On Double Summability of Double Conjugate Fourier Series

H. K. Nigam and Kusum Sharma Department of Mathematics, Faculty of Engineering and Technology, Mody Institute of Technology and Science, Deemed University, Laxmangarh 332311, Sikar, Rajasthan, India Correspondence should be addressed to Kusum Sharma, kusum31sharma@rediffmail.com Received 20 January 2012; Accepted 1 March 2012 Academic Editor: Ram U. Verma Copyright q 2012 H. K. Nigam and K. Sharma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For the first time, a theorem on double matrix summability of double conjugate Fourier series is established.


Introduction
The Fourier series of f x is given by a n cos nx b n sin nx .

1.1
Conjugate to the series 1.1 is given by and is known as conjugate Fourier series.It is well known that the corresponding conjugate function of 1.2 is defined as Let f x, y is integrable L over the square Q −π, −π; π, π and is periodic with period 2π in each variable.

International Journal of Mathematics and Mathematical Sciences
The double Fourier series of a function f x, y which is analogue for two variables of the series 1.1 , is given by f x, y  x, y corresponding to 1.7 , and 1.8 are defined as 1.9 We will consider the symmetric square partial sum of series 1.8 .The double series ∞ m 0 ∞ n 0 p m,n with the sequence of m, n th partial sums s m,n is said to summable by double matrix summability method or summable T, S if t m,n tends to a limit s as m → ∞ and n → ∞.

International
The regularity conditions of double matrix summability means are given by iii N, p m , q n summability mean 2 if a m,j p m−j /P m and b n,k q n−k /Q n , provided P m m j 0 p j / 0 and Q n n k 0 q k / 0. Double matrix summability method T, S is assumed to be regular throughout this paper.

Main Theorem
Rajagopal 1 previously proved a theorem on the N örlund summability of Fourier series.Result of Rajagopal 1 contained various results due to Hardy 2 , Hirokowa 3 , Hirokowa and Kayashima 4 , Pati 5 , Siddiqui 6 , and Singh 7 .Thereafter Sharma 8 proved a theorem dealing with the harmonic summability of double Fourier series.The result of Sharma 8 is a generalization of the theorem due to Hille and Tamarkin 9 for double Fourier series and also is analogous to the theorem of Chow 10 for summability C, 1, 1 of the double Fourier series.The theorem of Sharma 8 was generalized by Mishra 11 for double N örlund summability.The result Mishra 11 was generalized by Okuyama and Miyamoto 12 .But nothing seems to have been done so far to study double matrix summability of conjugate Fourier series.Therefore, the purpose of this paper is to establish the following theorem.Let T a m,j and S b n,k be two infinite triangular matrices with

2.1
If the conditions International Journal of Mathematics and Mathematical Sciences 5 hold, then the double conjugate Fourier series 1.8 is double matrix T, S summable to, f x, y , where at every point where these integrals exist provided ξ x and χ y are two positive monotonic increasing functions of x and y such that ξ m → ∞, as m → ∞ and χ n → ∞, as n → ∞, 2.5

3.2
Lemma 3.2.One has Proof.This can be proved similar to Lemma 3.1.Proof.This is similar to Lemma 3.3.

3.7
Lemma 3.6.One has Proof.It can be proved similar to Lemma 3.5 but using Lemma 3.4.

Proof of The Theorem
The j, k th partial sums s j,k x, y of the series 1.8 is given by

4.12
Thus, we get

4.23
International Journal of Mathematics and Mathematical Sciences 13 Therefore,  This completes the proof of the theorem.

The conjugate functions f 1 x, y , f 2 x
n α m,n cos mx cos ny β m,n sin mx cos ny γ m,n cos mx sin ny δ m,n sin mx sin ny y cos mx cos nydxdy 1.5 with three similar expressions for m, n 0, 1, 2, . . .and for β m,n , γ m,n and δ m,n where Q represents the fundamental square −π, −π; π, π .One can associate three conjugate series to the double Fourier series 1.4 in the following way: n −β m,n cos mx cos ny α m,n sin mx cos ny − δ m,n cos mx sin ny γ m,n sin mx sin ny , n −γ m,n cos mx cos ny − δ m,n sin mx cos ny α m,n cos mx sin ny β m,n sin mx sin ny , n cos mx cos ny − γ m,n sin mx cos ny − β m,n cos mx sin ny α m,n sin mx sin ny , 1.8where λ m,n 1, λ m,0 λ 0,n 1/2, m, n ≥ 1, λ 0,0 1/4., y and f

0 ∞n
Journal of Mathematics and Mathematical Sciences 3 Let T a m,j and S b n,k be two infinite triangular matrices.Let ∞ m 0 p m,n be a double series with s m,n m j 0 n k 0 p j,k as its m, n th partial sums.The double matrix mean t m,n is given by

Theorem 2 . 1 .
Let a m,j m j 0 and b n,k n k 0 be two real nonnegative and nondecreasing sequences with j ≤ m and k ≤ n, respectively.

Lemma 3 .3 see 2 . 4 Lemma 3 . 4 .
If a m,u is non-negative and non-decreasing with μ, then, for 0 ≤ a ≤ b ≤ ∞, 0 ≤ s ≤ π and any m, b μ a a m,m−μ e i m−μ s O A m,σ .3.If b n,v is non-negative and non-decreasing with ν, then, for 0 ≤ a ≤ b ≤ ∞, 0 ≤ t ≤ π and any n,

m j 1 a
m,m−j Re e i m−j s e i s/2

•
ψ s, t K m s K n t ds dtI 1 I 2 I 3 I 4say .
International Journal of Mathematics and Mathematical Sciences o 1 , as m → ∞, n → ∞.