Approximation of periodic functions in the classes $H_q^\Omega$ by linear methods

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 $\varGamma(N,\Omega)$).
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