# DWT initialization from nonuniform samples

Let $D\subset\mathbb R$ be compact, let $f:\mathbb R\to\mathbb R$ be a contintinuous function with support $D$.

Let $\phi(x)$ be a well-defined scaling function, in the sense that it generates a standard MRA $\{V_j\}$.

Define the orthonormal functions $\phi_{jk} := 2^{j/2}\phi(2^j x-k)$, then we have that the approximation $f_J$ of $f$ at a scale $J$ is in $\overline{\text{span}}\left\{\phi_{Jk}\right\}_{k\in K}$ being $K$ a suitable subset of $\mathbb Z$ such that $\cup_{k\in K} supp(\phi_{Jk})$ covers $D$.

As a consequence there exist a family of coefficients $\{a_k\}_{k\in K}$ such that $$f_J = \sum_{k\in K} a_k\phi_{J k}$$ and they have the expression $$a_{k} = \langle f,\phi_{Jk}\rangle = \int_{\mathbb R}f(x)\phi_{Jk}(x) dx \text{ .}$$

My question is:

assume I have a sequence of samples $\{(f_i,x_i)\}_{i\in \mathbb N}$ in $\mathbb R ^2$ such that $f_i = f(x_i)$ and assume that no regularity is given on how the samples are taken (non uniform sampling), how can I compute an estimate of $a_{k}$?

Which conditions are sufficient and necessary for reconstruction?