Sparse trigonometric approximation of Besov classes of functions with small mixed smoothness

We consider problems concerned with finding order-exact estimates for a sparse trigonometric approximation, more exactly, for the best $m$-term trigonometric approximation $\sigma_m(F)_q$, where $F$ are the Nikol'skii–Besov classes $\mathbf{MB}^r_{p,\theta}$ of functions with mixed smoothness and classes of functions close to them. Attention is paid to relations between the parameters $p$ and $q$ for $1<p<q<\infty$ and $q>2$. In 2003 Romanyuk found order-exact estimates of $\sigma_m(\mathbf{MB}^r_{p,\theta})_q$ for $1\leq\theta\leq\infty$ (the upper estimates are nonconstructive) in the cases $1<p\leq 2<q<\infty$, $r>1/p-1/q$ and $2<p<q<\infty$, $r>1/2$. Complementing Romanyuk's studies, Temlyakov has recently found constructive upper estimates (provided by a constructive method based on a greedy algorithm) for $\sigma_m(\mathbf{MB}^r_{p,\theta})_q \asymp\sigma_m(\mathbf{MH}^r_{p,\theta})_q$, $1\leq\theta\leq\infty$, in the case of great smoothness, i.e., for $1<p<q<\infty$, $q>2$, and $r>\max\{1/p;1/2\}$; he considered wider classes $\mathbf{MH}^r_{p,\theta}$ ($\mathbf{MB}^r_{p,\theta}\subset\mathbf{MH}^r_{p,\theta}\subset\mathbf{MH}^r_{p}$, $1\leq\theta<\infty$). Less attention was paid to constructive upper estimates of the values $\sigma_m(\mathbf{MB}^r_{p,\theta})_q$ and $\sigma_m(\mathbf{MH}^r_{p,\theta})_q$ in the case of small smothness, i.e., for $1<p\leq 2<q<\infty$ and $1/p-1/q<r\leq 1/p$. For $1<p\leq 2<q<\infty$ Temlyakov found a constructive upper estimate for $\sigma_m(\mathbf{MB}^r_{p,\theta})_q$ in the cases $\theta=\infty$, $1/p-1/q<r<1/p$ and $\theta=p$, $(1/p-1/q)q'<r<1/p$, where $1/q+1/q'=1$, while the author found a constructive upper estimate for $\sigma_m(\mathbf{MH}^r_{p,\theta})_q$ if $r=1/p$ and $p\leq\theta\leq\infty$; it turned out that $\sigma_m(\mathbf{MH}_{p,\theta}^{r})_q\asymp \sigma_m(\mathbf{MB}_{p,\theta}^{r})_q (\log m)^{1/\theta}$ for $r=1/p$ and $p\leq\theta<\infty$. In the present paper, we derive a constructive upper estimate for $\sigma_m(\mathbf{MB}^r_{p,\theta})_q$ (or $\sigma_m(\mathbf{MH}^r_{p,\theta})_q$) for $1<p\leq 2<q<\infty$ and $(1/p-1/q)q'<r<1/p$ when $p<\theta<\infty$ (or $p\leq\theta<\infty$) as well as order-exact (though nonconstructive upper) estimates for the values $\sigma_m(\mathbf{MB}^r_{p,\theta})_q$, $2<p<q<\infty$, $\theta=1$, $r=1/2$, and $\sigma_m(\mathbf{MH}^r_{p,\theta})_q$, $1<p\leq 2<q<\infty$, $1\leq\theta<p$, $r=1/p$, which complement Romanyuk's results and the author's recent results, respectively.