Approximations of the classes \(\mathbf{MB}_{p,\theta}^{r}\) of periodic functions of several variables by polynomials according to the Haar system (Q904548)

The spaces \(\mathbf{MB}_{p,\theta}^r\) for \(r>0\), \(1\leq p\leq\infty\), \(1\leq\theta\leq\infty\) are analogs of the Nikol'skii-Besov spaces of functions with mixed smoothness and \(L_p\) \(([0,1]^d)\), \(1\leq p<\infty\), be the space of functions of many variables that are 1-periodic in each variable and summable in power \(p\) on a cube \([0,1]^d\). In this paper, the work has been divided in two parts. In the first part, the best approximation of functions from the classes \(\mathbf{MB}_{p,\theta}^{r}\) by polynomials according to the Haar system with indices from step-hyperbolic crosses, and the approximation by step-hyperbolic Fourier Haar sums in the metric \(L_q\) \(([0,1]^d)\), \(1<q<\infty\) are studied. In the second part, exact-in-order estimates of the best \(m\)-term approximation of the classes \(\mathbf{MB}_{p,\theta}^{r}\) by polynomials according to the Haar system in the space \(L_q([0,1]^d)\), \(1<q<\infty\) are investigated. The article is an extension of several papers (see [\textit{S. B. Stechkin}, Dokl. Akad. Nauk SSSR 102, 37--40 (1955; Zbl 0068.05104); \textit{A. S. Romanyuk}, Math. Notes 71, No. 1, 98--109 (2002; Zbl 1020.42012); translation from Mat. Zametki 71, No. 1, 109--121 (2002); Math. Notes 82, No. 2, 216--228 (2007; Zbl 1219.41006); translation from Mat. Zametki 82, No. 2, 247--261 (2007); \textit{S. A. Stasyuk}, Anal. Math. 35, No. 4, 257--271 (2009; Zbl 1212.41029); Ukr. Math. J. 63, No. 4, 638--645 (2011; Zbl 1242.41027); translation from Ukr. Mat. Zh. 63, No. 4, 549--555 (2011)] etc.).