Wavelet transform

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 $\psi \in L^{2}(\mathbb {R} )$ 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 $\{\psi _{jk}:j,k\in \mathbb {Z} \}$ by means of dyadic translations and dilations of $\psi \,$ ,

\psi _{jk}(x)=2^{j/2}\psi (2^{j}x-k)\,

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

\langle \psi _{jk},\psi _{lm}\rangle =\delta _{jl}\delta _{km}

where $\delta _{jl}\,$ is the Kronecker delta and $\langle f,g\rangle$ is the standard inner product on $L^{2}(\mathbb {R} )$ :

\langle f,g\rangle =\int _{-\infty }^{\infty }{\overline {f(x)}}g(x)dx

The requirement of completeness is that every function $f\in L^{2}(\mathbb {R} )$ may be expanded in the basis as

f(x)=\sum _{j,k=-\infty }^{\infty }c_{jk}\psi _{jk}(x)

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.