Research Article On Double Summability of Double Conjugate Fourier Series

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1 Inernaional Journal of Mahemaic and Mahemaical Science Volume 22, Aricle ID 4592, 5 page doi:.55/22/4592 Reearch Aricle On Double Summabiliy of Double Conjugae Fourier Serie H. K. Nigam and Kuum Sharma Deparmen of Mahemaic, Faculy of Engineering and Technology, Mody Iniue of Technology and Science, Deemed Univeriy, Laxmangarh 3323, Sikar, Rajahan, India Correpondence hould be addreed o Kuum Sharma, kuum3harma@rediffmail.com Received 2 January 22; Acceped March 22 Academic Edior: Ram U. Verma Copyrigh q 22 H. K. Nigam and K. Sharma. Thi i an open acce aricle diribued under he Creaive Common Aribuion Licene, which permi unrericed ue, diribuion, and reproducion in any medium, provided he original work i properly cied. For he fir ime, a heorem on double marix ummabiliy of double conjugae Fourier erie i eablihed.. Inroducion The Fourier erie of f x i given by f x a 2 a n co nx b n in nx. n. Conjugae o he erie. i given by a n co nx b n in nx n.2 and i known a conjugae Fourier erie. I i well known ha he correponding conjugae funcion of.2 i defined a f x π f x f x. 2an /2.3 Le f x, y i inegrable Lver he quare Q π, π; π, π and i periodic wih period 2π in each variable.

2 2 Inernaional Journal of Mahemaic and Mahemaical Science The double Fourier erie of a funcion f x, y which i analogue for wo variable of he erie., i given by f x, y [ λ m,n αm,n co mx co ny β m,n in mx co ny γ m,n co mx in ny δ m,n in mx in ny λ m,n A m,n x, y, m n m n α m,n π 2 f x, y co mx co nydxdy Q.4.5 wih hree imilar expreion for m, n,, 2,... and for β m,n,γ m,n and δ m,n where Q repreen he fundamenal quare π, π; π, π. One can aociae hree conjugae erie o he double Fourier erie.4 in he following way: m n [ λ m,n βm,n co mx co ny α m,n in mx co ny δ m,n co mx in ny γ m,n in mx in ny,.6 [ λ m,n γm,n co mx co ny δ m,n in mx co ny α m,n co mx in ny β m,n in mx in ny, m n.7 [ δm,n co mx co ny γ m,n in mx co ny β m,n co mx in ny α m,n in mx in ny, m n.8 where λ m,n, λ m, λ,n /2, m,n, λ, /4. The conjugae funcion f x, y, f 2 x, y and f 3 x, y correponding o.7,and.8 are defined a f x, y f x, y f x, y d, π 2an /2 f 2 x, y f x, y f x, y, π 2an /2 f x, y [ f x, y f x, y f x, y f x, y d. π 2 2an /2 2an /2.9 We will conider he ymmeric quare parial um of erie.8.

3 Inernaional Journal of Mahemaic and Mahemaical Science 3 Le T a m,j and S b n,k be wo infinie riangular marice. Le m n p m,n be a double erie wih m,n m n j k p j,k a i m, n h parial um. The double marix mean m,n i given by m n m,n a m,j b n,k j,k, j k. The double erie m n p m,n wih he equence of m, n h parial um m,n i aid o ummable by double marix ummabiliy mehod or ummable T, S if m,n end o a limi a m and n. The regulariy condiion of double marix ummabiliy mean are given by m j k lim m,n lim m,n n a m,j b n,k a m,n, n am,j b n,k for each j, 2, 3,..., k m a m,j b n,k for each k, 2, 3,... j. We wrie ψ x, y ψ x, y;, f x, y f x, y f x, y f x, y, Ψ x, y x y ψ, d, x Ψ x, ψ, d, y Ψ 2, y ψ,, τ inegral par of, σ inegral par of, K m m co j /2 a m,j, 2π in /2 j K n n co k /2 b n,k, 2π in /2 k.2 Imporan paricular cae of he double marix ummabiliy mehod are i C,, ummabiliy mean if a m,j / m for all m and b n,k / n for all n;

4 4 Inernaional Journal of Mahemaic and Mahemaical Science ii H,, ummabiliy mean 5 if a m,j / m j log m and b n,k / n k log n; iii N, p m,q n ummabiliy 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 p j / andq n n k q k /. Double marix ummabiliy mehod T, S i aumed o be regular hroughou hi paper. 2. Main Theorem Rajagopal previouly proved a heorem on he Nörlund ummabiliy of Fourier erie. Reul of Rajagopal conained variou reul due o Hardy 2, Hirokowa 3, Hirokowa and Kayahima 4, Pai 5, Siddiqui 6, and Singh 7. Thereafer Sharma 8 proved a heorem dealing wih he harmonic ummabiliy of double Fourier erie. The reul of Sharma 8 i a generalizaion of he heorem due o Hille and Tamarkin 9 for double Fourier erie and alo i analogou o he heorem of Chow for ummabiliy C,, f he double Fourier erie. The heorem of Sharma 8 wa generalized by Mihra for double Nörlund ummabiliy. The reul Mihra wa generalized by Okuyama and Miyamoo 2. Bu nohing eem o have been done o far o udy double marix ummabiliy of conjugae Fourier erie. Therefore, he purpoe of hi paper i o eablih he following heorem. Theorem 2.. Le a m,j m j and b n,k n k be wo real nonnegaive and nondecreaing equence wih j m and k n, repecively. Le T a m,j and S b n,k be wo infinie riangular marice wih a m,j, a m,j, j > m; b n,k, b n,k, k > n; σ A m,σ a m,j ; τ B n,τ b n,k ; j k A m,m m ; B n,n n. 2. If he condiion x y Ψ x, y ψ, d { x O ξ /x { } x ψ x, O, ξ /x { y ψ 2, y d O χ /y } y χ /y, 2.2 } 2.3

5 Inernaional Journal of Mahemaic and Mahemaical Science 5 hold, hen he double conjugae Fourier erie.8 i double marix T, S ummable o, f x, y, where f x, y d 4π 2 an /2 an /2 2.4 a every poin where hee inegral exi provided ξ x and χ y are wo poiive monoonic increaing funcion of x and y uch ha ξ m, a m and χ n, a n, m A m, ξ n d O, m, B n, O, n. χ Lemma Lemma 3.. One ha K m O for m. 3. Proof. For /m, in /2 /π and co m, K m m co j /2 2 a m,j in /2 j m co j /2 a 2 m,j in /2 j π m a m,j j O. Lemma 3.2. One ha K n O for n Proof. Thi can be proved imilar o Lemma 3.. Lemma 3.3 ee 2. If a m,u i non-negaive and non-decreaing wih μ, hen, for a b, π and any m, b a m,m μ e i m μ O A m,σ. μ a 3.4

6 6 Inernaional Journal of Mahemaic and Mahemaical Science Lemma 3.4. If b n,v i non-negaive and non-decreaing wih, hen, for a b, π and any n, b b n,n e i n O B n,τ. a 3.5 Proof. Thi i imilar o Lemma 3.3. Lemma 3.5. One ha K m O Am,σ for < < π. m 3.6 Proof. Since /m < π, in /2 /π, K m m co j /2 2π a m,j in /2 j m { 2 a m,m j Re e i m j /2 } j m 2 a m,m j Re e i m j e i /2 j m 2 Re m,m j e j a i m j O O A m,σ by Lemma 3.3 Am,σ O. 3.7 Lemma 3.6. One ha K n O Bn,τ, for < <π. 3.8 n Proof. I can be proved imilar o Lemma 3.5 bu uing Lemma Proof of The Theorem The j, k h parial um j,k x, yf he erie.8 i given by π j,k x, y f x, y 4π 2 ψ, co j /2 in /2 co k /2 d. in /2 4.

7 Inernaional Journal of Mahemaic and Mahemaical Science 7 Then, m j k n { } a m,j b n,k j,k x, y f x, y 4π 2 m n co j /2 φ, a m,j b n,k in /2 j k co k /2 d in /2 4.2 or π m,n x, y f x, y ψ, K m K n d η η η I I 2 I 3 I 4 ay. η ψ, K m K n d 4.3 We conider I /m /n /m /n /m η η ψ, K m K n d /m /n /n 4.4 I. I.2 I.3 I.4 ay, where, η and 2.2 hold. Now conider I. /m /n O /m /n O mn ψ, K m K n d /m /n ψ, d by Lemma 3. and3.2 ψ, d O mn ψ m, n { } O mn o by 2.2 mnξ m χ n, a m,n. 4.5

8 8 Inernaional Journal of Mahemaic and Mahemaical Science Now, I.2 /m /n { /n O n /n K n O n O ψ, Km K n d /m /m [ /n A m, / [ /n ψ, A m, / d by Lemma 3.5 ψ, A } m, / d by Lemma 3.2 [ /n ψ, O mn A m,m ψ m, /m d d I.2. I.2.2 I.2.3 ay. Am, / ψ, d 4.6 Then, [ /n [ /n I.2. I.2.2 O n ψ, O mn ψ m, O n Ψ, O mn Ψ n m, n O n o ξ / /n /mn O mn o χ n ξ m χ n, a m,n. 4.7 Nex, I.2.3 O n O n /n [ O n /m [ /m /n /m /n Ψ, A m, / d O n 2 Ψ, O n Ψ, /m n O n /m ξ / [ O n /m ξ / Ψ, Am, / 2 nχ n Am, / nχ n 2 d d d /n Am, / d /m Ψ, d O n Ψ, d Am, / d /m n d Am, / 2 d d d Am, / d d Am, / d d

9 Inernaional Journal of Mahemaic and Mahemaical Science 9 [ [ m χ n [ χ n / o [ A m, ξ d χ n [ ξ m χ n, a m,n. A m,n [ m / d ξ d A m, d 4.8 Thu, I.2, a m,n. 4.9 Similarly I.3, a m,n. 4. Now, [ I.4 O η ψ, A m, / Bn, / d /m /n [ Bn, /η A m, / Ψ, η O mb n, /η A m,m Ψ /m, η η η B n, /η ψ, η d Am, / d nb n,na m, / Ψ, /n η /m d mn B n,n A m,m Ψ m, nb n,n Ψ, d Am, / d n /m n d ξ Bn, / d ma m,m Ψ 2 /n m, Bn, / A m, / Ψ, d /m { η Ψ, d } Am, / d Bn, / d /m /n d I.4. I.4.2 I.4.3 I.4.4 I.4.5 I.4.6 I.4.7 I.4.8 I.4.9 ay. 4. Then, [ Bn, /η A m, / Ψ, η I.4. I.4.2 O mb n, /η A m,m Ψ /m, η η η [ [ { } Bn, /η A m, / α / β /η mb n, /η mξ m η χ /η [ [ Bn, /η A m, / ξ / χ /η B n, /η ξ m η χ /η

10 Inernaional Journal of Mahemaic and Mahemaical Science [ ξ m, a m,n by he regulariy of T, S, I.4.3 B n, /η Ψ, η d Am, / d η /m d Bn, /η Am, / η /m ξ / η χ /η d 2 Bn, /η o η /m ξ / η χ /η d Am, / d d Bn, /η χ /η A m, / Bn, /η d /m ξ / χ /η d Am, / d /m ξ / d Bn, /η m χ /η A m, Bn,/η m d / ξ χ /η / ξ d A m, Bn, /η χ /η Bn, /η O χ /η O A m,m, a m,n. 4.2 Thu, we ge I.4.3, a m,n. 4.3 Similarly I.4.4, a m,n. 4.4 Now, I.4.5 mnb n,n A m,m Ψ m, n mnb n,n A m,m mnξ m χ n, a m,n. 4.5

11 Inernaional Journal of Mahemaic and Mahemaical Science Conider I.4.6 nb n,n Ψ, d Am, / d /m n d nb n,n Ψ, Am, / d nb /m n 2 n,n Ψ, d Am, / d /m n d nb n,n o /m ξ / nχ η Am, / d 2 nb n,n /m ξ / nχ η d Am, / d d nbn,n A m, / d d Am, / d nχ n /m ξ / χ n /m ξ / d, a m,n. 4.6 A imilar o I.4.3, I.4.7, a m,n. 4.7 A imilar o I.4.6, I.4.8, a m,n. 4.8 Now, [{ η I.4.9 O Ψ, /m /n Am, / 2 d } { Bn, / Am, / d 2 d } Bn, / d I.4.9. I I I ay. 4.9 Then, [ η { Am, / I.4.9. I O Ψ, /m /n 2 [ O /m η /n [ η O o /m /n { Am, / Ψ, { ξ / } Bn, / d 2 2 χ / d } Bn, / d } Am, / 2 Bn, / d 2

12 2 Inernaional Journal of Mahemaic and Mahemaical Science [ η { O o /m /n ξ / χ / [ η A m, / B n, / d /m /n ξ / χ / [ η A m, / d ξ / χ / [ m / /m [ m /n A m, ξ d / n /η A n m, ξ d /η B n, χ } Am, / 2 d Bn, / d Bn, / d d Bn, / χ n O d Bn, / /η, a m,n. 4.2 Similarly, I.4.9.3, a m,n. 4.2 Now, [ I O [ [ η /m η /m η /m Ψ, d Am, / /n d /n ξ / d χ / d /n ξ / χ / d Bn, / d Am, / d Bn, / d d d Am, / Bn, / d d d Bn, / [ d η Am, / d /m ξ / d /n χ / [ O d η Am, / d O d Bn, / d /m d /n, a m,n Hence, I.4, a m,n. 4.23

13 Inernaional Journal of Mahemaic and Mahemaical Science 3 Therefore, I, a m,n Now, /m < < π, /n < η < π, huweobain I 3 K m d /n ψ, Kn K m d ψ, Kn /n 4.25 I 3. I 3.2 ay. Uing Lemma 3.2 and Lemma 3.5, we have I 3. A m,σ O n O n O χ n d /n A m,σ d Ψ 2, n ψ, /n ψ, d, a m,n Uing Lemma 3.5 and Lemma 3.6, [ I 3.2 O A m,σ [ O d [ O d { { η d /n ψ, B n, τ Ψ 2, B n, τ Ψ 2, η B n, /n η η /n η Ψ 2, n Ψ, d /n } nb n,n Bn, τ }

14 4 Inernaional Journal of Mahemaic and Mahemaical Science [ η d Ψ 2, d /n [ B n, /n O Ψ 2, η d η [ O Bn, τ η d Ψ 2, d /n [ O n Ψ 2, d n Bn, τ imilar o I.4.9, a m,n Hence, I 3, a m,n Similarly, I 2, a m,n By he regulariy condiion of marix mehod and Riemann-Lebegue heorem, I 4, a m,n. 4.3 Therefore, m,n x, y f x, y 4.3 Thi complee he proof of he heorem. Reference C. T. Rajagopal, On he Nörlund ummabiliy of Fourier erie, Mahemaical Proceeding of he Cambridge Philoophical Sociey, vol. 59, pp , G. H. Hardy, On he ummabiliy of Fourier erie, Proceeding London Mahemaical Sociey, vol. 2, pp , H. Hirokawa, On he Nörlund ummabiliy of Fourier erie and i conjugae erie, Proceeding of he Japan Academy, vol. 44, pp , H. Hirokawa and I. Kayahima, On a equence of Fourier coefficien, Proceeding of he Japan Academy, vol. 5, pp , T. Pai, A generalizaion of a heorem of Iyengar on he harmonic ummabiliy of Fourier erie, Indian Journal of Mahemaic, vol. 3, pp. 85 9, J. A. Siddiqui, On he harmonic ummabiliy of Fourier erie, Proceeding of he Naional Iniue of Science of India, vol. 28, pp , T. Singh, On Nörlund ummabiliy of Fourier erie and i conjugae erie, Proceeding of he Naional Iniue of Science of India A, vol. 29, pp , 963.

15 Inernaional Journal of Mahemaic and Mahemaical Science 5 8 P. L. Sharma, On he harmonic ummabiliy of double Fourier erie, Proceeding of he American Mahemaical Sociey, vol. 9, pp , E. Hille and J. D. Tamarkin, On he ummabiliy of Fourier erie. I, Tranacion of he American Mahemaical Sociey, vol. 34, no. 4, pp , 932. Y. S. Chow, On he Ceàro ummabiliy of double Fourier erie, The Tohoku Mahemaical Journal, vol. 5, pp , 954. K. N. Mihra, Summabiliy of double Fourier erie by double Nörlund mehod, Bullein of he Iniue of Mahemaic. Academia Sinica, vol. 3, no. 3, pp , Y. Okuyama and I. Miyamoo, On he double Nörlund ummabiliy of double Fourier erie, Tamkang Journal of Mahemaic, vol. 27, no. 2, pp , 996.

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