On Nörlund summability of double Fourier series

: In this research paper, the authors studied some problems related to harmonic summability of double Fourier series on Nörlund summability method. These results constitute substantial extension and generalization of related work of Moricz [1] and Rhodes et al. , [2]. We also constructed a new result on ( N , p ( 1 ) b , p ( 2 ) a ) by regular Nörlund method of summability.


Definition 5. [5] If for any
in such a manner that γ ≥ m n , γ ≥ n m , then the sequence {a lg } is said to be restrictedly summable N p at (x, y) to the same limit.

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; where χ(y)= f(v+y)+f(v-y)-2f(y) as v → 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 → +0, then the given Fourier series χ(z) of z = y is summable (N, p n ) to χ(y).

Main Results
In this present research paper, we have established the following theorems which are the extended form of [2,11].
then the double Fourier series of f (α, β) at α = x and β = y is summable N, p 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:   a } are positively and monotonically decreasing sequence of constants such that a ∞ and if then double Fourier series (1) of the function f (x, y) is summable to the sum S at α = m and β = n, when (N, p a ) 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 ≤ a ≤ b ≤ ∞, 0 ≤ t ≤ π and for any n, we have Lemma 2. 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 (α) and D 2 n (β) are respectively denote the Dirichlet kernels. Let V mn (x, y) denote the double Nörlund transform of the sequence V mn − f (x, y) , then where N (1) and N (2) n (β) = 1 2πP (2) n n ∑ g=0 p (2) g−n .

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.