How best to recover a function from its inaccurately given spectrum?
Suppose that \(W_2^n(R)\) is the Sobolev class of functions on \(R\), \(\sigma >0\), and \(\mathcal{M} _\sigma\) is the set of measured subsets of \(R\) whose measures are not greater than \(2\sigma\). Let \(\delta >0\). Assume that we know the Fourier transform \(Fx(\cdot )\) of a function \(x(\cdot ) \in W_2^n(R)\) on a set \(M_\sigma \in \mathcal{M}_\sigma\) and a function \(y(\cdot )\in L_p(M_\sigma )\) such that \(\| Fx(\cdot )-y(\cdot )\|_{L_p(M_\sigma )} \leq \delta\), \(\;2<p<\infty \). The quantity of the so-called optimal recovery error \(E(k,\sigma ,p,\delta )=\inf _{M_\sigma }\inf _m\sup _{x\in W_2^n(R)} \| x^{(k)}(\cdot )-m(y(\cdot ))(\cdot )\|_{L_2(R)}\) is calculated. Moreover, there are given the optimal set in the first infimum and the optimal recovery method in the second infimum.
scientific article; zbMATH DE number 6138361