Definition
Given a square integrable function, define the series by
for integers .
Such a function is called an R-function if the linear span of is dense in, and if there exist positive constants A, B with such that
for all bi-infinite square summable series . Here, denotes the square-sum norm:
and denotes the usual norm on :
By the Riesz representation theorem, there exists a unique dual basis such that
where is the Kronecker delta and is the usual inner product on . Indeed, there exists a unique series representation for a square integrable function f expressed in this basis:
If there exists a function such that
then is called the dual wavelet or the wavelet dual to ψ. In general, for some given R-function ψ, the dual will not exist. In the special case of, the wavelet is said to be an orthogonal wavelet.
An example of an R-function without a dual is easy to construct. Let be an orthogonal wavelet. Then define for some complex number z. It is straightforward to show that this ψ does not have a wavelet dual.
Read more about this topic: Dual Wavelet
Famous quotes containing the word definition:
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)