Approximation of (ψ, β)-differentiable functions by Weierstrass integrals

Abstract

We obtain asymptotic equalities for upper bounds of approximations of functions from the classes C ψ β,∞ and L ψ β,1 by Weierstrass integrals. Main Definitions and Auxiliary StatementsLet C be the space of 2π-periodic continuous functions with normspace of 2π-periodic, measurable, essentially bounded functions with norm f ∞ = ess sup t f (t) , and let L be the space of 2π-periodic functions summable on a period with normIn [1], classes of periodic functions were introduced as follows:be the Fourier series of f. Further, let ψ(k) be an arbitrary fixed function of natural argument and let β be a fixed real number. If the series ∞ k=1

Approximation of (ψ, β)-differentiable functions defined on the real axis by Weierstrass operators

“…The results of the present paper are closely related to the results obtained by Kharkevych and Kal'chuk in [11], where the Nikol'skii-Kolmogorov problem was solved for Weierstrass integrals on the classes C ψ β,∞ and L ψ β,1 .…”

“…Taking into account equality (34) from [11], we conclude that the following estimate holds for the second integral in (10):…”

“…According to inequality (79) in [11], the following estimate holds for the second integral on the right-hand side of inequality (56):…”