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

Abstract

n−(l−σ)(∑v=1nvp(l−σ)−1EV−1P)1/p≍(∑v=n+1∞vqσ−1ωlq(f;π/v)p)1/q,n∈N,\({n^{ - (l - \sigma )}}{(\sum\limits_{v = 1}^n {{v^{p(l - \sigma ) - 1}}E_{V - 1}^P} )^{1/p}}\asymp{(\sum\limits_{v = n + 1}^\infty {{v^{q\sigma - 1}}\omega _l^q{{(f;\pi /v)}_p}} )^{1/q}},n \in N,\)

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In the lower bound in equality (a), the second term nσωl(f; π/n)p generally cannot be omitted. However, if the sequence {ω_l(f; π/n)_p}_n=1^∞ or the sequence {E_n−1(f)_p}_n=1^∞ satisfies Bari’s (B_l^(p))-condition, which is equivalent to Stechkin’s (S_l)-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 \(M_p(\mathbb{T})\).