Trigonometric approximation of functions f(x,y)$f(x,y)$ 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) belonging to the generalized Lipschitz class Lip((ξ1,ξ2); r), r ≥ 1, by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from Lip((α,β); r) and Lip(α,β) in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and (C,γ ,δ) 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.