📂Fourier Analysis

Definition of Wavelets

Definition

Let’s denote as $\psi \in L^{2}(\mathbb{R})$. When $\psi$ satisfies the following two conditions, the function $\psi$ is called a wavelet.

(a) For an integer $j,k \in \mathbb{Z}$, $\psi_{j,k}$ is defined as follows.

$$\psi_{j,k} (x):=2^{\frac{j}{2}}\psi (2^{j}x-k),\quad x\in \mathbb{R}$$

(b) $\left\{ \psi _{j,k}\right\}_{j,k\in \mathbb{Z}}$ is an orthonormal basis of $L^{2}(\mathbb{R})$ space.

$\psi_{j,k}$ can also be represented by dilation D and translation $T_{k}$ as follows.

$$\psi_{j,k}=D^{j}T_{k}\psi,\quad j,k\in\mathbb{Z}$$

Explanation

Wavelet theory provides a theoretical basis for effectively compressing data by finding an orthonormal basis. It allows for the removal of noise from data and identification of the necessary signals. Therefore, wavelets are used in a wide range of fields including signal processing, image processing, physics, chemistry, geostatistics, oceanography, meteorology, medicine, and financial management. Research on wavelet analysis began in France in the mid-1980s. Compared to Fourier analysis, which has been researched since the 18th and 19th centuries, it can be said that wavelet research has only recently begun.[^1]