Approximation by Fourier Sums in the Classes of Weyl–Nagy Differentiable Functions with High Exponent of Smoothness

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We establish asymptotic estimates for the least upper bounds of approximations in the uniform metric by Fourier sums of order n − 1 in the classes of 2𝜋-periodic Weyl–Nagy differentiable functions $$ {W}_{\beta, p}^r $$ , 1 ≤ p ≤ ∞, β ∈ ℝ, with high exponents of smoothness $$ r\left(r-1\ge \sqrt{n}\right) $$ . We also establish similar estimates for the function classes $$ {W}_{\beta, 1}^r $$ in metrics of the spaces Lp, 1 ≤ p ≤ ∞.