## 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].

### MSC:

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

### Keywords:

optimal recovery; Besov spaces; sampling operators