The Direct Theorem of the Theory of Approximation of Periodic Functions with Monotone Fourier Coefficients in Different Metrics

Abstract

We study the problem of order optimality of an upper bound for the best approximation in $$L_q(\mathbb{T})$$ in terms of the lth-order modulus of smoothness (the modulus of continuity for l = 1): $${E_{n - 1}}{(f)_q} \leq C(l,p,q){(\sum\limits_{v = n + 1}^\infty {{v^{q\sigma - 1}}\omega _l^q{{(f;\pi /v)}_p}} )^{1/q}},n \in \mathbb{N},$$ on the class $$M_p(\mathbb{T})$$ of all functions $$f \in L_p(\mathbb{T})$$ whose Fourier coefficients satisfy the conditions a0(f) = 0, an(f) ↓ 0, and bn(f) ↓ 0 (n ↑ ∞), where $$l \in \mathbb{N} $$ , 1 σ = 1/p − 1/q, and $$\mathbb{T} = (-\pi, \pi]$$ . 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 $$f \in M_p(\mathbb{T})$$ belongs to $$L_q(\mathbb{T})$$ , where 1 < p < q < ∞, if and only if $$\sum\nolimits_{n - 1}^\infty {{n^{q\sigma - 1}}\omega _l^q{{(f;\pi /n)}_p}} < \infty $$ and the following order (a) $${E_{n - 1}}{(f)_q} + {n^\sigma }{\omega _l}(f;\pi /n)\asymp{(\sum\limits_{v = n + 1}^\infty {{v^{q\sigma - 1}}\omega _l^q{{(f;\pi /v)}_p}} )^{1/q}},n \in \mathbb{N},$$ (b) $${n^{ - (l - \sigma )}}{(\sum\limits_{v = 1}^n {{v^{p(l - \sigma ) - 1}}E_{V - 1}^P} )^{1/p}}\asymp{(\sum\limits_{v = n + 1}^\infty {{v^{q\sigma - 1}}\omega _l^q{{(f;\pi /v)}_p}} )^{1/q}},n \in N,$$ . In the lower bound in equality (a), the second term nσωl(f; π/n)p generally cannot be omitted. However, if the sequence {ωl(f; π/n)p} n=1 ∞ or the sequence {En−1(f)p} =1 ∞ satisfies Bari’s (Bl(p))-condition, which is equivalent to Stechkin’s (Sl)-condition, then $$E_{n-1}(f)_q\asymp(\sum_{\nu=n+1}^\infty \nu^{q\sigma-1}\omega_l^q (f; \pi/\nu)_{p})^{1/q}, n\in \mathbb{N}.$$ The upper bound in equality (b), which holds for every function $$f \in L_p(\mathbb{T})$$ if the series converges, is a strengthened version of the direct theorem. The order equality (b) shows that the strengthened version is order-optimal on the whole class $$M_p(\mathbb{T})$$ .