On Nörlund summability of double Fourier series

1. Introduction

Let \(f(\alpha,\beta)\) be Lebesgue integral in the square \(R(-\pi,\pi;-\pi,\pi)\) and be of period \(2\pi\) in each of the variables \(\alpha\) and \(\beta\). Then the series

is called the double Fourier series associated with the function \(f(\alpha,\beta);\) (see [2,3]) where Also, we have

Definition 1.[4,5] Let\( {\{p^{(1)}_{m}}\}\) and \({\{p^{(2)}_{n}}\}\) are two sequence of constants, real or complex. Let

We shall also consider a double Nörlund transform of \({ \{a_{mn}}\}\). Then the double Nörlund transform is

Definition 2.[4,5] The double sequence \({\{a_{lg}}\}\) is said to be Nörlund summable to a limit \(v\) if

It is also known as summable \((N,p^{(1)}_{m}p^{(2)}_{n})\).

Definition 3.[1,2,4,5,6] If

then the double Nörlund transform reduces to double Ces\(\grave{a}\)ro transform of order one. This summability method is known as Ces\(\grave{a}\)ro summability \((C,1,1).\)

Definition 4.[1,2,5,6] If \begin{align*} p^{(1)}_{m}=\frac{1}{m+1}; m=0,1,2,... \;\;\textrm{and}\;\;p^{(2)}_{n}=\frac{1}{n+1},n=0,1,2,..., \end{align*} then the double Nörlund summability \((N,p^{(1)}_{m},p^{(2)}_{n})\) becomes Harmonic summability and is denoted by \((H,1,1).\)

Definition 5.[5] If for any \(\gamma\geq1,V_{mn} \to V,(m,n) \to(-\infty,\infty)\) in such a manner that \(\gamma\geq\frac{m}{n},\gamma \geq\frac{n}{m} \), then the sequence \(\{{a_{lg}\}}\) is said to be restrictedly summable \(N_{p} \) at \((x,y)\) to the same limit.

2. Known results

In 1953, Chow [7], for the first time studied Cesáro summability of double Fourier series. In 1958, Sharma [3] extended the results of Chow for \((H,1,1)\) summability which is weaker than \((C,1,1)\) summability of double Fourier series. There are several results on Nörlund summability of Fourier series [1,2,3,4,5,6,7,8,9,10]. This motivates us to study on the Nörlund summability of Fourier series in more generalized as particular cases. Therefore, in an attempt to make an advance in this research work, we study the double Fourier series and its conjugate series by Nörlund method. In 1963, Sing [11] proved the following theorem;

Theorem 1. If

where \(\chi\)\((y)= f(v+y)+f(v-y)-2f(y)\) as \( v\to\) 0, then the Fourier series of \(f(u)\) at v=y is summable \((N,p_n)\) to \(f(y)\) where \({\{p_n}\}\) is a real non-negative, non-increasing sequence such that Dealing with this topics, Pati [12] proved the following theorem;

Theorem 2. If \((N, p_n)\) be a regular Nörlund methods defined by a positive, monotonic decreasing sequences of coefficients \({\{p_n}\}\) such that

then if as \(z \to +0\), then the given Fourier series \(\chi(z)\) of \(z = y \) is summable \((N, p_n)\) to \(\chi(y)\).

3. Main Results

In this present research paper, we have established the following theorems which are the extended form of [2,11].

Theorem 3. If \((\alpha,\beta) \to (0,0),\)

and then the double Fourier series of \(f(\alpha,\beta)\) at \(\alpha=x\) and \(\beta=y\) is summable \(\left(N,p^{(1)}_{m} p^{(2)}_{n}\right)\) to the sum\(f(x,y)\) where \({\{p_n^{(v)}\}}\), \((v=1,2)\) are real non-negative, non-increasing sequence of constants such that The objective of this present work is to generalize the theorem B to a more general class of Nörlund summability for double Fourier series. We prove the following theorem:

Theorem 4. Let us suppose that the sequence \(\{p_b^{(1)}\}\) and \(\{p_a^{(2)}\}\) are positively and monotonically decreasing sequence of constants such that

\(P_b^{(1)} \to \infty, P_a^{(2)} \infty\) and if then double Fourier series (1) of the function \( f(x, y)\) is summable to the sum S at \(\alpha = m\) and \( \beta = n\), when \((N,p_{b}^{(1)},p_{a}^{(2)})\) is regular Nörlund method.

The following lemmas are required in the proof of our theorems;

Lemma 1. If \({\{p_n\}}\) is non-negative and non-increasing, then for \(0\leq a \leq b \leq \infty, 0\leq t \leq \pi\) and for any n, we have

Lemma 2. Under the condition of Lemma 1,

Lemma 3. Under the condition of Lemma 1,

Lemma 4. Under the condition of Lemma 1,

they are uniformly in each of the intervals.

Proof of Theorem 3. Let \(U_{mn}(x,y;f)=U_{mn}\) denotes the rectangular \((m,n)^{th}\) partial sum of the series (1), then we must have

where and where \(D^{1}_{m}(\alpha)\) and \(D^{2}_{n}(\beta)\) are respectively denote the Dirichlet kernels.

Let \(\bigg\{V_{mn}(x,y)\bigg\} \) denote the double Nörlund transform of the sequence \(\bigg\{V_{mn}-f(x,y) \bigg\},\) then

where and Also Eq. (29) can be written as By hypothesis and using the results of (23) and (24), we easily obtain Also, for \(I_3\), we have Thus, Again, by (23) and (24) and hypothesis, where and \(I_{3,2,1}\) stands for two inner intervals. Thus, and Also, for For thus, Now, by hypothesis (19) and using the Eqs (43) and (29), we get Combining (34), (35), (36), (38), (39), (43) and (44), we get Similarly, we can show that Now, for \(I_1\), Then by (23) and (24), Similarly, and Thus, where \(O(1)\) corresponds to the integrated part in (49) and \({C_1,C_2}\) and \({C_3}\) are repetitively denote the remaining there integrals. Thus following the estimates in \(I_{3,2,}\), we have Again for \(C_4\) As in (43), using the estimates (52) and (53), we get from (51) that Thus, Combining (33), (44), (45), (55), we get (32). Which competes the proof of the theorem.

Proof of Theorem 4. Let \(U_{b,a}\) denote the rectangular \((b, a)^{th}\) partial sums of the series (1) Then,

where \(0< \delta< \pi \; \text{and} \; 0< \theta< \pi\). Now, and Thus Also, by Lemma 1, we have Similarly, Thus, combining (57), (58), (59), (60), (61), we obtain (56). This completes the proof of the theorem.

4. Conclusion

The negligible set of conditions has been obtained for the finite series in this paper. We also discuss how Nörlund summability includes a broader range of functions that can be approximated. The approximation is a broad field having a widespread application in signals. Approximation treats signal one variable system and image as to the variable system.

Author Contributions:

All authors contributed equally in this paper. All authors read and approved the final version of this paper.

Conflicts of Interest:

The authors declare no conflict of interest.

Data Availability:

All data required for this research is included within this paper.

Funding Information:

No funding is available for this research.

Acknowledgments :

The author would like to thank the referee for his/her valuable comments which lead to an improvement of the original manuscript.