The Extension of the Theorem of J. W. Garrett, C. S. Rees and C. V. Stanojevic from One Dimension to Two Dimension

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1 Int. Journal of Math. Analysis, Vol. 3, 29, no. 26, The Extension of the Theorem of J. W. Garrett, C. S. Rees and C. V. Stanojevic from One Dimension to Two Dimension Jatinderdeep Kaur School of Mathematics & Computer Applications Thapar University, Patiala-1474, India S.S. Bhatia School of Mathematics & Computer Applications Thapar University, Patiala-1474, India Abstract In this paper, we extend the result of Garrett, Rees and Stanojevic [1] on the L 1 -convergence of the double cosine series from one dimensional to two dimensional series. Also, some results obtained by Moricz ([3], [4]) are particular cases of our results for double cosine series. Mathematics Subject Classication: Primary 42A2; Secondary 42A32 Keywords: L 1 -convergence, Dirichlet kernel, Fejer kernel 1 Introduction Consider the double cosine series f(x, y) = λ j λ k a jk cos jxcos ky (1.1) j= k= on the positive quadrant Q =[,π] [,π] of the two dimensional torus, where λ = 1 and λ 2 j = 1 for j =1, 2, 3,... and {a jk } is a double sequence of real numbers. We denote the rectangular partial sum of the series (1.1) by S mn i.e. m n S mn (x, y) = λ j λ k a jk cos jxcos ky (m, n ) j= k=

2 1252 J. Kaur and S. S. Bhatia and f(x, y) = lim m,n S mn. Denition 1.1. We say that {a jk } belongs to the class BV 2 if a jk as j + k, (1.2) and 11 a jk <. (1.3) j= k= where 11 a jk = a j,k a j+1,k a j,k+1 + a j+1,k+1 Condition (1.2) implies that {a jk } is a null sequence while (1.3) implies that {a jk } is a sequence of bounded variation. In addition to the above class, we introduce another class C 2 of coefcient sequences as: Denition 1.2. A null sequence {a jk } belongs to class C 2 if for every ɛ> there exists δ> such that for all m M and n N We have M N C(m, M; n, N; δ) := D j (x)d k (y) 11 a jk ɛ (1.4) Where D δ D δ := Q (δ, π] (δ, π] ={(x, y) : x, y π & min(x, y) δ} or D δ D j (x)d k (y) 11 a jk ɛ m, n. Concerning the L 1 -convergence of the double cosine series Moricz [2] proved the following result: Theorem 1.3. [2] If {a jk } BV 2 C 2, then the sum f(x, y) of series (1.1) belongs to L 1 (Q) and (1.1) is a Fourier series of f(x, y). Garrett, Rees and Stanojevic [1] introduced an equivalent class S 2 of the class S of Sidon [5] for one dimensional coefcient sequence {a k } dened as:

3 Double cosine series 1253 Denition 1.4. A null sequence {a k } belongs to the class S 2 if there exists a null-sequence {A k } of non-negative numbers such that k A k < and a k A k for all k. In this paper, we shall extend the class S 2 [1] of coefcient sequences from one dimensional to a new class S 2 d of two dimensional coefcient sequence {a jk} dened as: Denition 1.5. A double null sequence {a jk } belongs to Sd 2 if there exists a null sequence {A jk } of non-negative numbers such that and j= k= k=1 jk 11 A jk <, (1.5) 11 a jk A jk j, k. (1.6) The aim of this paper is to extend the corresponding result of Garrett, Rees and Stanojevic [1] from one dimensional to two-dimensional series and to obtain necessary and sufcient condition of L 1 -convergence of double cosine series. 2 Main Result We prove the following result: Theorem 2.1. Let {a jk } S 2 d. Then f L1 (Q) and S mn f = o(1), as m,n if and only if a mn ln(m +2)ln(n +2) as m + n. (2.1) Proof. First we shall show that the point wise limit f of S mn (x, y) = m j= k= exists in Q and that f is a Fourier series i.e. f L 1 (Q). From {a jk } S 2 d it follows that {a jk} BV 2. Indeed n λ j λ k a jk cos jxcos ky (2.2) 11 A jk j= k= jk 11 A jk <. j= k=

4 1254 J. Kaur and S. S. Bhatia On the other hand mna mn mn 11 A jk jk 11 A jk = o(1). (2.3) Performing double summation by parts, we have m n m 1 n 1 m 1 n 1 A jk = jk 11 A jk jn 1 A jn km 1 A mk + mna mn j= k= j= k= j= k= but and n 1 A jn = k 11 A jk m 1 A mk = j 11 A jk So, we get that A jk j= k= A jk <. Since {a jk } Sd 2 implies that 11a jk j, k, it follows that {a jk } BV 2, and hence (2.2) converges in Q to the point wise limit f. Now from theorem 1.3 we have that if {a jk } BV 2 C 2, then f L 1 (Q). Thus it sufces to prove that {a jk } S 2 d {a jk} C 2. Therefore, consider δ δ 11 a jk D j (x)d k (y) 11 a jk D j (x)d k (y) A jk D j (x)d k (y) Using double summation by parts, we get 11 A jk (j +1)F j (x)(k +1)F k (y)

5 Double cosine series 1255 N 1 A jn (j +1)F j (x)(n +1)F N (y) M 1 A Mk (M +1)F M (x)(k +1)F k (y) M,N MNA MN F M (x)f N (y) (2.4) where D n (x) and F n (x) represent the Dirichlet and Féjer kernel respectively. π Since {a jk } Sd 2 and F n (x) dx = π, we have 11 A jk (j +1)F j (x)(k +1)F k (y) = π2 (j + 1)(k +1) 11 A jk = o(1), 1 A jn (j +1)F j (x)(n +1)F N (y) = π2 (j + 1)(k +1) 11 A jk = o(1), k=n Similarly, 1 A Mk (M +1)F M (x)(k +1)F k (y) = π2 (j + 1)(k +1) 11 A jk = o(1). j=m Further, from (2.3), the last term on the right hand side of the (2.4) is of o(1). Hence, f L 1 (Q). Now, it remains to show that S mn f = o(1), as m,n. Therefore, consider f S mn = a jk cos jxcos ky Applying double summation by parts, we get 11 a jk D j (x)d k (y) 1 a jn D j (x)d N (y) N

6 1256 J. Kaur and S. S. Bhatia M 1 a Mk D M (x)d k (y) M,N a MN D M (x)d N (y) (2.5) Since {a jk } Sd 2 and D n (x) dx log n, the 1st, 2nd and 3rd terms on the right hand side of inequality (2.5) are o(1) as m, n. Thus the conclusion of the theorem follows if and only if (2.1) holds. Remark 1. (a) We note that Theorem 2.1 is an extension of the corresponding result of Garrett, Rees and Stanojevic [1] from one dimensional to two dimensional cosine series. (b) Now, to show that the results of Moricz ([3], [4]) are particular cases of our results for double cosine series, we note that (1.5) implies that A jk <, j= k= (as proved in Theorem 2.1) which in turn is equivalent to the condition 2 m A 2 m, + 2 n A,2 n + 2 m+n A 2 m,2 n < (2.6) m= n= m= n= further, the condition A of [3] is nite by making use of {a jk } Sd 2 and (2.6) and consequently the condition (1.11) of [3] is satised for all p>. Thus the conclusion of theorem 1 and of corollary 1 of [3] hold true in case {a jk } Sd 2. (c) We know that a coefcient sequence {a jk } is said to be quasi-convex if (j + 1)(k +1) 22 a jk <, and thus by setting we get j= k= A jk = 11 a jk 11 A jk = 11 ( 11 a jk )= 22 a jk which implies that Sd 2 contains all quasi-convex null sequence. Therefore, corollary 3 of [3] and Theorem 2 (i) of [4] are also particular cases of our result. References [1] J.W. Garrett, C.S. Rees and C.V. Stanojevic, L 1 -convergence of Fourier Series with coefcients of bounded variation,poceedings of American Mathematical Sciences, 8(3),(198),

7 Double cosine series 1257 [2] F. Móricz, Integrability of double trigonometric series with special coefcients, Analysis Mathematica, 16,(199), [3] F. Móricz, On the Integrability and L 1 -convergence of double trigonometric series, Studia Mathematica, 98(3),(1991), [4] F. Móricz, On the Integrability and L 1 -convergence of double trigonometric series. II, Acta Mathematica Hungarica, 69(1-2), (1995), [5] S. Sidon, Hinreichende Bedingungen für den Fourier-Charakter einer trigonometrischen Reihe, J. London Math. Soc., 14(1939), Received: December, 28