Approximation of certain bivariate functions by almost Euler means of double Fourier series

In this paper, we study the degree of approximation of 2π-periodic functions of two variables, defined on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T^{2}=[-\pi,\pi]\times[-\pi,\pi]$\end{document}T2=[−π,π]×[−π,π] and belonging to certain Lipschitz classes, by means of almost Euler summability of their Fourier series. The degree of approximation obtained in this way depends on the modulus of continuity associated with the functions. We also derive some corollaries from our theorems.


Introduction
Let f (x, y) be a 2π -periodic function in each variable and Lebesgue integrable over the two-dimensional torus T 2 = [-π, π] × [-π, π]. Then the double trigonometric Fourier series of f (x, y) is defined by wherê f (x + u, y + v)D m (u)D n (v) du dv, ( 2 ) where D k (t) = sin(k+ 1 2 )t 2 sin(t/2) is the Dirichlet kernel. The concept of almost convergence of sequences was introduced and studied by G.G. Lorentz in 1948 [1]. A sequence {x n } is said to be almost convergent to a limit L, if The following function classes are well known in the literature (see [3,4]). For 0 < α ≤ 1, the Lipschitz class Lip α is defined by where ω(f , δ) is the modulus of continuity of f , defined by For 0 < α, β ≤ 1, the Lipschitz class Lip(α, β) is defined by where ω 1,x (f , u) and ω 1,y (f , v) are the partial moduli of continuity of f , defined by and For 0 < α, β ≤ 2, the Zygmund class Zyg(α, β) is defined by where ω 2,x (f , u) and ω 2,y (f , v) are the partial moduli of smoothness of f , defined by and Here, we generalize the definitions of Lip(α, β) and Zyg(α, β) given in [3] and [4], respectively, by introducing a new Lipschitz class Lip(α, β; p) and a Zygmund class Zyg(α, β; p). Let L p (T 2 ) (p ≥ 1) denote the spaces of Lebesgue functions on the torus T 2 , with the norm defined by Let f (x, y) be a 2π -periodic function in each variable belonging to L p (T 2 ) (p ≥ 1) class. Then the total integral modulus of continuity of f is defined by while the two partial integral moduli of continuity of f are defined by The Lipschitz class Lip(α, β; p) (p ≥ 1) for α, β ∈ (0, 1] is defined as We also use the notion of integral modulus of smoothness. The total integral modulus of smoothness of a function f is defined by The partial integral moduli of smoothness are defined by It is clear that ω ,y (f , v) are nondecreasing functions in u and v and that For 0 < α, β ≤ 2, the Zygmund class Zyg(α, β; p) (p ≥ 1) is defined as From (4) it is clear that Lip(α, β; p) ⊆ Zyg(α, β; p) for 0 < α, β ≤ 1, and similar to onedimensional case, Lip(α, β; p) = Zyg(α, β; p) for 0 < α, β < 1, but Lip(α, β; p) = Zyg(α, β; p) for max(α, β) = 1 (see, e.g., [5], p. 44).
A function f (x, y) is said to belong to the class Lip(ψ(u, v); p) (p > 1) if where ψ(u, v) is a positive increasing function of the variables u, v and M is a positive constant independent of x, y, u, and v (see [6][7][8]).
Here, we generalize the definition of Lip(ψ(u, v); p) (p > 1) class given above by introducing a new Lipschitz class Lip(ψ(u, v)) L p (p > 1) defined as Throughout this paper we shall use the following notations: Note 1 We can easily prove that φ x,y (u, v) satisfies the following inequalities: and Móricz and Xianlianc Shi [4] studied the rate of uniform approximation of a 2π -periodic continuous function f (x, y) in the Lipschitz class Lip(α, β) and in the Zygmund class Zyg(α, β), 0 < α, β ≤ 1, by Cesàro means σ γ δ mn of positive order of its double Fourier series. They also obtained the result for conjugate function by using the corresponding Cesàro means.
Further, Móricz and Rhoades [9] studied the rate of uniform approximation of f (x, y) in Lip α, 0 < α ≤ 1, class by Nörlund means of its Fourier series. After that, Móricz and Rhoades [10] studied the rate of uniform approximation of a continuous function f (x, y) in the Lipschitz class Lip(α, β) and in the Zygmund class Zyg(α, β), 0 < α, β ≤ 1, by Nörlund means of its Fourier series. In [10], they also obtained the result for a conjugate function by using the corresponding Nörlund means. [3] generalized the results of [9,10], and [4] for a 2π -periodic continuous function f (x, y) in the Lipschitz class Lip(α, β) and in the Zygmund class Zyg(α, β), 0 < α, β ≤ 1, by using rectangular double matrix means of its double Fourier series. Lal [11,12] obtained results for double Fourier series using double matrix means and product matrix means.

Mittal and Rhoades
Also, Khan [6] obtained the degree of approximation of functions belonging to the class Lip(ψ(u, v); p) (p > 1) by Jackson type operator. Further, Khan and Ram [8] determined the degree of approximation for the functions belonging to the class Lip(ψ(u, v); p) (p > 1) by means of Gauss-Weierstrass integral of the double Fourier series of f (x, y). Khan et al. [7] extended the result of Khan [6] for n-dimensional Fourier series. In [13], Krasniqi determined the degree of approximation of the functions belonging to the class Lip(ψ(u, v); p) (p > 1) by Euler means of double Fourier series of a function f (x, y). In fact, he generalized the result of Khan [14] for two-dimensional and for n-dimensional cases.

Main results
In this paper, we study the problem in more generalized function classes defined in Sect. 1 and determine the degree of approximation by almost Euler means of the double Fourier series. More precisely, we prove the following theorem.
x is of the first kind and ω p 2,y is of the second kind, then x is of the second kind and ω p 2,y is of the first kind, then For p = ∞, the partial integral moduli of smoothness ω p 2,x and ω p 2,y reduce to the moduli of smoothness ω 2,x and ω 2,y , respectively. Thus, for p = ∞, we have the following theorem.

Theorem 2.2 Let f (x, y) be a 2π -periodic function in each variable belonging to L ∞ (T 2 ). Then the degree of approximation of f (x, y) by almost Euler means of its double Fourier series is given by:
(i) If both ω 2,x and ω 2,y are of the first kind, then (ii) If ω 2,x is of the first kind and ω 2,y is of the second kind, then (iii) If ω 2,x is of the second kind and ω 2,y is of the first kind, then (iv) If both ω 2,x and ω 2,y are of the second kind, then then the degree of approximation of f (x, y) by almost Euler means of its double Fourier series is given by , π + (m + 1) -σ ψ π, 1 n + 1 For p = ∞, the class Lip(ψ(u, v)) L p reduces to the class Lip(ψ(u, v)) L ∞ , defined as Thus, for p = ∞, we have the following theorem. Lip(ψ(u, v)) L ∞ . If the positive increasing function ψ(u, v) satisfies the condition (uv) -σ ψ(u, v) is nondecreasing for some 0 < σ < 1,

Lemmas
We need the following lemmas for the proof of our theorems. (10) and (11), respectively. Then

Lemma 3.1 Let R r m (u) and R s n (v) be given by
Proof (i) For 0 < u ≤ 1 m+1 , using sin(u/2) ≥ u/π and sin mu ≤ m sin u, we have (ii) It can be proved similarly to part (i). (10) and (11), respectively. Then

Lemma 3.2 Let R r m (u) and R s n (v) be given by
Proof (i) For 1 m+1 < u ≤ π , using sin(u/2) ≥ u/π and sin u ≤ 1, we have (ii) It can be proved similarly to part (i).

Proof of Theorem 2.2 We have
Using (12) and following the proof of Theorem 2.1 with supremum norm, we will get the required result.