Given a square-integrable function $\psi\in L^2(\mathbb{R})$, define the series $\{\psi_{jk}\}$ by
$$\psi_{jk}(x) = 2^{j/2}\psi(2^jx-k)$$
for integers $j,k\in \mathbb{Z}$.
Such a function is called an R-function if the linear span of $\{\psi_{jk}\}$ is dense in $L^2(\mathbb{R})$, and if there exist positive constants ''A'', ''B'' with $0<A\leq B < \infty$ such that
$$A \Vert c_{jk} \Vert^2_{l^2} \leq
\bigg\Vert \sum_{jk=-\infty}^\infty c_{jk}\psi_{jk}\bigg\Vert^2_{L^2} \leq
B \Vert c_{jk} \Vert^2_{l^2}\,$$
for all bi-infinite-square summable-series $\{c_{jk}\}$. Here, $\Vert \cdot \Vert_{l^2}$ denotes the square-sum norm
$$\Vert c_{jk} \Vert^2_{l^2} = \sum_{jk=-\infty}^\infty \vert c_{jk}\vert^2$$
and $\Vert \cdot\Vert_{L^2}$ denotes the usual norm on $L^2(\mathbb{R})$
$$\Vert f\Vert^2_{L^2}= \int_{-\infty}^\infty \vert f(x)\vert^2 dx$$
By the Riesz representation theorem, there exists a unique dual basis $\psi^{jk}$ such that
$$\langle \psi^{jk} \vert \psi_{lm} \rangle = \delta_{jl} \delta_{km}$$
where $\delta_{jk}$ is the Kronecker delta and $\langle f \vert g \rangle$ is the usual inner product on $L^2(\mathbb{R})$. Indeed, there exists a unique series representation for a square-integrable function $f$ expressed in this basis
$$f(x) = \sum_{jk} \langle \psi^{jk} \vert f \rangle \psi_{jk}(x)$$
If there exists a function $\tilde{\psi} \in L^2(\mathbb{R})$ such that
$$\tilde{\psi}_{jk} = \psi^{jk}$$
then $\tilde{\psi}$ is called the "dual wavelet" or the "wavelet dual to $\psi$". In general, for some given R-function $\psi$, the dual will not exist. In the special case of $\psi = \tilde{\psi}$, the wavelet is said to be an orthogonal wavelet.
In the above justification for existence of dual basis $\psi^{jk}$ I don't understand how Riesz representation theorem is applied? Could someone clarify how Riesz representation theorem is used to show that $\langle \psi^{jk} \vert \psi_{lm} \rangle = \delta_{jl} \delta_{km}$?
Best Answer
Let $v_n$ be a collection of vectors in a Hilbert space $\mathcal{H},$ such that the linear span is dense and $$A\left \|\sum a_kv_k\right \|\le \|(a_k)\|_{\ell^2}\le B\left \|\sum a_kv_k\right \|\qquad (*)$$ for positive constants $A$ and $B$ and all sequences $(a_k)\in \ell^2.$ Consider the functional $$\varphi_n\left (\sum a_kv_k\right )= a_n$$ By $(*)$ the functional $\varphi_n$ is well defined and bounded. Therefore it extends to the entire space $\mathcal{H}$ (by the assumption that the linear span of $\{v_k\}$ is dense in $\mathcal{H}$). By the Riesz representation theorem $$\varphi_n(x)=\langle x,w_n\rangle$$ In particular $$\delta_{m,n}=\varphi_n(v_m)=\langle v_m,w_n\rangle \qquad (**) $$ The elements $w_n$ are uniquely determined by $(**)$ as the vectors $v_k$ are linearly dense.
In your question vectors are parametrized by two indices, but that does not change the reasoning as the collection is still countable.