Wavelet Transform - Formal Definition

Formal Definition

A function 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 of square integrable functions. The Hilbert basis is constructed as the family of functions by means of dyadic translations and dilations of ,

for integers . This family is an orthonormal system if it is orthonormal under the inner product

where is the Kronecker delta and is the standard inner product on The requirement of completeness is that every function 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.