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.
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1 Introduction
Let \(f(x,y)\) be a 2π-periodic function in each variable and Lebesgue integrable over the two-dimensional torus \(T^{2}=[-\pi,\pi]\times [-\pi,\pi]\). Then the double trigonometric Fourier series of \(f(x,y)\) is defined by
where
are the Fourier coefficients of the function f.
The double sequence of symmetric rectangular partial sums associated with Fourier series of f is given by
and its integral representation is given by
where \(D_{k}(t)=\frac{\sin(k+\frac{1}{2})t}{2\sin(t/2)}\) 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
Móricz and Rhoades [2] extended the definition of almost convergence to double sequences of real numbers \(\{x_{mn}\}\), almost converging to L, if
The Euler means \(E_{mn}(x,y)\) of the sequence \(\{s_{kl}(x,y)\}\) are defined by
and almost Euler means of the sequence \(\{s_{kl}(x,y)\}\) are defined by
where
The following function classes are well known in the literature (see [3, 4]). For \(0<\alpha\leq1\), the Lipschitz class Lipα is defined by
where \(\omega(f,\delta)\) is the modulus of continuity of f, defined by
For \(0<\alpha,\beta\leq1\), the Lipschitz class \(\operatorname {Lip}(\alpha,\beta)\) is defined by
where \(\omega_{1,x}(f,u)\) and \(\omega_{1,y}(f,v)\) are the partial moduli of continuity of f, defined by
and
For \(0< \alpha, \beta\leq2\), the Zygmund class \(\operatorname {Zyg}(\alpha,\beta)\) is defined by
where \(\omega_{2,x}(f,u)\) and \(\omega_{2,y}(f,v)\) are the partial moduli of smoothness of f, defined by
and
Here, we generalize the definitions of \(\operatorname{Lip}(\alpha,\beta )\) and \(\operatorname{Zyg}(\alpha,\beta)\) given in [3] and [4], respectively, by introducing a new Lipschitz class \(\operatorname{Lip}(\alpha,\beta; p)\) and a Zygmund class \(\operatorname {Zyg}(\alpha,\beta; p)\).
Let \(L^{p}(T^{2})\) (\(p\geq1\)) 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 \(L^{p}(T^{2})\) (\(p\geq1\)) 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
and
The Lipschitz class \(\operatorname{Lip}(\alpha,\beta;p)\) (\(p\geq1\)) for \(\alpha,\beta\in(0,1]\) 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
where
The partial integral moduli of smoothness are defined by
and
It is clear that \(\omega_{2}^{p}(f,u,v)\), \(\omega_{2,x}^{p}(f,u)\) and \(\omega _{2,y}^{p}(f,v)\) are nondecreasing functions in u and v and that
and
For \(0< \alpha, \beta\leq2\), the Zygmund class \(\operatorname {Zyg}(\alpha,\beta;p)\) (\(p\geq1\)) is defined as
From (4) it is clear that \(\operatorname{Lip}(\alpha,\beta ;p)\subseteq\operatorname{Zyg}(\alpha,\beta;p)\) for \(0<\alpha,\beta\leq 1\), and similar to one-dimensional case, \(\operatorname{Lip}(\alpha ,\beta;p)= \operatorname{Zyg}(\alpha,\beta;p)\) for \(0<\alpha,\beta<1\), but \(\operatorname{Lip}(\alpha,\beta;p)\neq\operatorname{Zyg}(\alpha ,\beta;p)\) for \(\max(\alpha,\beta)=1\) (see, e.g., [5], p. 44).
Let \(\omega(\delta)\) be a nondecreasing function of \(\delta\geq0\). Then \(\omega(\delta)\) is of the first kind if
and \(\omega(\delta)\) is of the second kind if
(see [3]).
A function \(f(x,y)\) is said to belong to the class \(\operatorname {Lip}(\psi(u,v);p)\) (\(p>1\)) if
where \(\psi(u,v)\) is a positive increasing function of the variables u, v and M is a positive constant independent of x, y, u, and v (see [6–8]).
Here, we generalize the definition of \(\operatorname{Lip}(\psi(u,v);p)\) (\(p>1\)) class given above by introducing a new Lipschitz class \(\operatorname{Lip}(\psi(u,v))_{L^{p}}\) (\(p>1\)) defined as
Throughout this paper we shall use the following notations:
Note 1
We can easily prove that \(\phi_{x,y}(u,v)\) 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 \(\operatorname{Lip}(\alpha,\beta)\) 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
Let \(f(x,y)\) be a 2π-periodic function in each variable belonging to \(L^{p}(T^{2})\) (\(1\leq p<\infty\)). Then the degree of approximation of \(f(x,y)\) by almost Euler means of its double Fourier series is given by:
-
(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
Let \(f(x,y)\) be a 2π-periodic function in each variable belonging to \(L^{\infty}(T^{2})\). Then the degree of approximation of \(f(x,y)\) by almost Euler means of its double Fourier series is given by:
-
(i)
If both \(\omega_{2,x}\) 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
Let \(f(x,y)\) be a 2π-periodic function in each variable belonging to the class \(\operatorname{Lip}(\psi(u,v))_{L^{p}}\) (\(p>1\)). If the positive increasing function \(\psi(u,v)\) satisfies the condition
then the degree of approximation of \(f(x,y)\) by almost Euler means of its double Fourier series is given by
For \(p=\infty\), the class \(\operatorname{Lip}(\psi(u,v))_{L^{p}}\) reduces to the class \(\operatorname{Lip}(\psi(u,v))_{L^{\infty}}\), defined as
Thus, for \(p=\infty\), we have the following theorem.
Theorem 2.4
Let \(f(x,y)\) be a 2π-periodic function in each variable belonging to the class \(\operatorname{Lip}(\psi(u,v))_{L^{\infty}}\). If the positive increasing function \(\psi(u,v)\) satisfies the condition
then the degree of approximation of \(f(x,y)\) by almost Euler means of its double Fourier series is given by
3 Lemmas
We need the following lemmas for the proof of our theorems.
Lemma 3.1
Let \(R_{m}^{r}(u)\) and \(R_{n}^{s}(v)\) be given by (10) and (11), respectively. Then
-
(i)
\(R_{m}^{r}(u)= O ((1+q_{1})^{m}(m+1) )\) for \(0< u\leq\frac {1}{m+1}\).
-
(ii)
\(R_{n}^{s}(v)= O ((1+q_{2})^{n}(n+1) )\) for \(0< v\leq \frac{1}{n+1}\).
Proof
(i) For \(0< u\leq\frac{1}{m+1}\), using \(\sin(u/2)\geq u/ \pi\) and \(\sin mu\leq m\sin u\), we have
(ii) It can be proved similarly to part (i). □
Lemma 3.2
Let \(R_{m}^{r}(u)\) and \(R_{n}^{s}(v)\) be given by (10) and (11), respectively. Then
-
(i)
\(R_{m}^{r}(u)= O \Big(\frac{(1+q_{1})^{m}}{(m+1)u^{2}} \Big)\) for \(\frac{1}{m+1}< u\leq\pi\).
-
(ii)
\(R_{n}^{s}(v)= O \Big(\frac{(1+q_{2})^{n}}{(n+1)v^{2}} \Big)\) for \(\frac{1}{n+1}< v\leq\pi\).
Proof
(i) For \(\frac{1}{m+1}< u\leq\pi\), using \(\sin(u/2)\geq u/ \pi\) and \(\sin u\leq1\), we have
(ii) It can be proved similarly to part (i). □
4 Proof of the main results
Proof of Theorem 2.1
Using the integral representation of \(s_{kl}(x,y)\) given in (2), we have
Therefore,
Now
which, on applying the generalized Minkowski inequality, gives
Proof of part (i): Using Lemma 3.1 and (13), we have
Using Lemma 3.1, Lemma 3.2, (5), and (13), we have
Similarly, we have
Using Lemma 3.2, (5), and (13), we have
which proves part (i).
Proof of part (ii): Using (19) and (20), we have
Using Lemma 3.1, Lemma 3.2, (6), and (13), we have
Using Lemma 3.2, (5), (6), and (13), we have
Collecting (18), (23)–(26), we have
which proves part (ii).
In a similar manner, we can prove part (iii) and part (iv). □
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. □
Proof of Theorem 2.3
Following the proof of Theorem 2.1, using the generalized Minkowski inequality and the fact that \(\phi _{x,y}(u,v)\in\operatorname{Lip}(\psi(u,v))_{L^{p}}\) (\(p>1\)), we have
Using Lemma 3.1, we have
Using Lemma 3.1 and Lemma 3.2, we have
Similarly, we have
Using Lemma 3.1 and Lemma 3.2, we have
□
Proof of Theorem 2.4
We have
Now we can follow the proof of Theorem 2.3 with supremum norm to get the result. □
5 Corollaries
If \(f\in\operatorname{Zyg}(\alpha,\beta;p)\), then
For \(0<\alpha,\beta<1\),
which implies that \(u^{\alpha}\) and \(v^{\beta}\) are of the first kind.
For \(\alpha=\beta=1\),
which implies that \(u^{\alpha}\) and \(v^{\beta}\) are of the second kind.
Thus, Theorem 2.1 reduces to the following corollary.
Corollary 1
If \(f\in\operatorname{Zyg}(\alpha,\beta;p)\), then
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
If \(f\in\operatorname{Zyg}(\alpha,\beta)\), then
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Rathore, A., Singh, U. Approximation of certain bivariate functions by almost Euler means of double Fourier series. J Inequal Appl 2018, 89 (2018). https://doi.org/10.1186/s13660-018-1676-0
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DOI: https://doi.org/10.1186/s13660-018-1676-0