Abstract
In this paper, we establish a new estimate for the degree of approximation of functions \(f(x,y)\) belonging to the generalized Lipschitz class \(Lip ((\xi _{1}, \xi _{2} );r )\), \(r \geq 1\), by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from \(Lip ((\alpha ,\beta );r )\) and \(Lip(\alpha ,\beta )\) in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and \((C, \gamma , \delta )\) means.
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1 Introduction
The study of various summability means of double Fourier series have been done by several authors, for example, Chow [2], Sharma [11], Łenski [6], and Ustina [15]. Dealing with the first arithmetic means of double Fourier series, Hasegawa [4] obtained the following:
Theorem A
If a continuous function \(f(x, y)\) of period 2π with respect to both x and y belongs to \(Lip (\alpha , \beta )\), where \(0<\alpha <l\) and \(0<\beta <1\), then
uniformly in \((x, y)\) as m and n independently tend to infinity.
If \(\alpha =\beta =1 \), then
uniformly in \((x, y)\) as m and n independently tend to infinity.
Siddiqui and Mohammadzadeh [12] investigated the approximation by Cesàro and B means of double Fourier series. Stepanets [13, 14] has established estimates of approximation for certain classes of periodic functions and differentiable periodic functions of two 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.
Theorem B
If \(f \in E(\alpha , \beta )\), \(0 < \alpha \), \(\beta \leq 1\), \(\gamma , \delta \geq 0\), then
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].
2 Definitions and preliminaries
Let \(\sum_{m=0}^{\infty }\sum_{n=0}^{\infty } g_{m,n}\) be double series with the sequence of \((m,n)\)th partial sums
A double Hausdorff matrix has the entries
where \(\{ \mu _{j,k} \} \) is any real or complex sequence, and
If \(t_{m,n}^{H} = \sum_{j=0}^{m}\sum_{k=0}^{n} h_{m,n}^{j,k} s_{j,k} \rightarrow g \) as \(m \rightarrow \infty \) and \(n \rightarrow \infty \), then \(\sum_{m=0}^{\infty }\sum_{n=0}^{\infty } g_{m,n}\) is said to be summable to the sum g by the double Hausdorff matrix summability method [15].
A necessary and sufficient condition for double Hausdorff matrix summability method to be regular is there exists a function \(\chi (s,t) \in BV[0,1]\times [0,1]\) such that
and
where \(\chi (s,0)=\chi (s,0^{+})=\chi (0^{+},t)=\chi (0,t) = 0\), \(0\leq s \), \(t \leq 1 \), and \(\chi (1,1)-\chi (1,0)-\chi (0,1)+\chi (0,0) = 1\) [10].
It is easy to see that the absolute value of the measure \(d \chi (s,t)\) 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 \(\mu _{m,n} = \frac{1}{(1+q_{1})^{m}}\frac{1}{(1+q_{2})^{n}}\).
-
2
\((E,1,1)\) means if \(q_{1}=1\) and \(q_{2}=1\) in \((E,q_{1},q_{2})\) means.
-
3
\((C, \gamma , \delta )\) means if \(\mu _{m,n} = \frac{1}{A^{\gamma }_{m}}\frac{1}{A^{\delta }_{n}}\), where \(\gamma , \delta \geq -1\) and \(A^{\gamma }_{m} = \binom{{\gamma +m} }{m }\), \(A^{\delta }_{n} = \binom{{\delta +n} }{n }\).
-
4
\((C,1,1)\) means if \(\gamma =\delta =1\) in \((C, \gamma , \delta )\) means.
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 \(Q:(-\pi ,\pi ) \times (-\pi ,\pi )\). The double Fourier series of \(f(x,y)\) is given by
with \((m,n)\)th partial sums \(s_{m,n}(f;(x,y))\), where
and similar expressions for \(b_{m,n}\), \(c_{m,n}\), and \(d_{m,n}\) [3].
We define the \(L^{r} \) norm by
The degree of approximation of a function \(f :\mathbb{R}^{2} \rightarrow \mathbb{R}\) by a trigonometric polynomial [17]
of order \((m+n)\) is defined by
A function \(f :\mathbb{R}^{2} \rightarrow \mathbb{R}\) of two variables x and y is said to belong to the class \(Lip(\alpha ,\beta )\) [4] if
to the class \(Lip ((\alpha ,\beta );r )\) if
and to the class \(Lip ((\xi _{1},\xi _{2});r )\) if
where \(\xi _{1}\) and \(\xi _{2}\) are moduli of continuity, that is, nonnegative nondecreasing continuous functions such that \(\xi _{1}(0) =\xi _{2}(0) = 0\), \(\xi _{1}(u_{1} + u_{2}) \le \xi _{1}(u_{1}) + \xi _{1}(u_{2})\), and \(\xi _{2}(v_{1} + v_{2}) \le \xi _{2}(v_{1}) + \xi _{2}(v_{2})\).
If \(\xi _{1}(u)=u^{\alpha }\) and \(\xi _{2}(v)=v^{\beta }\), \(0<\alpha \leq 1\), \(0 < \beta \leq 1\), then the class \(Lip ((\xi _{1},\xi _{2});r )\) coincides with \(Lip ((\alpha ,\beta );r )\). As \(r \rightarrow \infty \), \(Lip ((\alpha ,\beta );r )\) reduces to \(Lip(\alpha ,\beta )\). Clearly, \(Lip(\alpha ,\beta ) \subseteq Lip ((\alpha ,\beta );r ) \subseteq Lip ((\xi _{1},\xi _{2});r ) \).
We define the forward difference operator Δ as \(\Delta \mu _{k} = \mu _{k} - \mu _{k+1} \); also, \(\Delta ^{n+1}\mu _{k}=\Delta (\Delta ^{n} \mu _{k} )\), \(k\geq 0\). We denote
3 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 Fourier series:
Theorem 1
If \(f(x,y)\) is a 2π periodic function with respect to both variables x and y, Lebesgue integrable in \((-\pi ,\pi )\times (-\pi ,\pi )\) and belonging to the class \(Lip ((\xi _{1}, \xi _{2});r )\) (\(r \geq 1\)), then the degree of approximation of \(f(x,y)\) by double Hausdorff matrix summability means
of double Fourier series (1) satisfies
4 Lemmas
For the proof of our theorems, we need the following lemmas.
Lemma 1
\(\vert M_{m}^{H}(u) \vert = O (m+1 )\) for \(0< u \leq \frac{1}{m+1}\), and \(\vert K_{n}^{H}(v) \vert = O (n+1 )\) for \(0< v \leq \frac{1}{n+1}\).
Proof
Since \(\vert \sin mu \vert \leq mu\) for \(0< u\leq \frac{1}{m+1}\) and \(\sin (u/2)\geq (u/\pi )\), we have
Similarly, for \(0< v \leq \frac{1}{n+1}\),
□
Lemma 2
\(\vert M_{m}^{H}(u) \vert = O (\frac{1}{(j+1)u^{2}} )\) for \(\frac{1}{m+1} < u \leq \pi \), and \(\vert K_{n}^{H}(v) \vert = O (\frac{1}{(k+1)v^{2}} )\) for \(\frac{1}{n+1} < v \leq \pi \).
Proof
Since \(\sin (m+1) u \leq 1\) for \(\frac{1}{m+1} < u \leq \pi \) and \(\sin (u/2)\geq (u/\pi )\), we get
Equating the imaginary parts of both sides, we get
Therefore
Similarly, for \(\frac{1}{n+1} < v \leq \pi \),
□
Lemma 3
If \(f(x,y)\in Lip ((\xi _{1},\xi _{2});r )\) (\(r\geq 1\)), then \(\Vert \phi (u,v)) \Vert _{r} = O ( \xi _{1}(u) + \xi _{2}(v) )\).
Proof
Clearly,
□
5 Proof of Theorem 1
The \((m,n)\)th partial sum of the double Fourier series (1) is given by
Denoting the double Hausdorff matrix sums of \(s_{m,n} \) by \(t_{m,n}^{H}\), we have
Using Lemmas 1 and 3, we obtain
Similarly,
Also, using Lemmas 2 and 3, we get
Next,
Similarly,
Combining equations (5)–(10), we have
This completes the proof of Theorem 1.
6 Corollaries
From the main theorem we derive the following corollaries.
Corollary 1
If \(f(x,y)\) is a 2π periodic function with respect to both variables x and y, Lebesgue integrable in \((-\pi ,\pi )\times (-\pi ,\pi )\) and belonging to the class \(Lip ((\alpha , \beta );r )\) (\(r \geq 1 \)), then the degree of approximation of \(f(x,y)\) by means \(t_{m,n}^{H}\) of double Fourier series (1) satisfies
for \(m,n=0,1,2,\dots \).
Corollary 2
If \(f(x,y)\) is a 2π periodic function with respect to both variables x and y, Lebesgue integrable in \((-\pi ,\pi )\times (-\pi ,\pi )\) and belonging to the class \(Lip(\alpha ,\beta )\), then the degree of approximation of \(f(x,y)\) by double Hausdorff matrix summability means \(t_{m,n}^{H} \) of double Fourier series (1) satisfies
for \(m,n=0,1,2,\dots \).
Corollary 3
If \(f(x,y)\) is a 2π periodic function with respect to both variables x and y, Lebesgue integrable in \((-\pi ,\pi )\times (-\pi ,\pi )\) and belonging to the class \(Lip ((\xi _{1}, \xi _{2});r )\), then the degree of approximation of \(f(x,y)\) by almost Euler summability means
of double Fourier series (1) satisfies
for \(m,n=0,1,2,\dots \).
Corollary 4
For \(\gamma , \delta \geq -1\), the Cesàro means \(\sigma _{m,n}^{\gamma , \delta }\) of order γ and δ, that is, \((C, \gamma , \delta )\) means of double Fourier series, are given by
where \(A^{\gamma }_{m} = \binom{{\gamma +m} }{m }\) and \(A^{\delta }_{n} = \binom{{\delta +n} }{n }\).
If \(f(x,y)\) is a 2π periodic function with respect to both variables x and y, Lebesgue integrable in \((-\pi ,\pi )\times (-\pi ,\pi )\) and belonging to the class \(Lip ((\xi _{1}, \xi _{2});r )\), then the degree of approximation of \(f(x,y)\) by \((C, \gamma , \delta )\) means of double Fourier series (1), satisfies
for \(m,n=0,1,2,\dots \).
7 Conclusion
We established the degree of approximation of a function \(f(x,y)\) belonging to the generalized Lipschitz class by double Hausdorff matrix summability means of its double Fourier series in the form of equation (2). If \(\xi _{1}=u^{\alpha }\) and \(\xi _{2}=v^{\beta }\), then Theorem 1 reduces to Corollary 1, and as \(r \rightarrow \infty \), Corollary 1 reduces to Corollary 2. Independent proofs of Corollaries 1–4 can be developed along the same lines as that of Theorem 1. Results similar to Corollaries 3 and 4 can be derived for \((E,1,1)\) means and \((C,1,1)\) means of its double Fourier series. In this way, we can obtain some more different results by changing \(\xi _{1}\), \(\xi _{2}\), and \(\mu _{m,n}\) under given conditions. For functions \(f(x,y)\) belonging to the Zygmund classes \(Zyg(\alpha ,\beta )\) and \(Zyg(\alpha ,\beta ;p)\) discussed in [9], the degree of approximation using double Hausdorff matrix summability means and hence almost Euler means of its double Fourier series can be obtained similarly to Theorem 1.
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Mishra, A., Mishra, V.N. & Mursaleen, M. Trigonometric approximation of functions \(f(x,y)\) of generalized Lipschitz class by double Hausdorff matrix summability method. Adv Differ Equ 2020, 681 (2020). https://doi.org/10.1186/s13662-020-03124-8
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DOI: https://doi.org/10.1186/s13662-020-03124-8