Geodesic
Order equalities in different metrics for moduli of smoothness of various orders
Ural mathematical journal, Tome 4 (2018) no. 2, pp. 24-32

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IIn this paper, we obtain order equalities for the $k$th order $L_{q}(T)$-moduli of smoothness $\omega_{k}(f;\delta)_{q}$ in terms of expressions that contain the $l$th order $L_{p}(T)$-moduli of smoothness $\omega_{ l }(f;\delta)_{p}$ on the class of periodic functions $f\in L_{p}(T)$ with monotonically decreasing Fourier coefficients, where $1$ $k,l \in \mathbb{N},$ and $T=(-\pi,\pi].$
Keywords: Inequalities of different metrics for moduli of smoothness, Order equality, Trigonometric Fourier series with monotone coefficients. 
@article{UMJ_2018_4_2_a3,
     author = {Niyazi A. Il'yasov},
     title = {Order equalities in different metrics for moduli of smoothness of various orders},
     journal = {Ural mathematical journal},
     pages = {24--32},
     publisher = {mathdoc},
     volume = {4},
     number = {2},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UMJ_2018_4_2_a3/}
}
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Niyazi A. Il'yasov. Order equalities in different metrics for moduli of smoothness of various orders. Ural mathematical journal, Tome 4 (2018) no. 2, pp. 24-32. http://geodesic.mathdoc.fr/item/UMJ_2018_4_2_a3/