An analogue of Leindler's theorem for hexagonal Fourier series

Abstract

Abstract. Let ℳ α $\mathcal {M}_{\alpha }$ be the class of moduli of continuity defined by Leindler in [Studia Sci. Math. Hungar. 14 (1979), 431–439], and H ω α ( Ω ¯ ) $H^{\omega _{\alpha }}(\overline{\Omega }) $ be the generalized Hölder class of functions on the closure of the regular hexagon Ω, where 0 < α ≤ 1 $0<\alpha \le 1$ and ω α ∈ ℳ α $\omega _{\alpha }\in \mathcal {M}_{\alpha }$ . The difference f - 𝒱 n λ ( f ) $f-\mathcal {V}_n^{\lambda }(f) $ is estimated in the uniform norm ∥ · ∥ C ( Ω ¯ ) $\Vert \cdot \Vert _{C(\overline{\Omega })}$ and in the generalized Hölder norm ∥ · ∥ ω β $\Vert \cdot \Vert _{\omega _\beta }$ , where 𝒱 n λ ( f ) $\mathcal {V}_n^{\lambda }(f) $ is the nth generalized de la Vallée-Poussin mean of hexagonal Fourier series of f ∈ H ω α ( Ω ¯ ) $f\in H^{\omega _{\alpha }}(\overline{\Omega }) $ and 0 ≤ β < α ≤ 1 $0\le \beta <\alpha \le 1$ .