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

S. A. Stasyuk

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 3, pp. 244-252

Résumé

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 $1$ 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 $1$, $r>1/p-1/q$ and $2$, $r>1/2$. 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 $1$, $q>2$, 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 $1$ and $1/p-1/q$. For $1$ Temlyakov found a constructive upper estimate for $\sigma_m(\mathbf{MB}^r_{p,\theta})_q$ in the cases $\theta=\infty$, $1/p-1/q$ and $\theta=p$, $(1/p-1/q)q'$, 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 $1$ and $(1/p-1/q)q'$ when $p\theta\infty$ (or $p\leq\theta\infty$) as well as order-exact (though nonconstructive upper) estimates for the values $\sigma_m(\mathbf{MB}^r_{p,\theta})_q$, $2$, $\theta=1$, $r=1/2$, and $\sigma_m(\mathbf{MH}^r_{p,\theta})_q$, $1$, $1\leq\theta$, $r=1/p$, which complement Romanyuk's results and the author's recent results, respectively.

Détail

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

Bibliographie

[1] Romanyuk A.S., “Nailuchshie M-chlennye trigonometricheskie priblizheniya klassov Besova periodicheskikh funktsii mnogikh peremennykh”, Izv. RAN. Ser. matematicheskaya, 67:2 (2003), 61–100 | DOI | MR | Zbl

[2] Belinskii E.S., “Priblizhenie plavayuschei sistemoi eksponent na klasakh periodicheskikh funktsii s ogranichennoi smeshannoi proizvodnoi”, Issledovaniya po teorii funktsii mnogikhveschestvennykh peremennykh, Izd-vo Yarosl. un-ta, Yaroslavl, 1988, 16–33

[3] Temlyakov V.N., “Konstruktivnye razrezhennye trigonometricheskie priblizheniya i drugie zadachi dlya funktsii smeshannoi gladkosti”, Mat. sb., 206:11 (2015), 131–160 | DOI | MR | Zbl

[4] Bazarkhanov D.B., “Nelineinye trigonometricheskie priblizheniya klassov funktsii mnogikh peremennykh”, Tr. MIAN, 293 (2016), 8–42 | Zbl

[5] Temlyakov V.N., “Constructive sparse trigonometric approximation for functions with small mixed smoothness”, Constr. Approx., 45:3 (2017), 467–495 | DOI | MR | Zbl

[6] Stasyuk S.A., “Konstruktivnye razrezhennye trigonometricheskie priblizheniya dlya klassov funktsii s nebolshoi smeshannoi gladkostyu”, Tr. In-ta matematiki i mekhaniki UrO RAN, 22:4 (2016), 247–253 | MR

[7] Stasyuk S.A., “Naikrasche m-chlenne trigonometrichne nablizhennya periodichnikh funktsii maloï mishanoï gladkosti z klasiv tipu Nikolskogo - Besova”, Ukr. mat. zhurn., 68:7 (2016), 983–1003 | MR

[8] Romanyuk A.S., Approksimativnye kharakteristiki klassov periodicheskikh funktsii mnogikh peremennykh, Pratsi Institutu matematiki NAN Ukraïni, 92, Institut matematiki NAN Ukraïni, Kiïv, 2012, 353 pp.

[9] Dung D., Temlyakov V.N., Ullrich T., Hyperbolic sross approximation, 2016, 182 pp., arXiv: math.1601.03978v2

[10] Temlyakov V.N., “Priblizhenie funktsii s ogranichennoi smeshannoi proizvodnoi”, Tr. MIAN SSSR, 178 (1986), 1–112 | Zbl

[11] Lizorkin P.I., Nikolskii S.M., “Prostranstva funktsii smeshannoi gladkosti s dekompozitsionnoi tochki zreniya”, Tr. MIAN SSSR, 187 (1989), 143–161

[12] Stasyuk S.A., “Priblizhenie nekotorykh gladkostnykh klassov periodicheskikh funktsii mnogikh peremennykh polinomami po tenzornoi sisteme Khaara”, Tr. In-ta matematiki i mekhaniki UrO RAN, 21:4 (2015), 251–260 | MR

[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, 177 (2014), 1–16 | DOI | MR

[14] Kashin B.S., Saakyan A.A., Ortogonalnye ryady, Nauka, M., 1984, 496 pp. | MR