Orlicz strongly convergent sequences with some applications to Fourier series and statistical convergence

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1 NTMSCI 6, No., (208) 22 New Treds i Mathematical Scieces Orlicz strogly coverget sequeces with some applicatios to Fourier series ad statistical covergece uldip Raj ad Adem ilicma 2 Departmet of Mathematics Shri Mata Vaisho Devi Uiversity, atra-82320, J &, Idia 2 Departmet of Mathematics, Uiversity Putra Malaysia, UPM, Serdag, Selagor, Malaysia Received: 3 Jauary 208, Accepted: 4 March 208 Published olie: 6 March 208. Abstract: I the preset wor we itroduce ad study some strogly coverget sequece spaces of Orlicz fuctios usig ifiite matrix over -ormed spaces. We study some algebraic ad topological properties of these sequece spaces. A ecessary ad sufficiet coditio for strogly coverget sequeces is obtaied. Fially, we study some applicatios of these sequeces for Fourier series ad statistical covergece. eywords: Orlicz fuctio, paraorm space, Λ 2 -strogly coverget, statistical coverget, -ormed spaces. Itroductio ad preiaries Let l,c ad c 0, respectively be the Baach spaces of bouded, coverget ad ull sequeces x = (x ), ormed by x =sup x, where N, wheren=0,,2,... Mursalee ad Noma 20] itroduced the otio of λ -coverget ad λ -bouded sequeces. Let λ = λ =0 be a strictly icreasig sequece of positive real umbers tedig to ifiity. A sequece x = (x ) w is said to be λ -coverget to the umber L ad called the λ -it of x if Λ m (x) L as m, where Λ m (x)= λ m m =0 (λ λ )x. A sequece x=(x ) w is λ -bouded if sup Λ m (x) <. It is well ow 20] that if x m = a i the ordiary sese of m covergece, the This implies that m ( ( m )) m λ m (λ λ ) x a = 0. =0 m Λ m (x) a = m λ m m =0 (λ λ )(x a) =0, which gives that m Λ m (x) = a ad we say x = (x ) is λ -coverget to a. Here ad i the sequel, we shall use the covetio that ay term with a egative subscript is equal to zero, that is, λ = x = 0. Correspodig author ailic@upm.edu.my

2 23. Raj ad A. ilicma: Orlicz strogly coverget sequeces with some applicatios to Fourier series... I 4] órus gave a ew appropriate defiitio for the Λ 2 -strog covergece by geeralizig the origial Λ-strog covergece cocept give by Móricz 23]. Moreover, òrus 5] geeralized the results o the L covergece of Fourier series. Let Λ = λ : = 0,,... be a o-decreasig sequece of positive umbers tedig to. A sequece (x ) of complex umbers coverges Λ 2 -strogly to a complex umber x if Λ 2 (x) x= λ =0 λ (x x) λ 2 (x 2 x) =0, with the argumet λ = λ 2 = x = x 2 = 0. I this paper we use the idea of Λ 2 -strog covergece ad we study these type of covergece over -ormed spaces. I 97 Lidestrauss ad Tzafriri 6] first ivestigated Orlicz sequece spaces i detail with certai aims i Baach space theory. A Orlicz fuctio M : 0, ) 0, ) is a cotiuous, o-decreasig ad covex such that M(0) = 0, M(x) > 0 for x > 0 ad M(x) as x. A Orlicz fuctio M = (M ) is said to satisfy 2 -coditio if there exist costats a, > 0 ad a sequece c=(c ) = l + (the positive coe of l ) such that the iequality holds for all N ad u R +, wheever M (u) a. M (2u) M (u)+c Let w be the set of all real or complex sequeces. Lidestrauss ad Tzafriri 6] used the idea of Orlicz fuctio to costruct the sequece space, l M = x=(x ) w: = ( x ) M <, for some > 0 is ow as a Orlicz sequece space. The spacel M is a Baach space with the orm, x =if > 0 : = ( x ) M. Also it was show i 6] that every Orlicz sequece space l M cotais a subspace isomorphic to l p (p ). A sequece M =(M ) of Orlicz fuctios is said to be Musiela-Orlicz fuctio (see 7], 2]). Let X ad Y be two sequece spaces ad A=(a ) be a ifiite matrix of real or complex umbers a, where, N. The we say that A defies a matrix mappig from X ito Y if for every sequece x = (x ) X, the sequece Ax=A (x) is i Y, where A (x)=a x ( N), () coverges for each N. By(X,Y) we deote the class of all matrices A such that A : X Y. For a sequece space X, the matrix domai X A of a ifiite matrix A is defied by which is also a sequece space (see 30]). X A =x=(x ) w:ax X (2) A satisfactory theory of 2-ormed spaces was iitially developed by Gähler ] i the mid of 960 s, while that of

3 NTMSCI 6, No., (208) / ormed spaces oe ca see i Misia 8]. Sice the, this cocept has bee studied by may authors ad obtaied various results (see 2], 3]). Let N ad X be a liear space over the field of real umbersr of dimesio d, where d 2. A real valued fuctio,, o X satisfyig the followig four coditios: () x,x 2,,x =0 if ad oly if x,x 2,,x are liearly depedet i X, (2) x,x 2,,x is ivariat uder permutatio, (3) αx,x 2,,x = α x,x 2,,x for ay α R, ad (4) x+x,x 2,,x x,x 2,,x + x,x 2,,x is called a -orm o X, ad the pair(x,,, ) is called a -ormed space over the field R. For example, we may tae X =R beig equipped with the -orm x,x 2,,x E = the volume of the -dimesioal parallelopiped spaed by the vectors x,x 2,,x which may be give explicitly by the formula where x i =(x i,x i2,,x i ) R for each i=,2,,. x,x 2,,x E = det(x i j ), Let (X,,, ) be a -ormed space of dimesio d 2 ad a,a 2,,a be liearly idepedet set i X. The the followig fuctio,, o X as defied by x,x 2,,x = max x,x 2,,x,a i : i=,2,, is called a( )-orm o X with respect to a,a 2,,a. A sequece(x ) i a -ormed space(x,,, ) is said to coverges to some L X if x L,z,,z =0 for every z,,z X. A sequece(x ) i a -ormed space(x,,, ) is said to be Cauchy if x x p,z,,z =0 for every z,,z X.,p If every Cauchy sequece i X coverges to some L X, the X is said to be complete with respect to the -orm. Ay complete -ormed space is said to be -Baach space. Let X be a liear metric space. A fuctio p : X R is called paraorm, if () p(x) 0 for all x X, (2) p( x)= p(x) for all x X, (3) p(x+y) p(x)+ p(y) for all x,y X, (4) if(λ ) is a sequece of scalars with λ λ as ad(x ) is a sequece of vectors with p(x x) 0as, the p(λ x λ x) 0; as;. A paraorm p for which p(x)=0 implies x=0 is called total paraorm ad the pair (X, p) is called a total paraormed space. It is well ow that the metric of ay liear metric space is give by some total paraorm (see 33], Theorem 0.4.2, P-83). For more details about this type of sequece spaces (see ], 4], 22], 24], 25], 26], 27], 28], 32]) ad 5] - 9] refereces therei.

4 25. Raj ad A. ilicma: Orlicz strogly coverget sequeces with some applicatios to Fourier series... A sequece (x ) of complex umbers is said to Λ 2 -strogly coverget to a complex umber x with respect to a sequece of Orlicz fuctios ad a ifiite matrix if λ =0 ( λ (x x) λ 2 (x 2 x) )] p a u M,z,...,z = 0 for some > 0 with the agreemet λ = λ 2 = x = x 2 = 0. Let M =(M ) be a sequece of Orlicz fuctios, p=(p ) be a bouded sequece of positive real umbers ad u=(u ) be a sequece of strictly positive real umbers, A=(a ) be a ifiite matrix ad(x,,, ) is a ormed space. Let Λ =λ be a o-decreasig sequece of positive umbers tedig to. I the preset paper we defie the followig classes of sequeces: Λ 2,M,A,u, p,.,...,. ]= ad x=(x ) : λ =0 Λ 2,M,A,u, p,.,...,. ] 0 = x=(x ) : ( λ (x x) λ 2 (x 2 x) )] p a u M,z,...,z = 0;as;, λ =0 Λ 2,M,A,u, p,.,...,. ] = x=(x ) : sup ( λ x λ 2 x )] 2 p a u M,z,...,z = 0 as λ =0 ( λ x λ 2 x )] 2 p a u M,z,...,z <. Let us cosider a few special cases of the above classes of sequeces: If M(x)=x, the the sequecesλ 2,M,A,u, p,.,...,. ],Λ 2,M,A,u, p,.,...,. ] 0 adλ 2,M, A,u, p,.,...,. ] reduces toλ 2,A,u, p,.,...,. ],Λ 2,A,u, p,.,...,. ] 0 adλ 2,A,u, p,.,...,. ] as follows: ad Λ 2,A,u, p,.,...,. ]= x=(x ) : λ =0 Λ 2,A,u, p,.,...,. ] 0 = x=(x ) : ( λ (x x) λ 2 (x 2 x) )] p a u,z,...,z = 0 as, λ =0 Λ 2,A,u, p,.,...,. ] = x=(x ) : sup ( λ x λ 2 x )] 2 p a u,z,...,z = 0 as λ =0 If p = for all N, we shall write above sequeces as ad Λ 2,M,A,u,.,...,. ]= x=(x ) : Λ 2,M,A,u,.,...,. ] 0 = λ =0 x=(x ) : ( λ x λ 2 x )] 2 p a u,z,...,z <. ( λ (x x) λ 2 (x 2 x) a u M λ =0 Λ 2,M,A,u,.,...,. ] = x=(x ) : sup,z,...,z )]=0as, ( λ x λ 2 x 2 a u M,z,...,z )]=0as λ =0 ( λ x λ 2 x 2 a u M,z,...,z )]<.

5 NTMSCI 6, No., (208) / 26 The followig iequality will be used throughout the paper. If 0<h=if p p sup p = H, = max,2 H, the a + b p a p + b p (3) for all Nad a,b C. Also a p max, a H for all a C. The mai objective of this paper is to itroduce the cocept of strogly coverget sequece spaces of Orlicz fuctios usig ifiite matrix over -ormed spaces. We also mae a effort to study some topological properties ad prove some iclusio relatios betwee these sequece spaces. Fially, by usig the cocept of strog covergece we study statistical covergece ad results related to Fourier series. 2 Mai results Theorem. Let M =(M ) be a sequece of Orlicz fuctios, p=(p ) be a bouded sequece of positive real umbers ad u=(u ) be a sequece of strictly positive real umbers ad A=(a ) be a ifiite matrix. The the sequece spaces Λ 2,M,A,u, p,.,...,. ],Λ 2,M,A,u, p,.,...,. ] 0 adλ 2,M,A,u, p,.,...,. ] are liear over the complex fieldc. Proof. Suppose x=(x ) ad y=(y ) Λ 2,M,A,u, p,.,...,. ] 0. The ad λ =0 λ =0 a u M ( λ x λ 2 x 2,z,...,z )] p = 0 as for some > 0 a u M ( λ y λ 2 y 2 2,z,...,z )] p = 0 as for some 2 > 0. Let 3 = max(2 α,2 β 2 ). Sice M =(M ) is a o-decreasig ad covex so by usig iequality (3), we have λ =0 ( α(λ x λ 2 x 2 )+β(λ y λ 2 y 2 ) )] p a u M,z,...,z λ =0 2 p a + λ =0 2 p a λ a =0 3 u M ( α(λ x λ 2 x 2 ) u M ( β(λ y λ 2 y 2 ) 2,z,...,z )] p,z,...,z )] p ( α(λ x λ 2 x 2 ) )] p u M,z,...,z + ( β(λ y λ 2 y 2 ) )] p λ a u M,z,...,z =0 2 = 0 as. Thus, αx+β y Λ 2,M,A,u, p.,...,. ] 0. This proves that Λ 2,M,A,u, p.,...,. ] 0 is a liear space. Similarly, we ca prove that Λ 2,M,A,u, p,.,...,. ] adλ 2,M,A,u, p,.,...,. ] are liear spaces. Theorem 2. Let M =(M ) be a sequece of Orlicz fuctios, p=(p ) be a bouded sequece of positive real umbers ad u = (u ) a sequece of strictly positive real umbers ad A = (a ) be a ifiite matrix. The

6 27. Raj ad A. ilicma: Orlicz strogly coverget sequeces with some applicatios to Fourier series... Λ 2,M,A,u, p,.,...,. ] 0 is a paraormed space with paraorm g(x)=if () p : λ =0 a where H = sup p < ad = max,h. u M ( λ x λ 2 x 2,z,...,z )] p ], as, Proof. (i) Clearly g(x) 0 for x=(x ) Λ 2,M,A,u, p,.,...,. ] 0. By defiitio of Orlicz fuctio, we get g(0)=0. (ii) g( x)=g(x). (iii) Let x=(x ),y=(y ) Λ 2,M,A,u, p,.,...,. ] 0 there exist positive umbers ad 2 such that λ =0 ( λ x λ 2 x )] 2 p ] a u M,z,...,z ad λ =0 ( λ x λ 2 x )] 2 p ] a u M,z,...,z. 2 Let = + 2. The by usig Miowsi s iequality, we have ad thus λ =0 ( (λ x λ 2 x 2 )+(λ y λ 2 y 2 ) )] p a u M,z,...,z = λ a =0 = λ a =0 + λ a =0 ( (λ x λ 2 x 2 )+(λ y λ 2 y 2 ) )] p u M,z,...,z + 2 ( λ x λ 2 x )] 2 p u M,z,...,z + 2 ( λ y λ 2 y )] 2 p u M,z,...,z + 2 ) ( λ x λ 2 x )] 2 p + 2 λ a u M,z,...,z =0 ( λ y λ 2 y )] 2 p a u M,z,...,z ( ( 2 ) λ. =0 2 g(x+y)= if if + if () p : ( ) p : ( 2 ) p : λ =0 λ λ Therefore, g(x+y)= g(x)+g(y). =0 =0 a u M ( (λ x λ 2 x 2 )+(λ y λ 2 y 2 ) ( λ x λ 2 x )] 2 p ] a u M,z,...,z ( λ y λ 2 y )] 2 p ] a u M,z,...,z 2,z,...,z )] p ].

7 NTMSCI 6, No., (208) / 28 Fially, we prove that the scalar multiplicatio is cotiuous. Let λ be ay complex umber. By defiitio, g(µx)=if = if () p : ( µ t) p : λ =0 a λ =0 a where t = µ > 0. Sice µ p max(, µ sup p ), we have g(µx) max(, µ sup p )if t p : λ =0 u M ( µ(λ x λ 2 x 2 ) u M ( µ(λ x λ 2 x 2 ) a u M ( µ(λ x λ 2 x 2 ) t,z,...,z )] p ],z,...,z )] p ],z,...,z )] p ] So, the fact that the scalar multiplicatio is cotiuous follows from the above iequality. This completes the proof of the theorem. Theorem 3. Suppose M = (M ), M = (M ), M = (M ) are sequeces of Orlicz fuctios, p = (p ) be a bouded sequece of positive real umbers, u=(u ) be a sequece of strictly positive real umbers, A=(a ) be a ifiite matrix ad 0<h=if p p sup p = H <. The (i) Λ 2,M,A,u, p,.,...,. ] 0 Λ 2,M M,A,u, p,.,...,. ] 0, (ii) Λ 2,M,A,u, p,.,...,. ] 0 Λ 2,M,A,u, p,.,...,. ] 0 Λ 2,M +M,A,u, p,.,...,. ] 0.,. Proof. (i) Let x Λ 2,M,A,u, p,.,...,. ] 0. The, we have λ =0 ( λ x λ 2 x )] 2 p a u M,z,...,z = 0. Let ε > 0 ad ( choose δ > 0 with )] 0 < δ < such that f (t) < ε for 0 t δ. We write y = a u M λ x λ 2 x 2,z,...,z ad let us cosider =0 M (y )] p = M (y )] p +M (y )] p, where the first summatio is over y δ ad the secod over y > δ. Sice M =(M ) is cotiuous, we have ad for y > δ, we use the fact that By defiitio of Orlicz fuctio, we have for y > δ, Hece, 2 λ M (y )] p < ε H (4) ( λ M (y )] p max 2 y < y δ + y δ. M (y )<2M () y δ., ( 2M ()δ ) H) λ =0y ] p. (5)

8 29. Raj ad A. ilicma: Orlicz strogly coverget sequeces with some applicatios to Fourier series... So by equatios (4) ad (5), we haveλ 2,M,A,u, p,.,...,. ] 0 Λ 2,M M,A,u, p,.,...,. ] 0. (ii) Let x Λ 2,M,A,u, p,.,...,. ] 0 Λ 2,M,A,u, p,.,...,. ] 0. The usig iequality (3) it ca be show that x Λ 2,M +M,A,u, p,.,...,. ] 0. Hece,Λ 2,M,A,u, p,.,...,. ] 0 Λ 2,M,A,u, p,.,...,. ] 0 Λ 2,M +M,A,u, p,.,...,. ] 0. Corollary. Suppose M = (M ), M = (M ), M = (M ) are sequeces of Orlicz fuctios, p =(p ) be a bouded sequece of positive real umbers ad u=(u ) be a sequece of strictly positive real umbers. The (i) Λ 2,M,A,u, p,.,...,. ] Λ 2,M M,A,u, p,.,...,. ], (ii) Λ 2,M,A,u, p,.,...,. ] Λ 2,M,A,u, p,.,...,. ] Λ 2,M +M,A,u, p,.,...,. ], (iii) Λ 2,M,A,u, p,.,...,. ] Λ 2,M M,A,u, p,.,...,. ], (iv) Λ 2,M,A,u, p,.,...,. ] Λ 2,M,A,u, p,.,...,. ] Λ 2,M +M,A,u, p,.,...,. ]. Proof. It is easy to prove by usig Theorem (3), so we omit the details. Theorem 4. Let M =(M ) be a sequece of Orlicz fuctios. The for ay two sequeces p=(p ) ad t =(t ) of strictly positive real umbers, we have (i) Λ 2,M,A,u, p,.,...,. ] 0 Λ 2,M,A,u,t,.,...,. ] 0 φ, (ii) Λ 2,M,A,u, p,.,...,. ] Λ 2,M,A,u,t,.,...,. ] φ, (iii) Λ 2,M,A,u, p,.,...,. ] Λ 2,M,A,u,t,.,...,. ] φ. Proof. (i) Sice the zero elemet belogs to Λ 2,M,A,u, p,.,...,. ] 0 ad Λ 2,M,A,u,t,.,...,. ] 0, thus the itersectio is o-empty. Similarly, we ca prove (ii) ad (iii). Propositio. Let M =(M ) be a sequece of Orlicz fuctios, p=(p ) be a bouded sequece of positive real umbers ad u=(u ) be a sequece of strictly positive real umbers. The, we have (i) Λ 2,A,u, p,.,...,. ] 0 Λ 2,M,A,u, p,.,...,. ] 0, (ii) Λ 2,A,u, p,.,...,. ] Λ 2,M,A,u, p,.,...,. ], (iii) Λ 2,A,u, p,.,...,. ] Λ 2,M,A,u, p,.,...,. ]. Proof. It is obvious so, we omit the details. Theorem 5. Let 0< p r ad( r p ) be bouded, theλ 2,M,A,u,r,.,...,. ] Λ 2,M,A,u, p,.,...,. ]. ( )] Proof. Let x Λ 2 (λ,m,a,u,r,.,...,. ], t = a u M (x x) λ 2 (x 2 x)) r,z,...,z ad µ = N so that 0< µ µ. Defie the sequece(v ) ad(w ) as follows: For t, let v = t ad w = 0 ad for t <, let v = 0 ad w = t. The, clearly for all N, we have t = v + w,t µ = v µ + wµ,vµ v t ad w µ w µ. Therefore, λ =0 t µ ] µ. λ + =0t λ w =0 ( ) p r for all Hece, x Λ 2,M,A,u, p,.,...,. ]. Thus,Λ 2,M,A,u,r,.,...,. ] Λ 2,M,A,u, p,.,...,. ]. This completes the proof of the theorem.

9 NTMSCI 6, No., (208) / Some characterizatios of strogly coverget sequeces Lemma. Let M =(M ) be a sequece of Orlicz fuctios, p=(p ) be a bouded sequece of positive real umbers ad u=(u ) be a sequece of strictly positive real umbers. The a sequece(x ) of complex umbers coverges strogly to a umber x if ad oly if (i) M(x ) coverges to M(x) i the ordiary sese ad ( x x )] 2 p (ii) a u M λ 2,z,...,z = 0. λ =2 Proof. The represetatio implies both ad λ =0 λ (x x) λ 2 (x 2 x)=(λ λ 2 )(x x)+λ 2 (x x 2 ) ( λ (x x) λ 2 (x 2 x) )] p a u M,z,...,z ( x x )] λ a u M (λ λ 2 ) =0,z p,...,z + λ λ =2 a u M ( + λ a u M (λ λ 2 ) =0 =2 a u M ( x x )] 2 p λ 2,z,...,z (6) x x )] 2 p ( λ (x x) λ 2 (x 2 x) )] p λ 2,z,...,z λ a u M,z,...,z =0 ( x x )] p.,z,...,z (7) By usig these iequalities together with the fact that M(x ) covergig to M(x), we have λ =0 Hece, we get the ecessity ad sufficiecy of both (i) ad (ii). ( x x )] a u M (λ λ 2 ),z p,...,z = 0. Lemma 2. Let M =(M ) be a sequece of Orlicz fuctios, p=(p ) be a bouded sequece of positive real umbers ad u=(u ) be a sequece of strictly positive real umbers. The a sequece(x ) of complex umbers coverges strogly to a umber x if ad oly if (i) M(σ )= λ (ii) λ =2 ( x )] a u M (λ λ 2 ),z p,...,z coverges to F(x) i the ordiary sese ad 0 2 a u M ( x x )] 2 p λ 2,z,...,z = 0.

10 22. Raj ad A. ilicma: Orlicz strogly coverget sequeces with some applicatios to Fourier series... Proof. Clearly, Hece, M(x ) M(σ )= λ 0 2 = λ 0 2 = λ 2 j 2 j ( (x x ) )] p a u M (λ λ 2 ),z,...,z (λ λ 2 )] p = ( λ a u M 2 j 2 j sup M(x ) M(σ ) sup λ +2 j 2 j ( x j x )] j 2 p a u M,z,...,z ( x j x )] j 2 p a u M,z,...,z (λ λ 2 )] p 0 j 2 2 x j x )] j 2 p. λ j 2,z,...,z =2 a u M ( x x )] 2 p. λ 2,z,...,z Accordig to lemma 0, for the ecessity part, it is easy to see that M(σ ) = M(x) which comes from the above iequality, coditio (ii) of this lemma ad M(x ) = M(x). For the sufficiet part, we oly eed M(x ) = M(x) which also comes from the above iequality, coditio (ii) of this lemma ad M(σ )=M(x). 3 Statistical covergece The otios of statistical covergece ad covergece i desity for sequeces has bee i the literature uder differet guises, sice the early part of the last cetury. Statistical covergece was recetly ivestigated by Fast 8] ad Schoeberg 3] idepedetly. Later o it was ivestigated from the sequece space poit of view ad lied with summability theory by Fridy 9], Coor 3], Salat 29], Fridy ad Orha 0] ad may others. The otio of statistical covergece depeds o the desity of subsets ofn. A subset E ofnis said to have desity δ(e) if δ(e)= = χ E () exists, where χ E is the characteristics fuctio of E. A sequece x=(x ) is said to be statistically coverget to L if for every ε > 0, : x L ε =0, I this case, we write S x = L or x L(S). The set of all statistical coverget sequeces is deoted by S. Defiitio.A sequece x=(x ) is said to beλ 2,A,u, p,.,...,. ]-statistically coverget to x if for ay ε > 0, : λ (x x) λ 2 (x 2 x) ε =0,,z,...,z λ where the vertical bars idicate the umber of elemets i the closed set. I this case, we write S set of all statistically coverget sequeces is deoted by S(Λ 2 ). Λ 2 (x)=x ad the

11 NTMSCI 6, No., (208) / Theorem 6. Cosider M = (M ) be a sequece of Orlicz fuctios, p = (p ) be a bouded sequece of positive real umbers ad u = (u ) be a sequece of strictly positive real umbers ad sup p = H <. The Λ 2,M,A,u, p,.,...,. ] (S(Λ 2 )). Proof. Let x Λ 2,M,A,u, p,.,...,. ]. Tae ε > 0, deote the sum over with λ (x x) λ 2 (x 2 x),z,...,z ε ad deote the sum over with λ (x x) λ 2 (x 2 x),z,...,z < ε. The for each z,,z X, we obtai 2 ( λ (x x) λ 2 (x 2 x) )] p λ a u M,z,...,.z =0 = ( ( λ (x x) λ 2 (x 2 x) )] p λ a u M,z,...,.z ( λ (x x) λ 2 (x 2 x) )] p ) +a u M,z,...,.z 2 ( λ (x x) λ 2 (x 2 x) )] p λ a u M,z,...,.z λ a u M (ε)] p λ mi(a u M (ε)] h,u a M (ε)] H ) = : λ (x x) λ 2 (x 2 x) ε,z,...,.z λ mi(a u M (ε)] h,a u M (ε)] H ). Hece, x (S(Λ 2 )). This completes the proof of the theorem. Theorem 7. Cosider M =(M ) be bouded sequece of Orlicz fuctios, p=(p ) be a bouded sequece of positive real umbers ad u=(u ) be a sequece of strictly positive real umbers ad 0<if p p sup p = H <. The (S(Λ 2 )) Λ 2,M,A,u, p]. Proof. Suppose that M = (M ) be bouded. For give ε > 0, deote the sum over with λ (x x) λ 2 (x 2 x),z,...,z ε ad deote the sum over with λ (x x) λ 2 (x 2 x),z,...,z < ε. 2 Sice M =(M ) be bouded there exists a iteger D such that M (x)<d for all x 0. The for each z,,z X. ( λ (x x) λ 2 (x 2 x) )] p a u M,z,...,z ( ( λ (x x) λ 2 (x 2 x) )] p λ a u M,z,...,z ( λ (x x) λ 2 (x 2 x) )] p ) u M,z,...,z λ =0 +a 2 λ max(a u D h ],a u D H ])+ λ a u M (ε)] p 2 max(a u D h ],a u D H ]) : λ (x x) λ 2 (x 2 x) ε,z,...,z λ + max(a u M (ε)] h,a u M (ε)] H ). Hece, x Λ 2,M,A,u, p,.,...,. ]. This completes the proof of the theorem.

12 223. Raj ad A. ilicma: Orlicz strogly coverget sequeces with some applicatios to Fourier series... 4 Results for Fourier series The space of all 2π periodic complex-valued cotiuous fuctios is a Baach space edowed with the orm f C = max t f(t) ad it is deoted by C. Let 2 a 0( f)+ = (a ( f)cost+ b ( f)si t) (8) be the Fourier series of f C ad deoted by s ( f) the -th partial sum of the series (4.). We shall deote by U, A ad S(Λ 2 ) respectively the classes of fuctios f C whose Fourier series coverges uiformly, absolutely ad strogly o0,2π). Now, if f S(Λ 2 ) ad z,...,z X, the ( ( λ (s ( f) f) λ 2 (s 2 ( f) f) ) )] p C λ a u M,z,...,z = 0, (9) =0 where λ = λ 2 = x = x 2 = 0. The space U is a Baach space with the orm f U = sup ad the space A is also a Baach space with the orm ( s ( f) )] a u M,z p C,...,z f A = ( 2 a a 0 ( f) )] p+ ( u M,z a ( f)+b ( f) )] p.,...,z u a f,z,...,z = Oe ca easily prove that U ad A are Baach spaces. We shall give the proof oly for S(Λ 2 ) i the ext Theorem. Now, we defie the orm f S(Λ 2 ) = sup ( λ (s ( f) f) λ 2 (s 2 ( f) f) )] p ) C λ a u M,z,...,z, =0 which is fiite for every f S(Λ 2 ). By usig triagle iequality, we have f S(Λ 2 ) f C+ sup ( λ (s ( f) f) λ 2 (s 2 ( f) f) )] p ) C λ a u M,z,...,z =0 ad this sup is due to equatio 9. The orm iequalities correspodig to equatio (3.6) i 23] are f U f S(Λ 2 ) 2 f A, (0) which implies that A S(Λ 2 ) U. Lemma 3. Let M =(M ) be a sequece of Orlicz fuctios, p=(p ) be a bouded sequece of positive real umbers ad u=(u ) be a sequece of strictly positive real umbers. The f S(Λ 2 ) if ad oly if (i) s ( f) f C = 0 ad ( λ 2 (a ( f)cost+ b ( f)si t) )] p C (ii) a u M,z,...,z = 0. λ =2

13 NTMSCI 6, No., (208) / Proof. It follows from the Lemma, so we omit it. Let us deote σ ( f)= ( (λ λ 2 )s ( f) )] p λ a u M,z,...,z (=0,,...). () =0 Lemma 4. Let M =(M ) be a sequece of Orlicz fuctios, p=(p ) be a bouded sequece of positive real umbers ad u=(u ) be a sequece of strictly positive real umbers. The f S(Λ 2 ) if ad oly if (i) (ii) λ σ ( f) f C = 0 ad ( λ 2 (a ( f)cost+ b ( f)si t) )] p C a u M,z,...,z = 0. =2 Theorem 8. The set S(Λ 2 ) edowed with orm is a Baach space. f S(Λ 2 ) = sup ( λ (s ( f) f) λ 2 (s 2 ( f) f) )] p C λ a u M,z,...,z =0 Proof. The oly thig we have to prove is completeess. For this, let s j j be a Cauchy sequece i the orm. S(Λ 2 ). The by equatio (0), s j is a Cauchy sequece i the orm. U as well so there exists a sequece s S(Λ 2 ) such that s j s U = 0. Now, we show that s S(Λ 2 j ). Suppose ε > 0, the by assumptio there exists v = v(ε) such that s j s i S(Λ 2 ) ε for all i, j v. (2) Let s j =s j : =0,,... ad s=s : =0,,... We shall fix i,. Similarly by equatio (0), we have λ =0 ( λ (s j s ) λ 2 (s j( 2) s 2 ) a u M,z,...,z 2 )] p s j s U λ =0 (λ + λ 2 ) ε, (3) provided j is large eough, due to j s j s U = 0. Here j depeds o ad ε ad assume that j v. Applyig triagle iequality by taig equatios (2) ad (3) ito accout we obtai that λ =0 ( λ (s j s ) λ 2 (s j( 2) s 2 ) )] p a u M,z,...,z ( λ (s j s i ) λ 2 (s j( 2) s i( 2) ) )] p λ a u M,z,...,z =0 + ( λ (s i s ) λ 2 (s i( 2) s 2 ) )] p λ a u M,z,...,z =0 s j s i S(Λ 2 ) + ε = 2ε, for j v. Sice this hold for ay 0, by defiitio s j s i S(Λ 2 ) s j s j S(Λ 2 ) = 0 ad s S(Λ 2 ) which completes the proof. 2ε for j v. This proves

14 225. Raj ad A. ilicma: Orlicz strogly coverget sequeces with some applicatios to Fourier series... Lemma 5. Let M =(M ) be a sequece of Orlicz fuctios, p=(p ) be a bouded sequece of positive real umbers ad u=(u ) be a sequece of strictly positive real umbers. If a trigoometric series = (a cost+ b sit) coverges strogly for t belogig to a set of positive measure or of secod category, the λ =2 ( λ 2 (a + b ) )] p a u M,z,...,z = 0. Competig iterests The authors declare that they have o competig iterests. Authors cotributios All authors have cotributed to all parts of the article. All authors read ad approved the fial mauscript. Refereces ] A. Alotaibi, M. Mursalee ad S. A. Mohiuddie, Statistical approximatio for periodic fuctios of two variables, J. Fuct. Spaces Appl., 203, Art. ID 49768, 5 pp. 2] C. Bele ad S. A. Mohiuddie, Geeralized weighted statistical covergece ad applicatio, Applied Mathematics ad Computatio, 29(8)(203), ] J. S. Coor, The statistical ad strog p-cesaro covergece of sequeces, Aalysis, 8 (988), ] H. Dutta, Some classes of Cesàro-type differece sequeces over -ormed spaces, Adv. Differece Equ., 203, 203:286, 3 pp. 5] A.Esi, Strogly almost summable sequece spaces i 2-ormed spaces defied by ideal covergece ad a Orlicz fuctio, Stud.Uiv.Babe s-bolyai Math. 27() (202), ] A.Esi, Strogly lacuary summable double sequece spaces i -ormed spaces defied by ideal covergece ad a Orlicz fuctio, Advaced Modelig ad Optimizatio, 4()(202), ], A. Esi ad M.. Ozdemir, Λ-strogly summable sequece spaces i -ormed spaces defied by ideal covergece ad a Orlicz fuctio, Math. Slavoca,63(4), ] H. Fast, Sur la covergece statistique, Colloq. Math., 2 (95), ] J. A. Fridy, O statistical covergece, Aalysis, 5 (985), ] J. A. Fridy ad C. Orha, Lacuary statistical covergece, Pacific J. Math., 60 (993), ] S. Gähler, Liear 2-ormietre Rume, Math. Nachr., 28 (965), ] H. Guawa, O -Ier Product, -Norms, ad the Cauchy-Schwartz Iequality, Sci. Math. Japoicae, 5 (200), ] H. Guawa ad M. Mashadi, O -ormed spaces, It. J. Math. Math. Sci., 27 (200), ] P. órus, O Λ 2 -strog covergece of umerical sequeces revisited, Acta Math. Hugar., 48 (206), ] P. òrus, O the uiform covergece of double sie itegrals with geerated mootoe coefficiets, Period. Math. Hugar., textbf63 (20), ] J. Lidestrauss ad L. Tzafriri, A Orlicz sequece spaces, Israel J. Math., 0 (97), pp ] L. Maligrada, Orlicz spaces ad iterpolatio, Semiars i Mathematics, 5, Polish Academy of Sciece, (989). 8] A. Misia, -Ier product spaces, Math. Nachr., 40 (989), ] S. A. Mohiudie ad B. Hazaria, Some Classes of Ideal Coverget Sequeces ad Geeralized Differece Matrix Operator, Filomat, 3(6), (207),

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