Approximation of certain smoothness classes of periodic functions of several variables by polynomials with regard to the tensor Haar system

S. A. Stasyuk

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 4, pp. 251-260

Résumé

We investigate two kinds of approximation (linear and nonlinear) of certain smoothness classes (close to Nikol'skii-Besov type classes of mixed smoothness) of periodic functions of several variables by polynomials with regard to the tensor Haar system. For the functions of these classes we obtain upper order estimates for the approximation by step-hyperbolic Fourier-Haar sums and exact order estimates for the best $m$-term approximation with regard to the tensor Haar system.

Détail

Keywords: approximation of functions of several variables, tensor Haar system, best $m$-term approximation.

Bibliographie

[1] Kashin B.S., Saakyan A.A., Ortogonalnye ryady, 2-e izd., dop., AFTs, M., 1999, 550 pp. | MR

[2] Temlyakov V.N., “The best $m$-term approximation and greedy algorithms”, Adv. Comput. Math., 8:3 (1998), 249–265 | DOI | MR | Zbl

[3] Temlyakov V.N., “Some inequalities for multivariate Haar polynomials”, East J. Appr., 1:1 (1995), 61–72 | MR | Zbl

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

[5] Temlyakov V.N., Approximation of periodic functions, Comput. Math. and Anal. Ser., Nova Science Publishers, Inc., Commack, New York, 1993, 419 pp. | MR | Zbl

[6] Romanyuk A.S., Approksimativnye kharakteristiki klassov periodicheskikh funktsii mnogikh peremennykh, Pratsi Institutu matematiki NAN Ukraïni, 93, In-t matematiki NAN Ukraïni, Kiïv, 2012, 352 pp.

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

[8] Stechkin S.B., “Ob absolyutnoi skhodimosti ortogonalnykh ryadov”, Dokl. AN SSSR, 102:2 (1955), 37–40 | Zbl

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

[10] Kashin B.S., “Ob approksimatsionnykh svoistvakh polnykh ortonormirovannykh sistem”, Tr. MIAN SSSR, 172 (1985), 187–191 | MR | Zbl

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

[12] 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

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

[14] 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

[15] Temlyakov V.N., Constructive sparse trigonometric approximation for functions with small mixed smoothness, 2015, arXiv: math.1503.00282v1

[16] Temlyakov V.N., “Non-linear $m$-term approximation with regard to the multivariate Haar system”, East J. Appr., 4:1 (1998), 87–106 | MR | Zbl

[17] Andrianov A.V., Temlyakov V.N., “Best $m$-term approximation of functions from classes $MW_{r_{q,\alpha}}$”, Approx. Theory IX, v. 1, 1998, 7–14 | MR | Zbl

[18] Andrianov A.V., “Priblizhenie funktsii iz klassov $MH^r_p$ polinomami Khaara”, Mat. zametki, 66:3 (1999), 323–335 | DOI | MR | Zbl

[19] Andrianov A.V., “Nonlinear Haar approximation of functions with bounded mixed derivative”, Wavelet analysis and multiresolution methods (Urbana-Champaign, IL, 1999), Lecture Notes in Pure and Appl. Math., 212, New York, 2000, 27–47 | MR | Zbl

[20] Osvald P., “Ob $N$-chlennykh priblizheniyakh po sisteme Khaara v $H^s$-normakh”, Metricheskaya teoriya funktsii i smezhnye voprosy analiza, ed. pod red. S. M. Nikolskogo, AFTs, M., 1999, 137–163 | MR

[21] Stasyuk S.A., “Priblizhenie funktsii mnogikh peremennykh klassov $H_p^{\Omega}$ polinomami po sisteme Khaara”, Anal. Math, 35:4 (2009), 257–271 | DOI | MR | Zbl

[22] Stasyuk S.A., “Nailuchshee $m$-chlennoe priblizhenie klassov $B^{\,r}_{\infty,{\,\theta}}$ funktsii mnogikh peremennykh polinomami po sisteme Khaara”, Ukr. mat. zhurn., 63:4 (2011), 549–555 | MR | Zbl

[23] Stasyuk S.A., “Priblizheniya klassov $\mathbf{MB} ^r_{p,\theta }$ periodicheskikh funktsii mnogikh peremennykh polinomami po sisteme Khaara”, Ukr. mat. visn., 12:1 (2015), 97–109

[24] Golubov B.I., “Nailuchshie priblizheniya funktsii v metrike $L_p$ polinomami Khaara i Uolsha”, Mat. sb., 87(129):2 (1972), 254–274 | MR | Zbl

[25] Romanyuk V.S., “Konstruktivnaya kharakteristika klassov Geldera i $m$-chlennye priblizheniya po kratnomu bazisu Khaara”, Ukr. mat. zhurn., 66:3 (2014), 349–360 | MR | Zbl