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1 Int. J. Nonliner Anl. Appl. (0 No., 6 74 ISSN: (electronic ON ABSOLUTE GENEALIZED NÖLUND SUMMABILITY OF DOUBLE OTHOGONAL SEIES XHEVAT Z. ASNIQI Abstrct. In the pper Y. Ouym, On the bsolute generlized Nörlund summbility of orthogonl series, Tmng J. Mth. Vol. 33, No., (00, 6-65] the uthor hs found some sufficient conditions under which n orthogonl series is summble N, p, q lmost everywhere. These conditions re expressed in terms of coefficients of the series. It is the purpose of this pper to extend this result to double bsolute summbility N (, p, q, (.. Introduction nd Preliminries Let n0 n be given infinite series with sequence of prtil sums s n }. Then, let p denotes the sequence p n }. For two given sequences p nd q, the convolution (p q n is defined by (p q n p m q n m p n m q m. m0 When (p q n 0 for ll n, the generlized Nörlund trnsform of the sequence s n } is the sequence t p,q n } obtined by putting t p,q n p n m q m s m. (p q n The infinite series n0 n is bsolutely summble (N, p, q, if the series n0 t p,q n m0 t p,q n converges (t 0, nd we write in brief n N, p, q. n0 m0 The N, p, q summbility ws introduced by Tn ]. Dte: eceived: Jnury 0; evised: My Mthemtics Subject Clssifiction. Primry 4C5, 4B08. ey words nd phrses. Double orthogonl series, Double Nörlund summbility. : Corresponding uthor. 6

2 ON ABSOLUTE GENEALIZED NÖLUND SUMMABILITY 63 Let ϕ n (x} be n orthonorml system defined in the intervl (, b. We ssume tht f(x belongs to L (, b nd f(x n ϕ n (x, (. n0 where n f(xϕ n(xdx, (n 0,,,.... Let us write Also we put n : (p q n, j n : P n : (p n mj p n m q m, nd n+ n 0, 0 n n. p m nd Q n : ( q n m0 q m. m0 Y. Ouym 3] mong others proved the following two theorems: Theorem.. If the series } n j j n j n n n0 j converges, then the orthogonl series n ϕ n (x n0 is summble N, p, q lmost everywhere. Theorem.. Let Ω(n} be positive sequence such tht Ω(n/n} is nonincresing sequence nd the series n converges. Let p nω(n n } nd q n } be non-negtive. If the series n n Ω(nw(n converges, then the orthogonl series n0 nϕ n (x N, p, q lmost everywhere, where w(n is defined by w(j : j nj n n j n j n n. Let φ (x : m, n 0,,... } be n orthogonl system on (, b. The orthogonl development of ny rel function f(x of clss L with respect to the system φ (x} is given by φ (x, (. where f(xφ (xdx, (m, n 0,,.... The series (. shll be referred to s the double Fourier series of f(x. Our min purpose in this pper is to study the bsolute summbility with index,, of the series (., nd to deduce some corollries from the min results. Before doing this we introduce some notions nd nottions.

3 64 XHEVAT Z. ASNIQI Let be given double infinite series with its prtil sums s }. Then, let p denotes the sequence p }. For two given sequences p nd q, the convolution (p q we define by : (p q p i q m i,n p m i,n q i. i0 0 Liewise we need the following nottions: : p m i,n q i ; i ; ν,n m,n ν,n m,n 0, 0 ν m; m,µ m,n m,µ m,n 0, 0 µ n; m,n m,n + m,n. m,n m,n m,n : When (p q 0, the generlized Nörlund trnsform of the sequence s } is the sequence t p,q } defined by t p,q p m i,n q i s i. (p q i0 The double infinite series is bsolutely summble (N (, p, q,, if the series converges with the greement tht 0 ( t p,q t p,q m,n t p,q m,n + t p,q m,n t p,q m, t p,q,n t p,q, 0, m, n 0,,..., nd we write in brief N (, p, q. Throughout this pper denotes positive constnt tht depends on, nd it my be different in different reltions. The min result is the following.. Min esults Theorem.. If the series m ( m n( }, },

4 ON ABSOLUTE GENEALIZED NÖLUND SUMMABILITY 65 nd m ( } converge for, then the orthogonl series φ (x is summble N (, p, q lmost everywhere. Proof. Let < <. For the generlized Nörlund trnsform t(x p,q of the prtil sums of the orthogonl series φ (x we hve tht t p,q (x i0 i0 p m i,n q i s i (x 0 p m i,n q i 0 µ0 φ (x µ0 i µ0 φ (x, φ (x p m i,n q i where s i (x re prtil sums of order (i, of the series (.. Thus, since ν,n m,n ν,n m,n 0, 0 ν m, nd m,µ m,n m,µ m,n 0, 0 µ n, we obtin tht

5 66 XHEVAT Z. ASNIQI t p,q (x t p,q (x t p,q m,n (x tm,n(x p,q + t p,q φ (x µ0 m,n m,n + m,n µ0 µ0 µ0 m,n φ (x m,n φ (x m,n φ (x m,n (x m,n m,n + m,n m,n m,n m,n ( + m,n m,n + m,n m,n µν + µν m,n µν m,n + µν m,n m,n φ (x + φ (x + m,n m,n φ (x m,n m,n φ (x φ (x φ (x. Applying twice the inequlity α + β r r ( α r + β r for r, then the well-nown Hölder s inequlity with p orthogonlity we hve tht >, q such tht p + q pq, nd

6 ON ABSOLUTE GENEALIZED NÖLUND SUMMABILITY 67 p,q t (x dx φ (x dx ( φ (x dx ( + ( + φ (x dx φ (x dx φ (x dx m φ (x dx + + m } } }. Hence, the series ( + p,q t (x dx m ( m + n( m ( } } } (.

7 68 XHEVAT Z. ASNIQI p,q converges since the lst do by the ssumption. Since the functions t (x re nonnegtive, then B. Levi s theorem (see 4], pge 77 implies tht the series ( t p,q (x converges lmost everywhere. For, nd we do the sme resoning s bove using Schwrz s inequlity insted of Hölder s inequlity. The proof of the theorem. is completed. Now we shll prove generl theorem tht is consequence of the theorem.. Nmely, if we put w (0, (ν, µ; : w (,0 (ν, µ; : w (, (ν, µ; : n µ nµ m ν mν ( ] νn ; mν nµ νn mµ mµ ; ] (, then the following theorem holds true. Theorem.. Let Ω(m, n} be positive sequence such tht Ω(m, n/} is non-incresing sequence with respect to m nd n, nd the series Ω(m,n converges. Let p } nd q } be two non-negtive sequences. If the series m0 nω (m, nw (,0 (m, n;, nd 0n mω (m, nw (0, (m, n;, Ω(m, nw (, (m, n(m, n; converge, then the double orthogonl series m0 n0 φ (x N (, p, q, (, lmost everywhere.

8 Proof. From (. we hve ON ABSOLUTE GENEALIZED NÖLUND SUMMABILITY 69 ( + + p,q t (x dx m ( m n( m ( } } } : I + I + (. I 3. Applying Hölder s inequlity we obtin I ( (Ω(m, n Ω (m, n n n n Ω(m, n mω (m, n m } ( Ω (m, n mν Ω(ν, µ µ ν mν m µω (ν, µw (,0 (ν, µ; } } } mµ mµ }, (.3

9 70 XHEVAT Z. ASNIQI I ( (Ω(m, n Ω (m, n m Ω(m, n m nω (m, n m n nµ } Ω(ν, µ ν µ ( Ω (m, n nµ n νω (ν, µw (0, (ν, µ; } } νn νn } }, (.4 nd I 3 ( (Ω(m, n Ω (m, n Ω(m, n Ω (m, n mν nµ } ( Ω (m, n } }

10 ON ABSOLUTE GENEALIZED NÖLUND SUMMABILITY 7 Ω(ν, µ mν nµ Ω (ν, µw (, (ν, µ; ( ( } }. (.5 Using (., (.3, (.4, nd (.5 we hve tht ( p,q t (x dx, is finite by the ssumptions. This completes the proof bsed on the sme resoning s in the proof of theorem.. emr.3. For in theorems. nd. we exctly obtin the two dimensionl versions of the theorems. nd.. 3. Corollries A double infinite sequence u } will be clled fctorble if there exist sequences c m } nd d n } such tht u c m d n, nd we focus on this cse below. Corollry 3.. If the series p m p n( m P m P m P np n p m νp n µ ( Pm P ( m ν P n P ] } n µ p m p m ν p n p n µ converges for, then the orthogonl series φ (x is summble N (, p n p m lmost everywhere.

11 7 XHEVAT Z. ASNIQI Proof. The proof of the corollry follows from theorem. nd the fct tht µν m,n µν m,n + µν m,n m,n µν m,n m,n P m νp n µ P m νp n µ P m νp n µ P m P n P m P n P m P n ( Pm ν P m ν P n µ P m P m P n P m µ P n (P P m P m P np m P m ν P m P m ν n P m P m P np n ( ] (P n p n P n µ P n P n µ p n µ ( p m p n Pm P ( m ν P n P m P m P np n p m p m ν p n + P m νp n µ P m P n ( P n P n µ P np m µ ] (P m p m P m ν P m (P m ν p m ν P n µ p p m ν p n µ, n µ nd 0, 0, for ll q. Corollry 3.. If the series q m q n( m } Qν Q µ Q m Q m Q nq n converges for, then the orthogonl series φ (x is summble N (, q n q m lmost everywhere.

12 ON ABSOLUTE GENEALIZED NÖLUND SUMMABILITY 73 Proof. We hve (Q m Q ν Q n Q µ (Q m Q ν Q n Q µ Q m Q n Q m Q n (Q m Q ν ( Q n Q µ Q m Q n ( Q n Q µ Q m Q ν Q m Q n + (Q m Q ν ( Q n Q µ Q m Q n Q n Q µ Q n Q ( m Q ν Q n Q µ Q n Q µ Q m Q n Q n ( Qm Q ν Q ( m Q ν Q n Q µ Q m Q m Q n ( Q ν Q µ Q m Q m q mq nq ν Q µ Q m Q m Q nq n Q n Q n Q n Q µ Q n for ll p. In this cse, with the greement Q Q 0, we lso note tht 0, 0. Now the proof is n immedite result of the theorem.. Let q i, for ll 0 m, 0 i n, α, β >, nd r + j p m i,n E α i E β, where Ej r. j Then the th term of the (C, α, β trnsform of sequence s i } is defined by (see ] σ α,β E α EmE α n β m i Eβ n s i. i0 In this wy, N (, p, q summbility reduces to the (C, α, β summbility, nd for α β the following holds true. Corollry 3.3. If the series 0 m converges for, then the orthogonl series φ (x is summble (C,, lmost everywhere. }

13 74 XHEVAT Z. ASNIQI Proof. By convention E r 0 we hve E α m i Eβ n n EmE α n β m E α m i Eβ n + Em E α n β ( E α m i E α m Eα m i Em α m E α m i Eβ n E α me β n n ( E β n E β n ( m + ( m n + n m ν + m(m + n µ + n(n + ( ν ( µ m + n +, for ll q i. With this we hve finished the proof. E α m i Eβ n Em E α β n Eβ n E β n eferences. B. E. hodes, Absolute comprison theorems for double weighted men nd double Cesàro mens, Mth. Slovc, Vol. 48 (998, No. 3, M. Tn, On generlized Nörlund methods of summbility, Bull. Austrl. Mth. Soc. 9, (978, Y. Ouym, On the bsolute generlized Nörlund summbility of orthogonl series, Tmng J. Mth. Vol. 33, No., (00, F. Móricz, On the.e. convergence of the rithmetic mens of double orthogonl series, Trns. Amer. Mth. Soc. Vol. 97, No., (984, Deprtment of Mthemtics nd Computer Sciences, University of Prishtin Avenue Mother Theres 5, Prishtinë, 0000, OSOVË E-mil ddress: xhei00@hotmil.com