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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

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