mathbb{R})$ the Lebesgue space of measurable functions $f:\mathbb{R}\to \mathbb{R} $ such that\begin{align*} \|f\|_1^2:=\int_{\mathbb{R}} |f(x)|dx<\infty.\end{align*}

Also, we recall that $\|\cdot\|_1$ is a norm on $ L^1(\mathbb{R})$ and that $( L^1(\mathbb{R}), \|\cdot\|_1 ) $ is a Banach space, no reflexive. On the other hand, the space of continuously differentiable functions with compact support $\mathcal{C}_c^1(\mathbb{R})$ is dense in the space $ L^1(\mathbb{R}) $.

As for any $t,x\in\mathbb{R}$, we have $|e^{-itx}f(x)|=|f(x)|$, then for $f\in L^1(\mathbb{R})$, the function $(x\mapsto e^{-itx}f(x) )\in L^1(\mathbb{R}) $. We then have the following definition:

Definition: For $f\in L^1(\mathbb{R}) $ we define the Fourier transform of $f$ by \begin{align*}\hat{f}(t)=\int_{\mathbb{R}} e^{-itx}f(x) dx,\qquad t\in \mathbb{R}.\end{align*}

Clearly $f\mapsto\hat{f}$ is a linear application on $ L^1(\mathbb{R})$. We have the following Riemann-Lebesgue result.

Theorem: The Fourier transform is an application from $ L^1(\mathbb{R})$ to $\mathcal{C}_0(\mathbb{R})$, where \begin{align*} \mathcal{C}_0(\mathbb{R}) :=\{f\in \mathcal{C}(\mathbb{R}) : \lim_{|t|\to +\infty}f(t)=0\}.\end{align*}

Proof: Here we use the density of $\mathcal{C}_c^1(\mathbb{R})$ is dense in the space $ L^1(\mathbb{R}) $. Hence, in the first step, we will prove the result for $f\in \mathcal{C}_c^1(\mathbb{R}) $. In this case, and by using and integration by part, we get for any $t\neq 0,$ \begin{align*}\hat{f}(t)=\frac{1}{it}\int_{\mathbb{R}}e^{-itx}f'(x)dx.\end{align*}This implies that\begin{align*} | \hat{f}(t) |\le \frac{1}{|t|}|f’|_1.\end{align*}

If $f\in L^1(\mathbb{R}) ,$ by density there exists a sequence $(f_n)_n\subset \mathcal{C}_c^1(\mathbb{R}) $ that converges to $f$ in $ L^1(\mathbb{R}) $. Thus, for any $\varepsilon>0,$ there exists $m\in\mathbb{N}$ such that $\|f_m-f\|_1\le \frac{\varepsilon}{2}$. Moreover, remark that\begin{align*} \hat{f}(t) = \hat{f_m}(t) + \hat{f}(t) – \hat{f_m}(t),\end{align*} so that \begin{align*} | \hat{f}(t) |&\le \frac{1}{|t|}\|f_m’\|_1+\|f-f_m\|_1\cr & \le \frac{1}{|t|}\|f_m’\|_1 + \frac{\varepsilon}{2}. \end{align*} This ends the proof.

Fourier Transform in the Schwartz space

We define the Schwartz space by\begin{align*}\mathcal{S}(\mathbb{R})=\{f\in \mathcal{C}^\infty(\mathbb{R}):\forall p,q\in\mathbb{N}&,\;\exists C_{p,q}>0:\cr & |x^pf^{(q)}(x)|\le C_{pq},\; \forall x\in\mathbb{R}\}.\end{align*} The first remark is that $ \mathcal{S}(\mathbb{R}) \subset L^1(\mathbb{R})$ continuously and densely. Hence we can prove all properties of Fourier transform in $\mathcal{S}(\mathbb{R})$. These properties are translated to $L^1(\mathbb{R})$ by density.

As example we have $(x\mapsto e^{-x^2})\in \mathcal{S}(\mathbb{R}) $. On the other hand, we can easily prove that if $f\in \mathcal{S}(\mathbb{R}) ,$ then also $f’$ and $x\mapsto xf(x)$ are in $ \mathcal{S}(\mathbb{R}) $.

Theorem: If $f\in \mathcal{S}(\mathbb{R}) $ then the Fourier transform $\hat{f}\in \mathcal{S}(\mathbb{R}) $.

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