Multiplicator type operators and approximation of periodic functions of one variable by trigonometric polynomials

The norms of the images of multiplier type operators generated by an arbitrary generator are estimated in terms of the best approximations of univariate periodic functions by trigonometric polynomials in the $L_p$-spaces, $1\le p\le+\infty$. As corollaries, estimates for the quality of approximation by Fourier means, an inverse theorem of approximation theory, comparison theorems, an analogue of the Marchaud inequality for generalized moduli of smoothness defined by a periodic generator, as well as some constructive sufficient conditions for generalized smoothness and Bernstein type inequalities for generalized derivatives of trigonometric polynomials are obtained.
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