Trigonometric approximation of functions f(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(x,y)$\end{document} of generalized Lipschitz class by double Hausdorff matrix summability method

In this paper, we establish a new estimate for the degree of approximation of functions f(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(x,y)$\end{document} belonging to the generalized Lipschitz class Lip((ξ1,ξ2);r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Lip ((\xi _{1}, \xi _{2} );r )$\end{document}, r≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r \geq 1$\end{document}, by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from Lip((α,β);r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Lip ((\alpha ,\beta );r )$\end{document} and Lip(α,β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Lip(\alpha ,\beta )$\end{document} in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and (C,γ,δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(C, \gamma , \delta )$\end{document} means.

variables by linear methods of summation of their Fourier sums. Móricz and Shi [8] proved the following result for the approximation to continuous functions by Cesàro means of double Fourier series.
The degree of approximation using Gauss-Weierstrass integrals was also investigated by Khan and Ram [5]. Recently, error and bounds of certain bivariate functions by almost Euler means of double Fourier series for the functions of Lipschitz and Zygmund classes was estimated by Rathor and Singh [9]. To find the approximation of functions of two-dimensional torus, in this paper, we obtain a new estimate for trigonometric approximation of functions f (x, y) of generalized Lipschitz class by double Hausdorff matrix summability method of double Fourier series. For other summability methods of approximation, see [1] and [7].

Definitions and preliminaries
Let ∞ where {μ j,k } is any real or complex sequence, and If t H m,n = m j=0 n k=0 h j,k m,n s j,k → g as m → ∞ and n → ∞, then ∞ m=0 ∞ n=0 g m,n is said to be summable to the sum g by the double Hausdorff matrix summability method [15].
It is easy to see that the absolute value of the measure dχ(s, t) can me majorized by K 1 K 2 ds dt for some constants K 1 and K 2 (see [16]).
The important particular cases of double Hausdorff matrix summability means are as follows: 1 Almost Euler summability means ((E, q 1 , q 2 ) means) if μ m,n = 1 Let f (x, y) be a Lebesgue-integrable function of period 2π with respect to both variables x and y and summable in the fundamental square Q : (-π, π)×(-π, π). The double Fourier series of f (x, y) is given by   [3]. We define the L r norm by The degree of approximation of a function f : R 2 → R by a trigonometric polynomial [17] t m,n (x, y) = m j=0 n k=0 λ m,n [a j,k cos mx cos ny + b j,k sin mx cos ny + c j,k cos mx sin ny + d j,k sin mx cos ny] of order (m + n) is defined by A function f : R 2 → R of two variables x and y is said to belong to the class Lip(α, β) [4] if and to the class where ξ 1 and ξ 2 are moduli of continuity, that is, nonnegative nondecreasing continuous We define the forward difference operator as

Result
The object of this paper is obtaining the degree of approximation of functions f (x, y) of generalized Lipschitz class by double Hausdorff matrix summability means of its double for m, n = 0, 1, 2, . . . .

Lemmas
For the proof of our theorems, we need the following lemmas.

Corollaries
From the main theorem we derive the following corollaries.