Approximate recovery of functions and Besov spaces of dominating mixed smoothness. (English) Zbl 1082.42004

Bojanov, B.D., Constructive theory of functions. Proceedings of the international conference, Varna, Bulgaria, June 19–23, 2002. Sofia: DARBA (ISBN 954-90126-6-2/hbk). 404-411 (2003).
In this interesting paper, the author investigates rates of optimal recovery by generalized sampling operators. Let \(\mathbb T^2 = [0,2\pi)\times [0,2\pi)\) and \(f\) be a real valued function defined on \(\mathbb T^2\). Let \(\xi^j\in \mathbb T^2\), \(1\leq j \leq m\), and \(\{\psi_j\}^m_{j=1}\) be functions defined on \(\mathbb T^2\). Form the sampling operator \[ \Psi_m (f,\xi)=\sum^m_{j=1}f (\xi^j)\psi_j. \] Let \(F\) be a class of \(2\pi\)-periodic continuous functions on \(\mathbb T^2\), and form the error over \(F\) in \(L_p\), \[ \Psi_m(f,\xi)=\sup_{f\in F}\| \Psi_m(f,\xi)-f\|_{L_p(\mathbb T^2)}. \] The quantity \[ \rho_m(F)=\inf_{\{\psi_j\}}\inf_{\{\xi^j\}}\Psi_m(F,\xi) \] measures the optimal approximate error in recovery of functions from \(F\).
The author’s main result is a constructive proof of the (new) estimate \[ \rho_m(F)\leq Cm^{-r} (\log m)^{r+1-1/q}, \] where \(C\) is independent of \(m\). Here \(1 < p< \infty\), \(1 < q <\infty\), \(r > 1/p\), and \(F\) is the unit ball of a Besov space \(S^r_{p,q}B(\mathbb T^2)\). The latter is defined in terms of the weighted sections of the Fourier series expansions of functions. The \(\{\Psi_j\}\) are tensor products of shifted one dimensional Dirichlet kernels, and so are trigonometric polynomials.
For the entire collection see [Zbl 1012.00038].


42A10 Trigonometric approximation
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems