Approximation of certain bivariate functions by almost Euler means of double Fourier series
Abstract
In this paper, we study the degree of approximation of 2π-periodic functions of two variables, defined on \(T^{2}=[-\pi,\pi]\times[-\pi,\pi]\) 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.
Keywords
Double Fourier series Symmetric partial sums Modulus of continuity Modulus of smoothness Lipschitz class Zygmund class Almost Euler meansMSC
40C05 41A25 26A151 Introduction
Note 1
Móricz and Xianlianc Shi [4] studied the rate of uniform approximation of a 2π-periodic continuous function \(f(x,y)\) in the Lipschitz class \(\operatorname{Lip}(\alpha,\beta)\) and in the Zygmund class \(\operatorname{Zyg}(\alpha,\beta)\), \(0<\alpha,\beta\leq1\), by Cesàro means \(\sigma_{mn}^{\gamma\delta}\) 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 Lipα, \(0<\alpha \leq1\), 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 \(\operatorname{Lip}(\alpha,\beta)\) and in the Zygmund class \(\operatorname{Zyg}(\alpha,\beta)\), \(0<\alpha,\beta\leq1\), by Nörlund means of its Fourier series. In [10], they also obtained the result for a conjugate function by using the corresponding Nörlund means.
Mittal and Rhoades [3] generalized the results of [9, 10], and [4] for a 2π-periodic continuous function \(f(x,y)\) in the Lipschitz class \(\operatorname{Lip}(\alpha,\beta)\) and in the Zygmund class \(\operatorname{Zyg}(\alpha,\beta)\), \(0<\alpha,\beta \leq1\), 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.
Also, Khan [6] obtained the degree of approximation of functions belonging to the class \(\operatorname{Lip}(\psi(u,v);p)\) (\(p > 1\)) by Jackson type operator. Further, Khan and Ram [8] determined the degree of approximation for the functions belonging to the class \(\operatorname{Lip}(\psi(u,v);p)\) (\(p > 1\)) 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 \(\operatorname{Lip}(\psi(u,v);p)\) (\(p>1\)) 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.
2 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.
Theorem 2.1
- (i)If both\(\omega_{2,x}^{p}\)and\(\omega_{2,y}^{p}\)are of the first kind, then$$ \big\| \tau_{mn}^{rs}(x,y)-f(x,y)\big\| _{p}=O \biggl( \omega_{2,x}^{p} \biggl(f,\frac {1}{m+1} \biggr)+ \omega_{2,y}^{p} \biggl(f,\frac{1}{n+1} \biggr) \biggr). $$
- (ii)If\(\omega_{2,x}^{p}\)is of the first kind and\(\omega _{2,y}^{p}\)is of the second kind, then$$ \big\| \tau_{mn}^{rs}(x,y)-f(x,y)\big\| _{p}=O \biggl( \omega_{2,x}^{p} \biggl(f,\frac {1}{m+1} \biggr)+\log\bigl( \pi(n+1)\bigr)\omega_{2,y}^{p} \biggl(f,\frac{1}{n+1} \biggr) \biggr). $$
- (iii)If\(\omega_{2,x}^{p}\)is of the second kind and\(\omega _{2,y}^{p}\)is of the first kind, then$$ \big\| \tau_{mn}^{rs}(x,y)-f(x,y)\big\| _{p}=O \biggl(\log \bigl(\pi(m+1)\bigr)\omega _{2,x}^{p} \biggl(f, \frac{1}{m+1} \biggr)+\omega_{2,y}^{p} \biggl(f, \frac {1}{n+1} \biggr) \biggr). $$
- (iv)If both\(\omega_{2,x}^{p}\)and\(\omega_{2,y}^{p}\)are of the second kind, then$$\begin{aligned} \big\| \tau_{mn}^{rs}(x,y)-f(x,y)\big\| _{p}={}&O \biggl(\log \bigl(\pi(m+1)\bigr)\omega _{2,x}^{p} \biggl(f, \frac{1}{m+1} \biggr) \\ &+\log\bigl(\pi(n+1)\bigr)\omega_{2,y}^{p} \biggl(f,\frac{1}{n+1} \biggr) \biggr).\end{aligned} $$
For \(p=\infty\), the partial integral moduli of smoothness \(\omega _{2,x}^{p}\) and \(\omega_{2,y}^{p}\) reduce to the moduli of smoothness \(\omega_{2,x}\) and \(\omega_{2,y}\), respectively. Thus, for \(p=\infty\), we have the following theorem.
Theorem 2.2
- (i)If both\(\omega_{2,x}\)and\(\omega_{2,y}\)are of the first kind, then$$ \big\| \tau_{mn}^{rs}(x,y)-f(x,y)\big\| _{\infty}=O \biggl( \omega_{2,x} \biggl(f,\frac {1}{m+1} \biggr)+\omega_{2,y} \biggl(f,\frac{1}{n+1} \biggr) \biggr). $$
- (ii)If\(\omega_{2,x}\)is of the first kind and\(\omega_{2,y}\)is of the second kind, then$$ \big\| \tau_{mn}^{rs}(x,y)-f(x,y)\big\| _{\infty}=O \biggl( \omega_{2,x} \biggl(f,\frac {1}{m+1} \biggr)+\log\bigl(\pi(n+1)\bigr) \omega_{2,y} \biggl(f,\frac{1}{n+1} \biggr) \biggr). $$
- (iii)If\(\omega_{2,x}\)is of the second kind and\(\omega_{2,y}\)is of the first kind, then$$ \big\| \tau_{mn}^{rs}(x,y)-f(x,y)\big\| _{\infty}=O \biggl(\log \bigl(\pi(m+1)\bigr)\omega _{2,x} \biggl(f,\frac{1}{m+1} \biggr)+ \omega_{2,y} \biggl(f,\frac{1}{n+1} \biggr) \biggr). $$
- (iv)If both\(\omega_{2,x}\)and\(\omega_{2,y}\)are of the second kind, then$$\begin{aligned} \big\| \tau_{mn}^{rs}(x,y)-f(x,y)\big\| _{\infty}={}&O \biggl(\log \bigl(\pi(m+1)\bigr)\omega _{2,x} \biggl(f,\frac{1}{m+1} \biggr) \\ &+ \log\bigl(\pi(n+1)\bigr)\omega_{2,y} \biggl(f,\frac{1}{n+1} \biggr) \biggr).\end{aligned} $$
Theorem 2.3
Theorem 2.4
3 Lemmas
We need the following lemmas for the proof of our theorems.
Lemma 3.1
Proof
(ii) It can be proved similarly to part (i). □
Lemma 3.2
Proof
(ii) It can be proved similarly to part (i). □
4 Proof of the main results
Proof of Theorem 2.1
In a similar manner, we can prove part (iii) and part (iv). □
Proof of Theorem 2.2
Proof of Theorem 2.3
Proof of Theorem 2.4
5 Corollaries
Thus, Theorem 2.1 reduces to the following corollary.
Corollary 1
For \(p=\infty\), the Zygmund class \(\operatorname{Zyg}(\alpha,\beta;p)\) reduces to \(\operatorname{Zyg}(\alpha,\beta)\). In this case, from Theorem 2.2 we have the following corollary.
Corollary 2
Notes
Acknowledgements
The authors are grateful to the reviewers for their suggestions and comments for improving the manuscript of this paper.
Authors’ contributions
Both the authors contributed equally to this work. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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