On statistical convergence in generalized Lacunary sequence spaces

Transcription

1 BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Number 2(87), 208, Pages 7 29 ISSN On statstcal convergence n generalzed Lacunary sequence spaces K.Raj, R.Anand Abstract. In the present paper we ntroduce and study some generalzed Lacunary sequence spaces of Muselak-Orlcz functon usng nfnte matrx over n-normed spaces. We also make an effort to study some ncluson relatons, topologcal and geometrc propertes of these spaces. Fnally, we study statstcal convergence on these spaces. Mathematcs subject classfcaton: 40A05, 40B50, 46A9, 46A45. Keywords and phrases: Muselak-Orlcz functon, Lacunary sequence, soldty, nfnte matrx, statstcally convergent, n-normed space. Introducton and Prelmnares Let w be the set of all real or complex sequences and l,c and c 0 respectvely, be the Banach spaces of bounded, convergent and null sequences x = (x k ), normed by x = sup x k, where k N. Let X and Y be two sequence spaces and A = (a k ) k be an nfnte matrx of real or complex numbers a k, where,k N. Then we say that A defnes a matrx mappng from X nto Y f for every sequence x = (x ) X, the sequence Ax = {A (x)}, the A-transform of x, s n Y, where A (x) = a k x k ( N). () k= The matrx doman X A of an nfnte matrx A n a sequence space X s defned by X A = {x = (x k ) : Ax X}. (2) The approach of constructng a new sequence space by means of the matrx doman of a partcular lmtaton method has been employed by several authors (see [24] and references theren). The noton of dfference sequence spaces was ntroduced by Kızmaz [4], who studed the dfference sequence spaces l ( ), c( ) and c 0 ( ). The noton was further generalzed by Et and Çolak [7] by ntroducng the spaces l ( n ), c( n ) and c 0 ( n ). Another type of generalzaton of the dfference sequence spaces s due to Trpathy and Es [27], who studed the spaces l ( m ), c( m ) and c 0 ( m ). c K. Raj, R. Anand, 208 7

2 8 K.RAJ, R.ANAND Trpathy et al. [26] generalzed the above notons and unfed these as follows: Let m, n be non-negatve ntegers, then for Z = l, c and c 0, we have sequence spaces Z( n (m) ) = {x = (x k) w : ( n (m) x k) Z}, where n (m) x = ( n (m) x k) = ( n (m) x k n (m) x k m) and 0 (m) x k = x k for all k N, whch s equvalent to the followng bnomal representaton n (m) x k = n ( ( ) n =0 ) x k+m. Let m and n be non-negatve ntegers and v = (v k ) be a sequence of non-zero scalars. Then for Z, a gven sequence space, we have Z( n (mv) ) = {x = (x k) w : ( n (mv) x k) Z}, for Z = l,c and c 0, where n (mv) x k = ( n (mv) x k n (mv) x k m) and 0 (mv) x k = v k x k for all k N, whch s equvalent to the followng bnomal representaton: n (mv) x k = n ( ( ) n =0 ) v k m x k m. In ths expanson t s mportant to note that we take v k m = 0 and x k m = 0 for non-postve values of k m. Dutta [6] showed that these spaces can be made BK-spaces under the norm x = sup n (mv) x k. k For n = and v k = for all k N, we get the spaces l ( m ),c( m ) and c 0 ( m ). If m = and v k = for all k N, we get the spaces l ( n ),c( n ) and c 0 ( n ). Takng m = n = and v k = for all k N, we get the spaces l ( ),c( ) and c 0 ( ). Parasar and Choudhary [20], Güngör et al. [2], Çolak et al. [4], and others used Orlcz functons for defnng some new sequence spaces. The concept of 2-normed spaces was ntally developed by Gähler [] n the md of 960 s, whle that of n-normed spaces can be seen n Msak [6]. Snce then many others have studed ths concept and obtaned varous results (see [3]). Let n N and X be a lnear space over the feld R of reals of dmenson d, where d n 2. A real valued functon,, on X n satsfyng the followng four condtons:. x,x 2,,x n = 0 f and only f x,x 2,,x n are lnearly dependent n X, 2. x,x 2,,x n s nvarant under permutaton, 3. αx,x 2,,x n = α x,x 2,,x n for any α R, and

3 ON STATISTICAL CONVERGENCE 9 4. x + x,x 2,,x n x,x 2,,x n + x,x 2,,x n s called an n-norm on X and the par (X,,, ) s called a n-normed space over the feld R. Example. We may take X = R n beng equpped wth the n-norm x,x 2,,x n E = the volume of the n-dmensonal parallelepped spanned by the vectors x,x 2,,x n whch may be gven explctly by the formula x,x 2,,x n E = det(x j ), where x = (x,x 2,,x n ) R n for each =,2,,n. Let (X,,, ) be an n-normed space of dmenson d n 2 and {a,a 2,,a n } be lnearly ndependent set n X. Then the functon,, on X n defned by x,x 2,,x n = max{ x,x 2,,x n,a : =,2,,n} defnes an (n )-norm on X wtespect to {a,a 2,,a n }. A sequence (x k ) n a n-normed space (X,,, ) s sad to converge to some L X f lm x k L,z,,z n = 0 for every z,,z n X. k An Orlcz functon M s a functon whch s contnuous, non-decreasng and convex wth M(0) = 0, M(x) > 0 for x > 0 and M(x) as x. If the convexty of an Orlcz functon s replaced by subaddtvty, we call t a modulus functon ntroduced by Nakano [9]. Lndenstrauss and Tzafrr [5] used the dea of Orlcz functon to defne the followng sequence space, l M = { x = (x k ) w : ( xk ) } M <, for some > 0, k= whch s called an Orlcz sequence space. The space l M s a Banach space wth the norm { ( xk ) } x = nf > 0 : M. k= A sequence M = (M ) of Orlcz functons s called a Muselak-Orlcz functon. A Muselak-Orlcz functon M = (M ) s sad to satsfy 2 -condton f there exst constants a, K > 0 and a sequence c = (c ) = l + (the postve cone of l ) such that the nequalty M (2u) KM (u) + c

4 20 K.RAJ, R.ANAND holds for all N and u R +, whenever M (u) a. For more detals about sequence spaces see [2,7,2 23] and references theren. An ncreasng sequence of non-negatve ntegers = ( r r ) as r can be made through lacunary sequence θ = ( r ), r = 0,,2,, where 0 = 0. The ntervals determned by θ are denoted by I r = ( r, r ] and the rato r / r wll be denoted by q r. Freedman [9] defned the space of lacunary strongly convergent sequences N θ as: N θ = { } x = (x k ) : lm x k L = 0 for some L. k I r Let X be a sequence space. Then X s called () Sold (or normal) f (α x ) X whenever (x ) X and for all sequences (α ) of scalars wth α, for all N; () Monotone provded X contans the canoncal premages of all ts step spaces. If X s normal, then X s monotone. Let A = (a k ) be an nfnte matrx of complex numbers. Let M = (M ) be a Muselak-Orlcz functon, p = (p ) be a bounded sequence of postve real numbers, u = (u ) be a sequence of strctly postve real numbers. We defne the followng sequence spaces n the present paper: ν[a,u, n (mv),θ,p,m,.,...,. ] = { x : lm I r M A (x) s )] p,z,...,z n = 0 for some s, > 0 }, ν 0 [A,u, n (mv),θ,p,m,.,...,. ] = { x : lm I r and ν [A,u, n (mv),θ,p,m,.,...,. ] = M A (x) )] p },z,...,z n = 0, for some > 0 { ( x : sup M A (x) )] p,z,...,z n <, for some > 0 }. r I r The followng nequalty wll be used throughout the paper. If 0 < p supp = H, K = max(,2 H ), then a + b p K{ a p + b p } (3)

5 ON STATISTICAL CONVERGENCE 2 for all and a,b C. Also, a p max(, a H ), for all a C. The man objectve of ths paper s to ntroduce the concept of generalzed Lacunary sequence spaces of Muselak-Orlcz functon usng nfnte matrx over n-normed spaces. We also make an effort to study some topologcal propertes and prove some ncluson relatons between these sequence spaces. Fnally, by usng these concepts we study statstcal convergence of these spaces. 2 Man Results Theorem. Let M = (M ) be a Muselak-Orlcz functon, p = (p ) be a bounded sequence of postve real numbers, u = (u ) be a sequence of strctly postve real numbers and A = (a k ) be an nfnte matrx of complex numbers. Then, the spaces ν[a,u, n (mv),θ,p,m,.,...,. ], ν 0 [A,u, n (mv),θ,p,m,.,...,. ] and ν [A,u, n (mv),θ,p,m,.,...,. ] are lnear spaces over the complex feld C. Proof. Let x and y ν 0 [A,u, n (mv),θ,p,m,.,...,. ] and α,β C. Then there exst postve real numbers and 2 such that and lm lm M A (x) )] p,z,...,z n = 0 I r M A (y) )] p,z,...,z n = 0. 2 I r Defne 3 = max(2 α,2 β 2 ). Snce.,...,. s a n-norm on X and M = (M ) s non-decreasng and convex functon for each, so by nequalty (3), we have ( lm M A (αx + βy) )] p,z,...,z n 3 I r lm I r )] p z n K lm 2 p I r + K lm = 0. [u M ( n (mv) A (αx) 2 p I r 3,z,...,z n + n (mv) A (βy) 3,z,..., M A (x) )] p,z,...,z n M A (y) )] p,z,...,z n 2

6 22 K.RAJ, R.ANAND Thus, we have αx+βy ν 0 [A,u, n (mv),θ,p,m,.,...,. ]. Hence, ν 0 [A,u, n (mv),θ,p, M,.,...,. ] s a lnear space. Smultaneously, t can be proved that ν[a,u, n (mv),θ, p,m,.,...,. ] and ν [A,u, n (mv),θ,p,m,.,...,. ] are lnear spaces. Theorem 2. Let M = (M ) be a Muselak-Orlcz functon, p = (p ) be a bounded sequence of postve real numbers, u = (u ) be a sequence of strctly postve real numbers and A = (a k ) be an nfnte matrx of complex numbers. If sup(m (x)) p < fxed x > 0, then ν[a,u, n (mv),θ,p,m,.,...,. ] ν [A,u, n (mv),θ,p,m,.,...,. ]. Proof. Let x ν[a,u, n (mv),θ,p,m,.,...,. ]. Then there exsts some postve such that lm M A (x) s )] p,z,...,z n = 0. I r Defne = 2. Snce (M ) s non-decreasng, convex and by usng nequalty (3), we have ( lm M A (x) )] p,z,...,z n I r = lm I r K lm 2 p I r + K lm 2 p I r < K lm M A (x) s + s )] p,z,...,z n 2 I r M A (x) s )] p,z,...,z n [ u M ( s,z,...,z n )] p M A (x) s )] p,z,...,z n [ ( s )] p + K lm u M,z,...,z n I r <. Hence, x ν [A,u, n (mv),θ,p,m,.,...,. ]. Ths completes the proof. Theorem 3. The sequence spaces ν 0 [A,u, n (mv),θ,p,m,.,...,. ] and ν [A,u, n (mv), θ,p,m,.,...,. ] are sold and so monotone. Proof. Suppose x ν [A,u, n (mv),θ,p,m,.,...,. ]. Then sup r I r M A (x) )] p,z,...,z n <, for some > 0.

7 ON STATISTICAL CONVERGENCE 23 Let α = (α ) be a sequence of scalars such that α for all N. Then, we have ( sup M A (αx) )] p,z,...,z n r I r sup r <, I r whch leads us to the desred result. M A (x) )] p,z,...,z n Theorem 4. Let M = (M ) and M = (M ) be Muselak-Orlcz functons. Then ν 0 [A,u, n (mv),θ,p,m,.,...,. ] ν 0[A,u, n (mv),θ,p,m,.,...,. ] ν 0[A,u, n (mv), θ,p,(m + M ),.,...,. ]. Proof. Suppose that x ν 0 [A,u, n (mv),θ,p,m,.,...,. ] ν 0[A,u, n (mv),θ,p,m,.,...,. ]. Then lm [ u (M + M ) I r ( n (mv) A (x) )] p,z,...,z n [ ( = lm u M n (mv) A (x) I r [ ( n (mv) A (x) + lm I r K lm I r + K lm I r 0 as r. Thus, x ν 0 [A,u, n (mv),θ,p,(m u M [ u M [ u M,z,...,z n )] p,z,...,z n )] p ( n (mv) A (x) )] p,z,...,z n ( n (mv) A (x) )] p,z,...,z n +M ),.,...,. ]. Hence, the proof s complete. Theorem 5. Let M = (M ) and M = (M ) be two Muselak-Orlcz functons satsfyng 2 condton. Then () ν 0 [A,u, n (mv),θ,p,m,.,...,. ] ν 0[A,u, n (mv),θ,p,m M,.,...,. ]. () ν[a,u, n (mv),θ,p,m,.,...,. ] ν[a,u, n (mv),θ,p,m M,.,...,. ]. () ν [A,u, n (mv),θ,p,m,.,...,. ] ν [A,u, n (mv),θ,p,m M,.,...,. ]. Proof. We consder the frst case only. Rests can be proved n a smlar way. Let x ν 0 [A,u, n (mv),θ,p,m,.,...,. ]. Then lm [ u M I r ( n (mv) A (x) )] p,z,...,z n = 0.

8 24 K.RAJ, R.ANAND Let ǫ > 0 and choose δ wth 0 < δ < such that M (t) < ǫ for 0 t δ. [ ( Let y = u M n (mv) A (x),z,...,z n )]. Thus, we can wrte lm [M (y )] p = lm I r [M (y )] p + lm [M (y )] p. y δ Snce M = (M ) satsfes 2 condton, we have lm y δ y >δ [M (y )] p max{,m () H } lm [(y )] p. (4) For y > δ, we use the fact that y < y δ < + y δ. Snce M = (M ) s non-decreasng and convex, t follows that M (y ) < M ( + y δ ) < 2 M (2) + 2 (2y δ ). Snce M = (M ) satsfes 2 condton and y δ >, there exsts K > 0 such that y δ M (y ) < 2 K y δ M (2) + 2 K y δ M (2) = K y δ M (2). Therefore, we have lm [u M (y )] p max{,(k M (2) δ y >δ Hence, by equatons (4) and (5), we have ( [u (M M h ) n (mv) A (x) )] p,z,...,z n r I r lm = lm [u M (y )] p I r D lm [y ] p y δ + G lm [y ] p, y >δ ) H } lm [(y )] p. (5) where D = max{,m () H } and G = max{,(k M (2) δ ) H }. Hence, ν 0 [A,u, n (mv),θ,p,m,.,...,. ] ν 0[A,u, n (mv),θ,p,m M,.,...,. ]. y >δ Theorem 6. Let 0 p q for all and ( q p ) be bounded. Then ν[a,u, n (mv),θ,q, M,.,...,. ] ν[a,u, n (mv),θ,p,m,.,...,. ].

9 ON STATISTICAL CONVERGENCE 25 Proof. Let x ν[a,u, n (mv),θ,q,m,.,...,. ]. Wrte t = [u M ( n (mv) A (x) s,z, )] q...,z n and µ = p q for all N. Then 0 < µ for every N. Take 0 < µ < µ for every N. Defne the sequences (a ) and (b ) as follows: For t, let a = t and b = 0 and for t <, let a = 0 and b = t. Then clearly for all N, we have t = a + b,t µ = a µ + b µ. Now, t follows that a µ a t and b µ b µ. Therefore, lm t µ = lm (a µ + b µ ) I r I r lm t + lm b µ h. r I r I r Now for each, lm and so ( b µ ) µ ( ) µ = lm b I r I r lm ( lm I r = lm ( I r b [( ) µ ] ) µ ( µ b. lm ) µ I r ( ) µ. t µ lm t + lm b I r I r I r Hence, x ν[a,u, n (mv),θ,p,m,.,...,. ]. [( ) µ ] µ ) µ Theorem 7. () If 0 nf p p for all, then ν[a,u, n (mv),θ,m,.,...,. ] ν[a,u, n (mv),θ,p,m,.,...,. ]. () If < p supp = H <, then ν[a,u, n (mv),θ,p,m,.,...,. ] ν[a,u, n (mv),θ,m,.,...,. ]. Proof. () Let x ν[a,u, n (mv),θ,m,.,...,. ]. Snce 0 < nf p, we get lm I r M A (x) s )],z,...,z n lm I r M A (x) s )] p,z,...,z n and hence x ν[a,u, n (mv),θ,p,m,.,...,. ]. () Let p supp = H < and x ν[a,u, n (mv),θ,p,m,.,...,. ]. Then for

10 26 K.RAJ, R.ANAND each 0 < ǫ <, there exsts a postve nteger s 0 such that lm I r Ths mples that lm I r M A (x) s )] p,z,...,z n ǫ < for all r s 0. M A (x) s )] p,z,...,z n lm I r Therefore, x ν[a,u, n (mv),θ,m,.,...,. ]. M A (x) s,z,...,z n )]. 3 Statstcal Convergence The noton of statstcal convergence was ntroduced by Fast [8] and Schoenberg [25] ndependently. Over the years and under dfferent names, statstcal convergence has been dscussed n the theory of Fourer analyss, ergodc theory and number theory. Later on, t was further nvestgated from the sequence space pont of vew and lnked wth summablty theory by Frdy [0], Connor [3], Mursaleen et al.[8] and many others. In recent years, generalzatons of statstcal convergence have appeared n the study of strong ntegral summablty and the structure of deals of bounded contnuous functons on locally compact spaces. Statstcal convergence and ts generalzatons are also connected wth subsets of the Stone-Cech compactfcaton of natural numbers. Moreover, statstcal convergence s closely related to the concept of convergence n probablty. A complex number sequence x = (x ) s sad to be statstcally convergent to the number l f for every ǫ > 0, lm K(ǫ) = 0, where K(ǫ) denotes the number of n n elements n K(ǫ) = { N : x l ǫ}. The set of statstcally convergent sequences s denoted by S. A sequence x = (x ) s sad to be S[A,u, n (mv),θ,.,...,. ]-statstcally convergent to s f { lm I r : n (mv) A (x) s },z,...,z n ǫ = 0. In ths secton we ntroduce S[A,u, n (mv),θ,.,...,. ]-statstcal convergence and gve some relatons between S[A,u, n (mv),θ,.,...,. ]-statstcally convergent sequences and ν[a,u, n (mv),θ,p,m,.,...,. ]-convergent sequences.

11 ON STATISTICAL CONVERGENCE 27 Theorem 8. Let M = (M ) be a Muselak-Orlcz functon. Then ν[a,u, n (mv),θ,p, M,.,...,. ] S[A,u, n (mv),θ,.,...,. ]. Proof. Let x ν[a,u, n (mv),θ,p,m,.,...,. ]. Take ǫ > 0, denotes the sum over n wth n (mv) A (x) s,z,...,z n ǫ and denotes the sum over n wth n (mv) A (x) s lm I r,z,...,z n < ǫ. Then for each z,...,z n, we obtan M A (x) s )] p,z,...,z n = lm M A (x) s )] p,z,...,z n ( + lm M A (x) s 2 ( lm M A (x) s lm lm { = lm n : {u M (ǫ)} H]. [ ] p u M (ǫ) mn [{u M (ǫ)} h, {u M (ǫ)} H] 2,z,...,z n )] p,z,...,z n )] p n (mv) A (x) s },z,...,z n ǫ mn [{u M (ǫ)} h, Hence, x S[A,u, n (mv),θ,.,...,. ]. Ths completes the proof of the theorem. Theorem 9. Let M = (M ) be a bounded Muselak-Orlcz functon. Then S[A,u, n (mv),θ,.,...,. ] ν[a,u, n (mv),θ,p,m,.,...,. ]. Proof. Suppose that M = (M ) be bounded. For gven ǫ > 0, denotes the sum over n wth n (mv) A (x) s,z,...,z n ǫ and denotes the sum over n 2 wth n (mv) A (x) s,z,...,z n < ǫ. Snce M = (M ) s bounded, there exsts an nteger N such that M < N for all. Then for each z,...,z n, we obtan ( lm M A (x) s )] p,z,...,z n I r ( = lm M A (x) s )] p,z,...,z n

12 28 K.RAJ, R.ANAND ( + lm M A (x) s )] p,z,...,z n 2 [ ] lm max {u N h }, {u N H } [ ] u M (ǫ) p + lm 2 [ max + max ] {u N h }, {u N H } lm [ {u M (ǫ) h }, {u M (ǫ) H } Hence, x ν[a,u, n (mv),θ,p,m,.,...,. ]. { n : n (mv) A (x) s },z,...,z n ǫ ]. Acknowledgement. The authors thank the referee for hs/her valuable suggestons. References [] Başarır M., Sonalcan O. On some double sequence spaces. J. Indan Acad. Math., 999, 2, [2] Blgn T. Some sequence spaces defned by a modulus. Int. Math. J., 2003, 3, [3] Connor J. S. The statstcal and strong p-cesaro convergence of sequeces. Analyss, 988, 8, [4] Çolak R., Trpathy B. C., Et M. Lacunary strongly summable sequences and q-lacunary almost statstcal convergence. Vetnam J. Math., 2006, 34, [5] Das N. R., Choudhary A. Matrx transformaton of vector valued sequence spaces. Bull. Calcutta Math. Soc., 992, 84, [6] Dutta H. Characterzaton of certan matrx classes nvolvng generalzed dfference summablty spaces. Appl. Sc., 2009,, [7] Et M., Çolak R. On some generalzed dfference sequence spaces. Soochow. J. Math., 995, 2, [8] Fast H. Sur la convergence statstque. Colloq. Math., 95, 2, [9] Freedman A. R., Sember J. J., Raphael M. Some Cesàro-type summablty spaces. Proc. London Math. Soc., 978, 37, [0] Frdy J. A. On the statstcal convergence. Analyss, 985, 5, [] Gähler S. Lnear 2-normetre Rume. Math. Nachr., 965, 28, 43. [2] Güngör M., Et M., Altn Y. Strongly (V σ; λ; q) summable sequences defned by Orlcz functons. Appl. Math. Model. Anal., 2007, 2, [3] Gunawan H., Mashad M. On n-normed spaces. Int. J. Math. Math. Sc., 200, 27, [4] Kızmaz H. On certan sequence spaces. Canad. Math-Bull., 98, 24, [5] Lndenstrauss J., Tzafrr L. On Orlcz sequence spaces. Israel J. Math., 97, 0, [6] Msak A. n-inner product spaces. Math. Nachr., 989, 40,

13 ON STATISTICAL CONVERGENCE 29 [7] Mursaleen M. λ statstcal convergence. Math. Slovaca, 2000, 50, 5. [8] Mursaleen M., Edely O. H. H. Statstcal convergence of double sequences. J. Math. Anal. Appl., 2003, 288, [9] Nakano H. Concave modulars, J. Math. Soc. Japan, 5 (953), [20] Parasar S. D., Choudhary B. Sequence spaces defned by Orlcz functons. Indan. J. Pure Appl. Math., 994, 25, [2] Raj K., Sharma C. Applcatons of strongly convergent sequences to Fourer seres by means of modulus functons. Acta Math. Hungar., 206, 50, [22] Raj K., Jamwal S. On some generalzed statstcal convergent sequence spaces. Kuwat J. Sc., 205, 42, [23] Raj K., Jamwal S., Sharma S. K. New classes of generalzed sequence spaces defned by an Orlcz functon. J. Comput. Anal. Appl., 203, 5, [24] Savaş E., Mursaleen M. Matrx transformatons n some sequence spaces. Istanb. Unv. Fen Fak. Mat. Derg., 993, 52, 5. [25] Schoenberg I. J. The ntegrablty of certan functons and related summablty methods. Amer. Math. Monthly, 959, 66, [26] Trpathy B. C., Es A., Trpathy B. K. On a new type of generalzed dfference Cesáro sequence spaces. Soochow J. Math., 2005, 3, [27] Trpathy B.C., Es A. A new type of dfference sequence spaces. Int J. Sc. Technol, 2006,, 4. K.Raj, R.Anand Department of Mathematcs Shr Mata Vashno Dev Unversty Katra-82320, J&K, Inda E-mal: kuldpraj68@gmal.com, renuanand7@gmal.com Receved Aprl 20, 207