We study the problem of order optimality of an upper bound for the best approximation in Lq(T)\documentclass[12pt]{minimal}
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\begin{document}$$L_q(\mathbb{T})$$\end{document} in terms of the lth-order modulus of smoothness (the modulus of continuity for l = 1): En−1(f)q≤C(l,p,q)(∑v=n+1∞vqσ−1ωlq(f;π/v)p)1/q,n∈N,\documentclass[12pt]{minimal}
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\begin{document}$${E_{n - 1}}{(f)_q} \leq C(l,p,q){(\sum\limits_{v = n + 1}^\infty {{v^{q\sigma - 1}}\omega _l^q{{(f;\pi /v)}_p}} )^{1/q}},n \in \mathbb{N},$$\end{document} on the class Mp(T)\documentclass[12pt]{minimal}
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\begin{document}$$M_p(\mathbb{T})$$\end{document} of all functions f∈Lp(T)\documentclass[12pt]{minimal}
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\begin{document}$$f \in L_p(\mathbb{T})$$\end{document} whose Fourier coefficients satisfy the conditions a0(f) = 0, an(f) ↓ 0, and bn(f) ↓ 0 (n ↑ ∞), where l∈N\documentclass[12pt]{minimal}
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\begin{document}$$l \in \mathbb{N} $$\end{document}, 1 < p < q < ∞, l > σ = 1/p − 1/q, and T=(−π,π]\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb{T} = (-\pi, \pi]$$\end{document}. For l = 1 and p ≥ 1, the bound was first established by P. L.Ul’yanov in the proof of the inequality of different metrics for moduli of continuity; for l > 1 and p ≥ 1, the proof of the bound remains valid in view of the Lp-analog of the Jackson–Stechkin inequality. Below, we formulate the main results of the paper. A function f∈Mp(T)\documentclass[12pt]{minimal}
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\begin{document}$$f \in M_p(\mathbb{T})$$\end{document} belongs to Lq(T)\documentclass[12pt]{minimal}
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\begin{document}$$L_q(\mathbb{T})$$\end{document}, where 1 < p < q < ∞, if and only if ∑n−1∞nqσ−1ωlq(f;π/n)p<∞\documentclass[12pt]{minimal}
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\begin{document}$$\sum\nolimits_{n - 1}^\infty {{n^{q\sigma - 1}}\omega _l^q{{(f;\pi /n)}_p}} < \infty $$\end{document} and the following order (a)En−1(f)q+nσωl(f;π/n)≍(∑v=n+1∞vqσ−1ωlq(f;π/v)p)1/q,n∈N,\documentclass[12pt]{minimal}
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\begin{document}$${E_{n - 1}}{(f)_q} + {n^\sigma }{\omega _l}(f;\pi /n)\asymp{(\sum\limits_{v = n + 1}^\infty {{v^{q\sigma - 1}}\omega _l^q{{(f;\pi /v)}_p}} )^{1/q}},n \in \mathbb{N},$$\end{document}(b)n−(l−σ)(∑v=1nvp(l−σ)−1EV−1P)1/p≍(∑v=n+1∞vqσ−1ωlq(f;π/v)p)1/q,n∈N,\documentclass[12pt]{minimal}
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\begin{document}$${n^{ - (l - \sigma )}}{(\sum\limits_{v = 1}^n {{v^{p(l - \sigma ) - 1}}E_{V - 1}^P} )^{1/p}}\asymp{(\sum\limits_{v = n + 1}^\infty {{v^{q\sigma - 1}}\omega _l^q{{(f;\pi /v)}_p}} )^{1/q}},n \in N,$$\end{document}.
