Moduli of smoothness and best approximation of functions with singularities

Error estimates for approximation of functions λ,α,0(x) = λ,α,1(x) + i λ,α,2(x) = xλ exp(iAx−a), λ > 0, α > 0, A ε R are given. Let E(f, B, Lp(Ω)) denote the error of approximation of f by elements from B in the Lp-metric. Then, it is shown that for polynomial approximation E( λ,α,i, Pn, Lp (−a, a)) n−(λ+1/p)/(1+α) holds true for 1 ≤ p ≤ ∞, 0 ≤ i ≤ 2. The similar estimates are also valid for the errors of approximation by entire functions of exponential type, trigonometric polynomials, and periodic splines. The proofs are based on exact estimates of the moduli of smoothness of λ,α,i and a general Stechkin-type theorem.