Geodesic
Generalized Smoothness and Approximation of Periodic Functions in the Spaces~$L_p$, $1$
Matematičeskie zametki, Tome 106 (2019) no. 3, pp. 436-449

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Norms of images of operators of multiplier type with an arbitrary generator are estimated by using best approximations of periodic functions of one variable by trigonometric polynomials in the scale of the spaces $L_p$, $1$. A Bernstein-type inequality for the generalized derivative of the trigonometric polynomial generated by an arbitrary generator $\psi$, sufficient constructive $\psi$-smoothness conditions, estimates of best approximations of $\psi$-derivatives, estimates of best approximations of $\psi$-smooth functions, and an inverse theorem of approximation theory for the generalized modulus of smoothness generated by an arbitrary periodic generator are obtained as corollaries.
Keywords: best approximation, modulus of smoothness, generalized derivative. 
@article{MZM_2019_106_3_a9,
     author = {K. V. Runovskii},
     title = {Generalized {Smoothness} and {Approximation} of {Periodic} {Functions} in the {Spaces~}$L_p$, $1<p<+\infty$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {436--449},
     publisher = {mathdoc},
     volume = {106},
     number = {3},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2019_106_3_a9/}
}
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K. V. Runovskii. Generalized Smoothness and Approximation of Periodic Functions in the Spaces~$L_p$, $1