Approximation of functions of several variables by spherical Riesz means

Abstract

For even N ≥ 2 and δ 2N-3 (for N-2 or 4 we assume that δ > (N-1)/2) we find asymptotic approximations for the quantity

$$E_R^\delta (H_{\rm N}^\omega ) = \mathop {sup}\limits_{f \in H_{\rm N}^\omega } \parallel f(x) - S_R^\omega (x,f)\parallel _ \in (R \to \infty ),$$

, where S ^δ_R (x,f) is the spherical Riesz mean of order δ of the Fourier kernel of the functionf(x), and H ^ω_N is the class of periodic functions of N variables whose moduli of continuity do not exceed a given convex modulus of continuity ω(δ). For N 2 and δ > 1/2 the result is known.