Given a square-integrable function ${\displaystyle \psi \in L^{2}(\mathbb {R} )}$ , define the series ${\displaystyle \{\psi _{jk}\}}$  by

${\displaystyle \psi _{jk}(x)=2^{j/2}\psi (2^{j}x-k)}$ 

for integers ${\displaystyle j,k\in \mathbb {Z} }$ .

Such a function is called an R-function if the linear span of ${\displaystyle \{\psi _{jk}\}}$  is dense in ${\displaystyle L^{2}(\mathbb {R} )}$ , and if there exist positive constants A, B with ${\displaystyle 0<A\leq B<\infty }$  such that

${\displaystyle A\Vert c_{jk}\Vert _{l^{2}}^{2}\leq {\bigg \Vert }\sum _{jk=-\infty }^{\infty }c_{jk}\psi _{jk}{\bigg \Vert }_{L^{2}}^{2}\leq B\Vert c_{jk}\Vert _{l^{2}}^{2}\,}$ 

for all bi-infinite square summable series ${\displaystyle \{c_{jk}\}}$ . Here, ${\displaystyle \Vert \cdot \Vert _{l^{2}}}$  denotes the square-sum norm:

${\displaystyle \Vert c_{jk}\Vert _{l^{2}}^{2}=\sum _{jk=-\infty }^{\infty }\vert c_{jk}\vert ^{2}}$ 

and ${\displaystyle \Vert \cdot \Vert _{L^{2}}}$  denotes the usual norm on ${\displaystyle L^{2}(\mathbb {R} )}$ :

${\displaystyle \Vert f\Vert _{L^{2}}^{2}=\int _{-\infty }^{\infty }\vert f(x)\vert ^{2}dx}$ 

By the Riesz representation theorem, there exists a unique dual basis ${\displaystyle \psi ^{jk}}$  such that

${\displaystyle \langle \psi ^{jk}\vert \psi _{lm}\rangle =\delta _{jl}\delta _{km}}$ 

where ${\displaystyle \delta _{jk}}$  is the Kronecker delta and ${\displaystyle \langle f\vert g\rangle }$  is the usual inner product on ${\displaystyle L^{2}(\mathbb {R} )}$ . Indeed, there exists a unique series representation for a square-integrable function f expressed in this basis:

${\displaystyle f(x)=\sum _{jk}\langle \psi ^{jk}\vert f\rangle \psi _{jk}(x)}$ 

If there exists a function ${\displaystyle {\tilde {\psi }}\in L^{2}(\mathbb {R} )}$  such that

${\displaystyle {\tilde {\psi }}_{jk}=\psi ^{jk}}$ 

then ${\displaystyle {\tilde {\psi }}}$  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 ${\displaystyle \psi ={\tilde {\psi }}}$ , the wavelet is said to be an orthogonal wavelet.

An example of an R-function without a dual is easy to construct. Let ${\displaystyle \phi }$  be an orthogonal wavelet. Then define ${\displaystyle \psi (x)=\phi (x)+z\phi (2x)}$  for some complex number z. It is straightforward to show that this ψ does not have a wavelet dual.

• Multiresolution analysis

• Charles K. Chui, An Introduction to Wavelets (Wavelet Analysis & Its Applications), (1992), Academic Press, San Diego, ISBN 0-12-174584-8