Wavelet approximation and Fourier widths of classes of periodic functions of several variables.~I

D. B. Bazarkhanov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and differential equations, Tome 269 (2010), pp. 8-30.

Résumé

We obtain characterizations (and prove the corresponding equivalence of norms) of function spaces $\mathbf B^{sm}_{pq}(\mathbb I^k)$ and $\mathbf L^{sm}_{pq}(\mathbb I^k)$ of Nikol'skii–Besov and Lizorkin–Triebel types, respectively, in terms of representations of functions in these spaces by Fourier series with respect to a multiple system $\mathcal W^\mathbb I_m$ of Meyer wavelets and in terms of sequences of the Fourier coefficients with respect to this system. We establish order-sharp estimates for the approximation of functions in $B^{sm}_{pq}(\mathbb I^k)$ and $L^{sm}_{pq}(\mathbb I^k)$ by special partial sums of these series in the metric of $L_r(\mathbb I^k)$ for a number of relations between the parameters $s,p,q,r$, and $m$ ($s=(s_1,\dots,s_n)\in\mathbb R^n_+$, $1\leq p,q,r\leq\infty$, $m=(m_1,\dots,m_n)\in\mathbb N^n$, $k=m_1+\dots+m_n$, and $\mathbb I= \mathbb R$ or $\mathbb T$). In the periodic case, we study the Fourier widths of these function classes.

Bibliographie

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

[2] Besov O.V., Ilin V.P., Nikolskii S.M., Integralnye predstavleniya funktsii i teoremy vlozheniya, 2-e izd., Nauka, M., 1996 | MR

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

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

[5] Amanov T.I., Prostranstva differentsiruemykh funktsii s dominiruyuschei smeshannoi proizvodnoi, Nauka, Alma-Ata, 1976 | MR

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

[7] Temlyakov V.N., Approximation of periodic functions, Nova Sci. Publ., Commack (NY), 1993 | MR | Zbl

[8] Schmeisser H.-J., “On spaces of functions and distributions with mixed smoothness properties of Besov–Triebel–Lizorkin type. I”, Math. Nachr., 98 (1980), 233–250 ; “On spaces of functions and distributions with mixed smoothness properties of Besov–Triebel–Lizorkin type. II”, Math. Nachr., 106 (1982), 187–200 | DOI | MR | Zbl | DOI | MR | Zbl

[9] Yamazaki M., “Boundedness of product type pseudodifferential operators on spaces of Besov type”, Math. Nachr., 133 (1987), 297–315 | DOI | MR | Zbl

[10] Schmeisser H.-J., “Recent developments in the theory of function spaces with dominating mixed smoothness”, Nonlinear analysis, function spaces and applications, Proc. Spring School (Prague, May 30–June 6, 2006), v. 8, Czech Acad. Sci., Math. Inst., Prague, 2007, 145–204 | Zbl

[11] Bazarkhanov D.B., “Kharakterizatsii funktsionalnykh prostranstv Nikolskogo–Besova i Lizorkina–Tribelya smeshannoi gladkosti”, Tr. MIAN, 243, 2003, 53–65 | MR | Zbl

[12] Bazarkhanov D.B., “$\varphi $-Transform characterization of the Nikol'skii–Besov and Lizorkin–Triebel function spaces with mixed smoothness”, East J. Approx., 10:1–2 (2004), 119–131 | MR | Zbl

[13] Bazarkhanov D.B., “Ekvivalentnye (kvazi)normirovki nekotorykh funktsionalnykh prostranstv obobschennoi smeshannoi gladkosti”, Tr. MIAN, 248, 2005, 26–39 | MR | Zbl

[14] Bazarkhanov D.B., “Razlichnye predstavleniya i ekvivalentnye normirovki prostranstv Nikolskogo–Besova i Lizorkina–Tribelya obobschennoi smeshannoi gladkosti”, DAN, 402:3 (2005), 298–302 | MR | Zbl

[15] Vybiral J., Function spaces with dominating mixed smoothness, (Diss. math.; V. 436)., 436, Inst. Math., Pol. Acad. Sci., Warszawa, 2006 | MR

[16] Hansen M., Vybiral J., “The Jawerth–Franke embedding of spaces with dominating mixed smoothness”, Georgian Math. J., 16:4 (2009), 667–682 | MR | Zbl

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

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

[19] Wojtaszczyk P., A mathematical introduction to wavelets, Cambridge Univ. Press, Cambridge, 1997 | MR | Zbl

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

[21] Hernández E., Weiss G., A first course on wavelets, Stud. Adv. Math., CRC Press, Boca Raton (FL), 1996 | DOI | MR | Zbl

[22] Triebel H., Theory of function spaces. III, Birkhäuser, Basel, 2006 | MR | Zbl

[23] Fefferman C., Stein E.M., “Some maximal inequalities”, Amer. J. Math., 93 (1971), 107–115 | DOI | MR | Zbl

[24] Stein E.M., Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, Princeton (NJ), 1993 | MR | Zbl

[25] DeVore R.A., Konyagin S.V., Temlyakov V.N., “Hyperbolic wavelet approximation”, Constr. Approx., 14:1 (1998), 1–26 | DOI | MR

[26] Wang H., “Representation and approximation of multivariate functions with mixed smoothness by hyperbolic wavelets”, J. Math. Anal. and Appl., 291 (2004), 698–715 | DOI | MR | Zbl

[27] Lizorkin P.I., “O bazisakh i multiplikatorakh v prostranstvakh $B^r_{p,\theta }(\Pi )$”, Tr. MIAN, 143, 1977, 88–104 | MR | Zbl

[28] OrlovskiĭD.G., “On multipliers in the space $B^r_{p,\theta }$”, Anal. Math., 5 (1979), 207–218 | DOI | MR | Zbl

[29] Schmeisser H.-J., “An unconditional basis in periodic spaces with dominating mixed smoothness properties”, Anal. Math., 13 (1987), 153–168 | DOI | MR | Zbl

[30] Dinh Dũng., “Stability in periodic multi-wavelet decompositions and recovery of functions”, East J. Approx., 11:4 (2005), 447–479 | MR | Zbl

[31] Andrianov A.V., Temlyakov V.N., “O dvukh metodakh rasprostraneniya svoistv sistem funktsii ot odnoi peremennoi na ikh tenzornoe proizvedenie”, Tr. MIAN, 219, 1997, 32–43 | MR | Zbl

[32] Galeev E.M., “Priblizhenie klassov periodicheskikh funktsii neskolkikh peremennykh yadernymi operatorami”, Mat. zametki, 47:3 (1990), 32–41 | MR | Zbl

[33] Galeev E.M., “Poperechniki klassov Besova $B_{p,\theta }^r(\mathbb T^d)$”, Mat. zametki, 69:5 (2001), 656–665 | DOI | MR | Zbl

[34] Romanyuk A.S., “Nailuchshie priblizheniya i poperechniki klassov periodicheskikh funktsii mnogikh peremennykh”, Mat. sb., 199:2 (2008), 93–114 | DOI | MR | Zbl

[35] Schmeisser H.-J., Sickel W., “Spaces of functions of mixed smoothness and approximation from hyperbolic crosses”, J. Approx. Theory, 128 (2004), 115–150 | DOI | MR | Zbl

[36] Ullrich T., “Smolyak's algorithm, sampling on sparse grids and Sobolev spaces of functions of dominating mixed smoothness”, East J. Approx., 14 (2008), 1–38 | MR | Zbl

[37] Bazarkhanov D.B., “Otsenki poperechnikov Fure klassov tipa Nikolskogo–Besova i Lizorkina–Tribelya periodicheskikh funktsii mnogikh peremennykh”, Mat. zametki, 87:2 (2010), 305–308 | DOI | Zbl