A sharp constant in the Jackson-Stechkin inequality in the space L ² on the period

Abstract

In the space L ² of real-valued measurable 2π-periodic functions that are square summable on the period [0, 2π], the Jackson-Stechkin inequality

$$ E_n (f) \leqslant \mathcal{K}_n (\delta ,\omega )\omega (\delta ,f), f \in L^2 $$

, is considered, where E _n (f) is the value of the best approximation of the function f by trigonometric polynomials of order at most n and ω(δ, f) is the modulus of continuity of the function f in L ² of order 1 or 2. The value

$$ \mathcal{K}_n (\delta ,\omega ) = \sup \left\{ {\frac{{E_n (f)}} {{\omega (\delta ,f)}}:f \in L^2 } \right\} $$

is found at the points δ = 2π/m (where m ∈ ℕ) for m ≥ 3n ² + 2 and ω = ω ₁ as well as for m ≥ 11n ⁴/3 − 1 and ω = ω ₂.