Geodesic
The inverse theorem in various metrics of approximation theory for periodic functions with monotone Fourier coefficients
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 22 (2016) no. 4, pp. 153-162

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We prove the exactness with respect to order of an upper bound for the $k$th-order modulus of smoothness in $L_q({\mathbb T})$ in terms of the elements of a sequence of best approximations in $L_p({\mathbb T})$ on the class of all functions with monotonically decreasing Fourier coefficients, where $1$ and $k\in {\mathbb N}$.
Keywords: modulus of smoothness, best approximation, inverse theorem in various metrics, trigonometric Fourier series with monotone coefficients, order-sharp inequality on a class. 
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     author = {N. A. Il'yasov},
     title = {The inverse theorem in various metrics of approximation theory for periodic functions with monotone {Fourier} coefficients},
     journal = {Trudy Instituta matematiki i mehaniki},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2016_22_4_a14/}
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N. A. Il'yasov. The inverse theorem in various metrics of approximation theory for periodic functions with monotone Fourier coefficients. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 22 (2016) no. 4, pp. 153-162. http://geodesic.mathdoc.fr/item/TIMM_2016_22_4_a14/