On the informativeness of linear functionals.
The paper deals with the optimal recovery of functions \(f\) from the values \(\tau_i=\ell_i(f)\), \(i=1, \ldots , N\) of some \(N\) functionals. The most general setting includes a set \(F\) to which \(f\) belongs, an algorithm by which \(f\) must be recovered from \(\tau_1, \ldots , \tau_N\), and a normed linear space \(Y\) in which the error of recovery must be measured. For natural numbers \(r,s\) and \(1 \leq p \leq \infty\), let \(F=W_p^r\) be the Sobolev class \(W_p^r\) on the cube \((0,1)^s\). Here for \(2\leq p \leq q \leq \infty\) and \(1 \leq q\leq p \leq 2\) the authors show that \(f \in W_p^r\) can be recovered from some \(N\) of its trigonometric Fourier coefficients with a maximal \(Y=L_q(0,1)^s\) error of order \(N^{-\lambda}\), \( \lambda=r/s-(1/p-1/q)_+\). No other set of \(N\) linear functionals can provide a better order of recovery. Similar results are established for the Besov classes \(B_{p, \theta}^r(0,1)^s\) and for the classes \(SW_2^r\) of functions with dominating mixed derivative. The set of the optimal \(N\) Fourier coefficients is different in each case.
scientific article; zbMATH DE number 2120989