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 Trudy Inst. Mat. i Mekh. UrO RAN, 2017, Volume 23, Number 3, Pages 244–252 (Mi timm1454)

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

S. A. Stasyuk

Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev

Abstract: 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.

Keywords: nonlinear approximation, sparse trigonometric approximation, mixed smoothness, Besov classes, exact order bounds.

DOI: https://doi.org/10.21538/0134-4889-2017-23-3-244-252

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Bibliographic databases:

UDC: 517.518
MSC: 41À60, 41À65, 42À10, 46Å30, 46Å35

Citation: S. A. Stasyuk, “Sparse trigonometric approximation of Besov classes of functions with small mixed smoothness”, Trudy Inst. Mat. i Mekh. UrO RAN, 23, no. 3, 2017, 244–252

Citation in format AMSBIB
\Bibitem{Sta17} \by S.~A.~Stasyuk \paper Sparse trigonometric approximation of Besov classes of functions with small mixed smoothness \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2017 \vol 23 \issue 3 \pages 244--252 \mathnet{http://mi.mathnet.ru/timm1454} \crossref{https://doi.org/10.21538/0134-4889-2017-23-3-244-252} \elib{https://elibrary.ru/item.asp?id=29938016}