Geodesic
Approximation of periodic functions in the classes~$H_q^\Omega$ by linear methods
Sbornik. Mathematics, Tome 203 (2012) no. 1, pp. 88-110

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The following result is proved: if approximations in the norm of $L_\infty$ (of $H_1$) of functions in the classes $H_\infty^\Omega$ (in $H_1^\Omega$, respectively) by some linear operators have the same order of magnitude as the best approximations, then the set of norms of these operators is unbounded. Also Bernstein's and the Jackson-Nikol'skiǐ inequalities are proved for trigonometric polynomials with spectra in the sets $Q(N)$ (in $\varGamma(N,\Omega)$). Bibliography: 15 titles.
Keywords: modulus of continuity, linear approximations, Bernstein's inequalities, Nikol'skiǐ's inequalities, functions of several variables. 
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     title = {Approximation of periodic functions in the classes~$H_q^\Omega$ by linear methods},
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N. N. Pustovoitov. Approximation of periodic functions in the classes~$H_q^\Omega$ by linear methods. Sbornik. Mathematics, Tome 203 (2012) no. 1, pp. 88-110. http://geodesic.mathdoc.fr/item/SM_2012_203_1_a4/