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.
Read more about this topic: Wavelet Transform
Famous quotes containing the words formal and/or definition:
“True variety is in that plenitude of real and unexpected elements, in the branch charged with blue flowers thrusting itself, against all expectations, from the springtime hedge which seems already too full, while the purely formal imitation of variety ... is but void and uniformity, that is, that which is most opposed to variety....”
—Marcel Proust (1871–1922)
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)