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

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 {c_{jk}}. Here, denotes the square-sum norm: and denotes the usual norm on : By the Riesz representation theorem, there exists a unique dual basis \psi^{jk} such that where \delta_{jk} 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 \phi 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.