Definitions

# Wavelet series

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.

## Formal definition

A function $psiin L^2\left(mathbb\left\{R\right\}\right)$ 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 $L^2\left(mathbb\left\{R\right\}\right)$ of square integrable functions. The Hilbert basis is constructed as the family of functions $\left\{psi_\left\{jk\right\}:j,kinZ\right\}$ by means of dyadic translations and dilations of $psi,$,

$psi_\left\{jk\right\}\left(x\right) = 2^\left\{j/2\right\} psi\left(2^jx-k\right),$

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

$langlepsi_\left\{jk\right\},psi_\left\{lm\right\}rangle = delta_\left\{jl\right\}delta_\left\{km\right\}$

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

$f\left(x\right)=sum_\left\{j,k=-infty\right\}^infty c_\left\{jk\right\} psi_\left\{jk\right\}\left(x\right)$

with convergence of the series understood to be convergence in the 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

$left\left[W_psi fright\right]\left(a,b\right) = frac\left\{1\right\}\left\{sqrt$
>}
int_{-infty}^infty overline{psileft(frac{x-b}{a}right)}f(x)dx,

The wavelet coefficients $c_\left\{jk\right\}$ are then given by

$c_\left\{jk\right\}= left\left[W_psi fright\right]\left(2^\left\{-j\right\}, k2^\left\{-j\right\}\right)$

Here, $a=2^\left\{-j\right\}$ is called the binary dilation or dyadic dilation, and $b=k2^\left\{-j\right\}$ is the binary or dyadic position.

## General remarks

Unlike the Fourier transform, which is an integral transform in both directions, the wavelet series is an integral transform in one direction, and a series in the other, much like the Fourier series.

The canonical example of an orthonormal wavelet, that is, a wavelet that provides a complete set of basis elements for $L^2\left(mathbb\left\{R\right\}\right)$, is the Haar wavelet.