Trigonometric approximation of functions f ( x , y ) $f(x,y)$ of generalized Lipschitz class by double Hausdorff matrix summability method

Abstract

Abstract In this paper, we establish a new estimate for the degree of approximation of functions f ( x , y ) $f(x,y)$ belonging to the generalized Lipschitz class L i p ( ( ξ 1 , ξ 2 ) ; r ) $Lip ((\xi _{1}, \xi _{2} );r )$ , r ≥ 1 $r \geq 1$ , by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from L i p ( ( α , β ) ; r ) $Lip ((\alpha ,\beta );r )$ and L i p ( α , β ) $Lip(\alpha ,\beta )$ in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and ( C , γ , δ ) $(C, \gamma , \delta )$ means.

Keywords

• Double Hausdorff matrix summability
• Double Fourier series
• Generalized Lipschitz class
• Modulus of continuity
• Cesàro summability
• Almost Euler summability means