A Note on Approximation by Trigonometric Polynomials

Abstract

Let \( E=\underset{k=1}{\overset{n}{\cup }}\left[{a}_k,{b}_k\right]\subset \mathbb{R} \); if n > 1, then we assume that the segments [a_k, b_k] are pairwise disjoint. Assume that the following property holds: E ∩ (E + 2πν) = ∅, ν ∈ ℤ, ν ≠ 0. Denote by H^ω + r(E) the space of functions f defined on E such that |f^(r)(x₂) − f^(r)(x₁)| ≤ c_fω(|x₂ − x₁|), x₁, x₂ ∈ E, f⁽⁰⁾ ≡ f. Assume that a modulus of continuity ω satisfies the condition

\( \underset{0}{\overset{x}{\int }}\frac{\omega (t)}{t} dt+x\underset{x}{\overset{\infty }{\int }}\frac{\omega (t)}{t^2} dt\le c\omega (x). \)

We find a constructive description of the space H^ω + r(E) in terms of the rate of nonuniform approximation of a function f ∈ H^ω + r(E) by trigonometric polynomials if E and ω satisfy the above conditions.