On Orders of Approximation Functions of Generalized Mixed Smoothness in Lorentz Spaces

Gabdolla  Akishev

doi:10.5644/SJM.15.01.08

Authors

Gabdolla Akishev

DOI:

https://doi.org/10.5644/SJM.15.01.08

Keywords:

Lorentz space, approximations of functions, Nikol’skii-Besov class, hyperbolic cross

Abstract

We consider the Lorentz space with mixed norm of periodic functions of many variables and Nikol’skii-Besov type classes of the generalized mixed smoothness. We have obtained estimates of the best approximation by trigonometric polynomials with the harmonics from the hyperbolic crosses of functions fromNikol’skii-Besov’s type classes of the generalized mixed smoothness in the Lorentz space with the mixed norm.

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References

G. Akishev, Approximation of function classes in Lorentz spaces with mixed norm, East Journal of Approx., 14 (2) (2008), 193--214.

G. Akishev, Approximation of function classes in spaces with mixed norms, Mat. Sb., 197 (8) (2006) , 17--40.

G. Akishev, On degree of approximation of function classes in the Lebesgue space with the anisotropic norm, Uchenie Zapiskii Kazan Univer., 148 (2) (2006), 5--17.

G. Akishev, On order of approximation of generalized Nikol'skii-Besov class in Lorentz space, Centre de Recerca Mat., Barcelona. Preprint, 1222 (2016), 1-22.

G. Akishev, Inequalities of distinct metric of polynomials in Lorentz spaces with mixed norm, First Erjanov reading, Pavlodar state universuty, (2004) 211-215.

T.I. Amanov, Representation and embedding theorems for the functional spaces $S_{p, theta}^{r}B(R^{n})$ and $S_{p, theta}^{r}*B$, Tr. Mat. Inst. Steklov., 77 (1965), 143--167.

K.I. Babenko, Approximation by trigonometric polynomials in a certain class of periodic functions of several variables, Dokl. Akad. Nauk SSSR., 132 (5) (1960), 982-985 (English transl. in Soviet math. Dokl. 1 (1960), 672--675 ).

Sh.A. Balgimbaeva, T.I. Smirnov, Bounds for Fourier widths of classes of periodic functions with a mixed modulus of smoothness, Tr. Inst. Mat. i Mekh. UrO RAN., 21 (4) (2015), 78--94.

N.K. Bary , S.B. Stechkin, The best approximations and differential properties of two conjugate functions, Tr. Moskov. Mat. Obshest. 5 (1956), 483--522.

D.B. Bazarkhanov, Approximation with wavelets and Fourier widths of classes of periodic functions of several variables I, Tr. Mat. Inst. Steklov., 269 (2010), 8--30.

K.A. Bekmaganbetov, On orders of approximations of the Besov class in the metric of the anisotropic Lorentz space, Ufimskii Mat. Zh, 1 (2) (2009), 9--16.

K.A. Bekmaganbetov, Orders of approximations of Besov classes in the metric of anisotropic Lorentz spaces, Meth. Fourier Anal. and Approx. Th., (2016), 149--158.

A.P. Blozinski, Multivariate rearrangements and Banach function spaces with mixed norms, Trans. Amer. Math. Soc., 263 (1) (1981), 146-167.

Ya. S. Bugrov, Approximation of function classes with the dominant mixed derivative, Mat. Sb., 64 (3) (1964), 410--418.

R.A. DeVore, S.V. Konyagin, V.N. Temlyakov, Hyperbolic wavelet approximation, Construc. Approx., 14 (1998), 1--26.

Dinh Dung, Approximation by trigonometric polynomials of functions of several variables on the torus, Mat. Sb., 131 (2) (1986), 251--271.

Dinh Dung , Vladimir N. Temlyakov, Tino Ullrich, Hyperbolic cross approximation, E-Preprint, arXiv:1601.03978v1[math.NA], (2016), 1--154.

E.M. Galeev, Approximation of some classes of periodic functions of several variables by Fourier sums in the metric of $tilde{L}_{p}$, Uspekhi Mat. Nauk., 32 (4) (1977), 251--252.

E.M. Galeev, Approximation by of Fourier sums of classes of functions with bounded derivatives, Mat. Zam., 23 (2) (1978), 197--212.

P.I. Lizorkin, S.M. Nikol'skii, Spaces of functions of mixed smoothness from the decomposition point of view, Proc. Stekov Inst. Math. , 187 (1989), 143--161.

B.S. Mityagin, Approximation of functions in the spaces $L_{p}$ and $C$ on the torus, Mat. Sb., 58 (4) (1962), 397--414.

S. M. Nikolskii, Functions with the dominant mixed derivative which satisfy multi Holder's condition, Sib. Mat. Zh., 4 (6) (1963), 1342--1364.

S. M. Nikolskii, Approximation of functions of several variables and embedding theorems, Nauka, Moscow, 1977.

N.S. Nikolskaya, The approximation differentiable functions of several variables by Fourier sums in the $L_{p}$-metric, Sib. Mat. Zh., 15 (2) (1974), 395--412.

N. N. Pustovoitov, Approximation of multidimensional functions with a given majorant of mixed moduli of continuity, Mat. Zam., 65 (1) (1999), 107--117.

N. N. Pustovoitov, On best approximations by analogs of „proper“ and „improper“ hyperbolic crosses, Math. Notes., 93 (3) (2013), 487--496.

A.S. Romanyuk, Approximation of the Besov classes of periodic functions of several variables in the space $L_{q}$, Ukrain . Mat. Zh., 43 (1) (1991), 1297--1306.

A.S. Romanyuk, On approximation of classes of periodic functions of several variables, Ukrain . Mat. Zh., 44 (5) (1992), 596--606.

A.S. Romanyuk, On estimates of approximation characteristics of the Besov classes of periodic functions of many variables, Ukrain . Mat. Zh., 9 (9) (1997), 1409--1422.

L.A. Sherstneva, On the properties of best Lorentz approximations and certain embedding theorems, Izv. Vyssh. Uchebn. Zav. Mat., 10 (1987), 48--58.

W. Sickel, T. Ullrich, Tensor products of Sobolev -- Besov spaces and applications to approximation from the hyperbolic cross, J. Approx. Th., 161(2), 748-- 786 (2009)

M.B. Sikhov, Approximation of functions of several variables with a given majorant in the Besov space, Mat. Zh., 2 (4) (2002), 95--100.

H.-J. Schmeisser, W. Sickel, Spaces of functions of mixed smoothness and approximation from hyperbolic crosses, J. Approx. Th., 128 (2) (2004), 115--150.

S. A. Stasyuk, The best approximations of periodic functions of several variables in the classes $B_{p, theta}^{Omega}$, Mat. Zam., 87 (1) (2010), 108--121.

S. A. Stasyuk, Approximation by Fourier sums and Kolmogorov widths for classes $MB_{p, theta}^{Omega}$ of periodic functions of several variables, Tr. Inst. Mat. i Mekh. UrO RAN., 20 (1) (2014), 247--257.

S. A. Stasyuk, S.Ya. Yanchenko, Approximation of functions from Nikol'skii-Besov type classes of generalized mixed smoothness, Anal. Math., 41 (4) (2015), 311--334.

Sun Yongsheng, Wang Heping, Representation and approximation of multivariate periodic functions with bounded mixed moduli of smoothness, Tr. Mat. Inst. Steklov., 219 (1997) , 356--377.

S.A. Telyakovskii, Some estimates for trigonometric series with quasiconvex coefficients, Mat. Sb., 63 (3) (1964), 426--444.

V.N. Temlyakov, Approximation of functions with bounded mixed derivative, Tr. Mat. Inst. Steklov., 178 (1986), 3--112.

V.N. Temlyakov, Approximation of periodic functions of several variables with bounded mixed differences, Mat. Sb., 113 (1) (1980), 65--80.