Nonlinear trigonometric approximations of multivariate function classes

D. B. Bazarkhanov

Informatics and Automation, Function spaces, approximation theory, and related problems of mathematical analysis, Tome 293 (2016), pp. 8-42.

Résumé

Order-sharp estimates are established for the best $N$-term approximations of functions from Nikol'skii–Besov type classes $\mathrm B^{sm}_{pq}(\mathbb T^k)$ with respect to the multiple trigonometric system $\mathfrak T^{(k)}$ in the metric of $L_r(\mathbb T^k)$ for a number of relations between the parameters $s,p,q,r$, and $m$ ($s=(s_1,\dots,s_n)\in\mathbb R^n_+$, $1\leq p,q,r\leq\infty$, $m=(m_1,\dots,m_n)\in\mathbb N^n$, $k=m_1+\dots+m_n$). Constructive methods of nonlinear trigonometric approximation –variants of the so-called greedy algorithms – are used in the proofs of upper estimates.

Bibliographie

[1] Bazarkhanov D.B., “Otsenki nekotorykh approksimativnykh kharakteristik prostranstv Nikolskogo–Besova obobschennoi smeshannoi gladkosti”, DAN, 426:1 (2009), 11–14 | MR | Zbl

[2] Bazarkhanov D.B., “Priblizhenie vspleskami i poperechniki Fure klassov periodicheskikh funktsii mnogikh peremennykh. I”, Tr. MIAN, 269 (2010), 8–30 | MR | Zbl

[3] Bazarkhanov D.B., “Priblizhenie vspleskami i poperechniki Fure klassov periodicheskikh funktsii mnogikh peremennykh. II”, Anal. math., 38:4 (2012), 249–289 | DOI | MR | Zbl

[4] Bazarkhanov D.B., “Nelineinye priblizheniya klassov periodicheskikh funktsii mnogikh peremennykh”, Tr. MIAN, 284 (2014), 8–37 | MR | Zbl

[5] Belinskii E.S., “Priblizhenie “plavayuschei” sistemoi eksponent na klassakh periodicheskikh gladkikh funktsii”, Tr. MIAN., 180 (1987), 46–47

[6] Belinskii E.S., “Priblizhenie “plavayuschei” sistemoi eksponent na klassakh gladkikh periodicheskikh funktsii”, Mat. sb., 132:1 (1987), 20–27 | MR

[7] Belinskii E.S., “Decomposition theorems and approximation by a “floating” system of exponentials”, Trans. Amer. Math. Soc., 350:1 (1998), 43–53 | DOI | MR | Zbl

[8] DeVore R.A., “Nonlinear approximation”, Acta numer., 7 (1998), 51–150 | DOI | MR | Zbl

[9] DeVore R.A., Temlyakov V.N., “Nonlinear approximation by trigonometric sums”, J. Fourier Anal. Appl., 2:1 (1995), 29–48 | DOI | MR | Zbl

[10] Dilworth S.J., Kutzarova D., Temlyakov V.N., “Convergence of some greedy algorithms in Banach spaces”, J. Fourier Anal. Appl., 8:5 (2002), 489–505 | DOI | MR | Zbl

[11] Ismagilov R.S., “Poperechniki mnozhestv v lineinykh normirovannykh prostranstvakh i priblizhenie funktsii trigonometricheskimi mnogochlenami”, UMN., 29:3 (1974), 161–178 | MR | Zbl

[12] Kashin B.S., Saakyan A.A., Ortogonalnye ryady, 2-e izd., Izd-vo AFTs, M., 1999 | MR

[13] Kashin B.S., Temlyakov V.N., “O nailuchshikh $m$-chlennykh priblizheniyakh i entropii mnozhestv v prostranstve $L^1$”, Mat. zametki., 56:5 (1994), 57–86 | MR | Zbl

[14] Korneichuk N.P., Ekstremalnye zadachi teorii priblizheniya, Nauka, M., 1976 | MR

[15] Meyer Y., Wavelets and operators, Cambridge Stud. Adv. Math., 37, Cambridge Univ. Press, Cambridge, 1992 | MR | Zbl

[16] Nikolskii S.M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya., 2-e izd., Nauka, M., 1977 | MR

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

[18] Romanyuk A.S., “Nailuchshie trigonometricheskie priblizheniya klassov periodicheskikh funktsii mnogikh peremennykh v ravnomernoi metrike”, Mat. zametki., 82:2 (2007), 247–261 | DOI | MR | Zbl

[19] Schmeisser H.-J., Triebel H., Topics in Fourier analysis and function spaces, J. Wiley Sons, Chichester, 1987 | MR | Zbl

[20] Stein I., Veis G., Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974

[21] Stechkin S.B., “Ob absolyutnoi skhodimosti ortogonalnykh ryadov”, DAN SSSR., 102:1 (1955), 37–40 | Zbl

[22] Temlyakov V.N., “O priblizhenii periodicheskikh funktsii mnogikh peremennykh”, DAN SSSR., 279:2 (1984), 301–305 | MR | Zbl

[23] Temlyakov V.N., Priblizhenie funktsii s ogranichennoi smeshannoi proizvodnoi, Tr. MIAN, 178, Nauka, M., 1986 | MR | Zbl

[24] Temlyakov V.N., “Nelineinye poperechniki po Kolmogorovu”, Mat. zametki., 63:6 (1998), 891–902 | DOI | MR | Zbl

[25] Temlyakov V.N., “Nonlinear methods of approximation”, Found. Comput. Math., 3:1 (2003), 33–107 | DOI | MR | Zbl

[26] Temlyakov V.N., “Greedy-type approximation in Banach spaces and applications”, Constr. Approx., 21:2 (2005), 257–292 | DOI | MR | Zbl

[27] Temlyakov V.N., “Greedy approximation”, Acta numer., 17 (2008), 235–409 | DOI | MR | Zbl

[28] Temlyakov V.N., Greedy approximation, Cambridge Monogr. Appl. Comput. Math., 20, Cambridge Univ. Press, Cambridge, 2011 | MR | Zbl

[29] Temlyakov V.N., “Sparse approximation and recovery by greedy algorithms in Banach spaces”, Forum Math. Sigma, 2 (2014), e12 | DOI | MR | Zbl

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

[31] Temlyakov V.N., Constructive sparse trigonometric approximation for functions with small mixed smoothness, E-print, 2015, arXiv: 1503.00282 [math.NA]

[32] Temlyakov V., Sparse approximation with bases, Birkhäuser, Basel, 2015 | MR | Zbl

[33] Tribel Kh., Teoriya funktsionalnykh prostranstv, Mir, M., 1986 | MR