# Wavelet compression

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In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform.

## Contents

### Formal definition

A function $\psi\in L^2(\mathbb{R})$ is called an orthonormal wavelet if it can be used to define a Hilbert basis, that is a complete orthonormal system, for the Hilbert space $L^2(\mathbb{R})$ of square integrable functions. The Hilbert basis is constructed as the family of functions $\{\psi_{jk}:j,k\in\Z\}$ by means of dyadic translations and dilations of $\psi\,$,

for integers $j,k\in \mathbb{Z}$. This family is an orthonormal system if it is orthonormal under the inner product

where $\delta_{jl}\,$ is the Kronecker delta and $\langle f,g\rangle$ is the standard inner product $\langle f,g\rangle = \int_{-\infty}^\infty \overline{f(x)}g(x)dx$ on $L^2(\mathbb{R}).$ The requirement of completeness is that every function $f\in L^2(\mathbb{R})$ may be expanded in the basis as

with convergence of the series understood to be convergence in norm. Such a representation of a function f is known as a wavelet series. This implies that an orthonormal wavelet is self-dual.

### Wavelet transform

The integral wavelet transform is the integral transform defined as