📂Fourier Analysis

Several Definitions and Notations of Fourier Transform

Overview

The definition and notation of the Fourier transform vary depending on the needs and preferences of the author. Therefore, before dealing with the Fourier transform in textbooks, lectures, research papers, etc., it is common to clarify the definition and notation. If you skip over the definition thinking it is a concept you know, you might find the equations odd, so it is necessary to check carefully. Of course, the most important thing is that all these definitions are essentially the same, so there is no need to be too concerned about the notation or the definition itself. This document introduces the advantages and disadvantages, as well as the differences, of each definition.

Description¹

The Fourier transform can be naturally induced by thinking of a Fourier series of a function whose period is the entire set of real numbers. In this process, the Fourier transform and the inverse Fourier transform are defined as follows:

Fourier Transform	Inverse Fourier Transform
$\displaystyle \hat{f}(\xi):=\int _{-\infty} ^{\infty}f(x)e^{-i \xi x}dx$	$\displaystyle f(x):=\frac{1}{2\pi}\int _{-\infty} ^{\infty}\hat{f}(\xi)e^{i \xi x}d\xi$

Here, $\hat{}$ is read as “hat”. $\hat{f}(\xi)$ is read as “F hat of xi”. When one wants to emphasize the feeling of being an operator, the feeling of being an integral transform, or when one needs to use the ${}^{\prime}$ symbol for derivative together with the $\hat{}$ symbol, or to avoid confusion, it may be written in the following notation:

Fourier Transform	Inverse Fourier Transform
$\mathcal{F}:L^{1} \to L^{1}$	$\mathcal{F}^{-1}:L^{1} \to L^{1}$
$\displaystyle \mathcal{F}f(\xi):=\int _{-\infty} ^{\infty}f(x)e^{-i \xi x}dx$	$\displaystyle \mathcal{F}^{-1}f(x):=\frac{1}{2\pi}\int _{-\infty} ^{\infty}f(\xi)e^{i \xi x}d\xi$

The notation $\mathscr{F}$ is also used. Apart from the notation differences, the definition of the Fourier transform itself may differ as follows:

Fourier Transform	Inverse Fourier Transform
$\displaystyle \hat{f}(\xi):=\frac{1}{\sqrt{2\pi}}\int _{-\infty} ^{\infty}f(x)e^{-i \xi x}dx$	$\displaystyle f(x):=\frac{1}{\sqrt{2\pi}}\int _{-\infty} ^{\infty}\hat{f}(\xi)e^{i \xi x}d\xi$
$\displaystyle \hat{f}(\xi):=\int _{-\infty} ^{\infty}f(x)e^{-2\pi i \xi x}dx$	$\displaystyle f(x):=\int _{-\infty} ^{\infty}\hat{f}(\xi)e^{2\pi i \xi x}d\xi$

For the sake of clarity in the explanation, let’s denote each of the above definitions as follows.

$$ \tilde{f}(\xi):=\frac{1}{\sqrt{2\pi}}\int _{-\infty} ^{\infty}f(x)e^{-i \xi x}dx \quad \text{and} \quad \check{f}(\xi):=\int _{-\infty} ^{\infty}f(x)e^{-2\pi i \xi x}dx $$

The variety of definitions for the Fourier transform can be seen in the table below.

Plancherel Theorem	Convolution	Fourier Transform of Derivatives
$\| \hat{f} \|^{2} =2\pi\left\| f \right\|^{2}$	$(f \ast g)\hat{}=\hat{f}\hat{g}$	$(f^{\prime})\hat{} (\xi)=i\xi \hat{f}(\xi)$
$\| \tilde{f} \|^{2}=\left\| f \right\|^{2}$	$(f \ast g)\tilde{}=\sqrt{2\pi}\tilde{f}\tilde{g}$	$(f^{\prime})\tilde{} (\xi)=i\xi \tilde{f}(\xi)$
$\| \check{f} \|^{2}=\left\| f \right\|^{2}$	$(f \ast g)\check{}=\check{f}\check{g}$	$(f^{\prime})\check{} (\xi)=2\pi i\xi \check{f}(\xi)$

As can be seen in the table, depending on the definition, the formula where the constant $2\pi$ appears can differ. Therefore, the definition can change depending on what formula one wants to simplify. Empirically, in fields like signal and image processing, definitions like $\check{f}$ are often used. Also, in the definition of the Fourier transform, there might not be a minus sign $(-)$ in the exponent. In that case, since it will be on the inverse side, do not be confused even if it is different from the definition you know.