# S.A. Stasyuk. Sparse trigonometric approximation of Besov classes of functions with small mixed smoothness ... P. 244-252.

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.

The paper was received by the Editorial Office on July 26, 2017.

Sergej Andreevich Stasyuk, Cand. Sci. (Phys.-Math.), Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, 01601, Ukraine, e-mail: stasyuk@imath.kiev.ua

REFERENCES

1.   Romanyuk A.S. Best M-term trigonometric approximations of Besov classes of periodic functions of several variables. Izv. Math., 2003, vol. 67, no. 2, pp. 265–302.
doi: 10.1070/IM2003v067n02ABEH000427 .

2.   Belinskii E.S. Approximation by a “floating” system of exponentials on classes of periodic functions with a bounded mixed derivative. Studies in the theory of functions of several real variables. Matematika. Yaroslavl’: Yaroslav. Gos. Univ. Publ., 1988, pp. 16–33 (in Russian).

3.   Temlyakov V.N. Constructive sparse trigonometric approximation and other problems for functions with mixed smoothness, Sb. Math., 2015, vol. 206, no. 11, pp. 1628–1656.
doi: 10.1070/SM2015v206n11ABEH004507 .

4.   Bazarkhanov D.B. Nonlinear trigonometric approximations of multivariate function classes. Proc. Steklov Inst. Math., 2016, vol. 293, pp. 2–36. doi: 10.1134/S0081543816040027 .

5.   Temlyakov V.N. Constructive sparse trigonometric approximation for functions with small mixed smoothness. Constr. Approx., 2017, vol. 45, no. 3, pp. 467–495. doi: 10.1007/s00365-016-9345-3 .

6.   Stasyuk S.A. Constructive sparse trigonometric approximations of functions with small mixed smoothness. Trudy Inst. Mat. i Mekh. UrO RAN, 2016, vol. 22, no. 4, pp. 247–253 (in Russian).
doi: 10.21538/0134-4889-2016-22-4-247-253 .

7.   Stasyuk S.A. Best m-term trigonometric approximation for periodic functions with small mixed smoothness from Nikolskii–Besov type classes. Ukrain. Mat. Zh., 2016, vol. 68, no. 7, pp. 983–1003 (inUkrainian).

8.   Romanyuk A.S. Approksimativnye kharakteristiki klassov periodichestikh funktsii mnogikh peremennykh [Approximation characteristics of classes of periodic functions of several variables]. Pratsi InstytutuMatematyky Natsional’noЈ AkademiЈ Nauk UkraЈny. Matematyka ta ЈЈ Zastosuvannya 93. KyЈv: Instytut Matematyky NAN UkraЈny, 2012, 352 p. ISBN: 978-966-02-6692-6 .

9.   D. D~ung, Temlyakov V.N., Ullrich T. Hyperbolic cross approximation, arXiv: math.1601.03978v2 [math.NA] 2 Dec 2016, pp. 1–182. Available at: https://arxiv.org/abs/1601.03978v2 .

10.   Temlyakov V.N. Approximation of functions with bounded mixed derivative. Proc. Steklov Inst. Math., 1989, vol. 178, no. 1, 121 p.

11.   Lizorkin P.I., Nikol’skii S.M. Functional spaces of mixed smoothness from decompositional point of view. Proc. Steklov Inst. Math., 1990, vol. 187, pp. 163–184.

12.   Stasyuk S.A. Approximation of certain smoothness classes of periodic functions of several variables by polynomials with regard to the tensor Haar system. Trudy Inst. Mat. i Mekh. UrO RAN, 2015, vol. 21,no. 4, pp. 251–260 (in Russian).

13.   Stasyuk S.A. Best m-term trigonometric approximation of periodic functions of several variables from Nikol’skii–Besov classes for small smoothness. J. Approx. Theory., 2014, vol. 177, pp. 1–16. doi: 10.1016/j.jat.2013.09.006 .

14.   Kashin B.S., Saakyan A.A. Orthogonal series. Providence, RI: American Mathematical Society (AMS), 1989, Ser. Trans. Math. Monogr., vol. 75, 451 p. ISBN: 0821845276 . Original Russian textpublished in Ortogonal’nye ryady, Moscow, Nauka Publ., 1984, 496 p.