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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.