In mathematics, a dual wavelet is the dual to a wavelet. In general, the wavelet series generated by a square-integrable function will have a dual series, in the sense of the Riesz representation theorem. However, the dual series is not itself in general representable by a square-integrable function.

Definition

by

.

Such a function is called an R-function is dense, and if there exist positive constants A, B such that

for all bi-infinite square summable denotes the square-sum norm:

:

By the Riesz representation theorem such that

is the Kronecker delta is the usual inner product. Indeed, there exists a unique series representation for a square-integrable function f expressed in this basis:

such that

is called the dual wavelet or the wavelet dual to . In general, for some given R-function , the wavelet is said to be an orthogonal wavelet.

An example of an R for some complex number z. It is straightforward to show that this does not have a wavelet dual.

See also

• Multiresolution analysis

References

• Charles K. Chui, An Introduction to Wavelets (Wavelet Analysis & Its Applications), (1992), Academic Press, San Diego,