Şükran Konca * and Metin Başarır. 1 Introduction

Transcription

1 Koncaand Başarır Journal of Inequalities and Applications :8 R E S E A R C H Open Access On some spaces of almost lacunary convergent sequences derived by Riesz mean and weighted almost lacunary statistical convergence in a real n-normed space Şükran Konca * and Metin Başarır * Correspondence: skonca@sakarya.edu.tr Department of Mathematics Sakarya University Sakarya 5487 Turkey Abstract In this paper we introduce some new spaces of almost convergent sequences derived by Riesz mean and the lacunary sequence in a real n-normed space. By combining the definitions of lacunary sequence and Riesz mean we obtain a new concept of statistical convergence which will be called weighted almost lacunary statistical convergence in a real n-normed space. We examine some connections between this notion with the concept of almost lacunary statistical convergence and weighted almost statistical convergence where the base space is a real n-normed space. MSC: Primary 40C05; secondary 40A35; 46A45; 40A05; 40F05 Keywords: Riesz mean; weighted lacunary statistical convergence; almost convergence; lacunary sequence; n-norm Introduction The concept of 2-normed space has been initially introduced by Gähler []. Later this concept was generalized to the concept of n-normedspacesbymisiak[2]. Since then many others have studied these concepts and obtained various results [3 0]. The idea of statistical convergence was given by Zygmund [] in 935 in order to extend the convergence of sequences. The concept was formally introduced by Fast [2] and Steinhaus [3] and later on by Schoenberg [4] and also independently by Buck [5]. Many years later it has been discussed in the theory of Fourier analysis ergodic theory and number theory under different names. In 993 Fridy and Orhan [6]introducedthe concept of lacunary statistical convergence. Statistical convergence has been generalized to the concept of a 2-normed space by Gürdal and Pehlivan [3] and to the concept of an n-normed space by Reddy [9]. Moricz and Orhan [7] have defined the concept of statistical summability (R p r ). Later on Karakaya and Chishti [8] haveused(r p r )-summability to generalize the concept of statistical convergence and have called this new method weighted statistical convergence. Mursaleen et al. [9] have altered the definition of weighted statistical convergence and have found its relation with the concept of statistical (R p r )-summability. In general the statistical convergence of weighted mean is studied as a regular matrix transformation. In 204 Konca and Başarır; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited.

2 Koncaand Başarır Journal of Inequalities and Applications :8 Page 2 of [8]and[9] the concept of statistical convergence is generalized by using a Riesz summability method and it is called weighted statistical convergence. For more details related to this topic we may refer to [ ]. In this paper we introduce some new spaces of almost convergent sequences derived by Riesz mean and lacunarysequence in a real n-normed space. By combining the definitions of lacunary sequence and Riesz mean we obtain a new concept of statistical convergence which will be called weighted almost lacunary statistical convergence in a real n-normed space. We examine some connections between this notion with the concept of almost lacunary statistical convergence and weighted almost statistical convergence where the base space is a real n-normed space. 2 Definitions and preliminaries Let K be a subset of natural numbers N and we denote the set K n = j K : j n. The cardinality of K n is denoted by K n. The natural density of K is given by δ(k):=lim r r K r if it exists. The sequence x =(x j ) is statistically convergent to ξ provided that for every ε >0thesetK = K(ε):=j N : x j ξ ε has natural density zero. Let ( ) be a sequence of non-negative real numbers and P r = p + p p r for r N. Then the Riesz transformation of x =(x k )isdefinedas t r := P r r x k. (2.) k= If the sequence t r has a finite limit ξ then the sequence x is said to be (R p r )-convergent to ξ. Let us note that if P r as r then the Riesz transformation is a regular summability method that is it transforms every convergent sequence to convergent sequence and preserves the limit. If =forallk N in (2.) then the Riesz mean reduces to the Cesaro mean C of order one. By a lacunary sequence θ =(k r ) where k 0 = 0 we will mean an increasing sequence of non-negative integers with k r k r as r.theintervalsdeterminedbyθ will be denoted by I r =(k r k r ]. We write = k r k r.theratio k r k r will be denoted by q r. Throughoutthepaperwewillusethefollowingnotationswhichhavebeendefinedin [24]. Let θ =(k r )bealacunarysequence( ) be a sequence of positive real numbers such that := P kr := k (0k r ] P kr := k (0k r ] Q r := P kr P P 0 = 0 and the intervals kr determined by θ and ( )aredenotedbyi r =(P k r P kr ] = P kr P kr.if =forall k Nthen P kr P kr Q r and I r reduce to k r k r q r and I r respectively. If θ =(k r )isalacunarysequenceandp r as r thenθ =(P kr )isalacunary sequence that is P 0 =00<P kr < P kr and = P kr P kr as r. Throughout the paper we will take P r as r unlessotherwisestated. Lorentz [25] hasprovedthatasequencex is almost convergent to a number ξ if and only if t km (x) ξ as k uniformlyinmwhere t km (x)= x m + x m+ + + x m+k k N m 0. (2.2) k

3 Koncaand Başarır Journal of Inequalities and Applications :8 Page 3 of We write f lim x = ξ if x is almost convergent to ξ. Maddox [26] hasdefinedx =(x j )to be strongly almost convergent to a number ξ if and only if t km ( x ξe ) 0ask uniformly in mwherex ξe =(x j ξ) for all j and e =(...). Let n N and X be a real vector space of dimension d n 2. A real-valued function... : X n R satisfying the following conditions is called an n-norm on X and the pair (X... ) is called a linear n-normed space: () x...x n =0if and only if x...x n are linearly dependent (2) x...x n is invariant under permutation (3) αx...x n x n = α x...x n x n for any α R (4) x...x n y + z x...x n y + x...x n z for all y z x...x n X. Asequencex =(x j )inann-normed space (X... ) is said to be convergent to some ξ X in the n-norm if for each ε > 0 there exists a positive integer j 0 = j 0 (ε) suchthat x j ξ z...z n < ε for all j j 0 and for every nonzero z...z n X. Asequencex =(x j ) is said to be statistically convergent to ξ if for every ε >0thesetK := j N : x j ξ z...z n ε has natural density zero for every nonzero z...z n X in other words x =(x j ) is statistically convergent to ξ in n-normed space (X... ) if lim j j j N : x j ξ z...z n ε = 0 for every nonzero z...z n X.Forξ =0 we say this is statistically null. 3 Main results Throughout the paper w(x) l (X) denote the spaces of all and bounded X valued sequence spaces respectively where (X... )isarealn-normed space. The set of all almost convergent sequences and strongly almost convergent sequences witespect to the n-norm are denoted by F and [F] respectively as follows: x l (X):lim k t km (x ξe) z...z n =0 uniformlyinm F = for every nonzero z...z n X and x l (X):lim k t km ( x ξe z...z n )=0 uniformlyinm [F]= for every nonzero z...z n X where t km (x) isdefinedasin(2.2). We write F lim x = ξ if x is almost convergent to ξ witespect to the n-norm and [F] lim x = ξ if x is strongly almost convergent to ξ with respect to the n-norm. It is easy to see that the inclusions [F] F l (X)hold. Now we define some new sequence spaces in a real n-normed space as follows: [ R Hr k I p r θ] n = r t km (x ξe) z...z n =0 uniformlyinm for some ξ and for every nonzero z...z n X ( R p r θ) n = t km (x ξe) z...z n =0 uniformlyinm for some ξ and for every nonzero z...z n X R p r θ n = t km ( x ξe z...z n )=0 uniformlyinm. for some ξ and for every nonzero z...z n X The following results are obtained for some special cases:

4 Koncaand Başarır Journal of Inequalities and Applications :8 Page 4 of () If we take m =0then the sequence spaces above are reduced to the sequence spaces [C θ] n (C θ) n C θ n respectively as follows: [C θ] n = t k0 (x ξe) z...z n =0 for some ξ and for every nonzero z...z n X (C θ) n = t k0 (x ξe) z...z n =0 for some ξ and for every nonzero z...z n X C θ n = t k0 x ξe z...z n =0. for some ξ and for every nonzero z...z n X (2) If we take =for all k N then the sequence spaces above are reduced to the following spaces: hr k I [w θ ] n = r t km (x ξe) z...z n =0 uniformly in m for some ξ and for every nonzero z...z n X (w θ ) n = t km (x ξe) z...z n =0 uniformly in m for some ξ and for every nonzero z...z n X w θ n = t km ( x ξe z...z n )=0 uniformly in m. for some ξ and for every nonzero z...z n X (3) Let us choose θ =(k r )=2 r for r >0 then these sequence spaces above are reduced to the following spaces: [ R r p r ] n = P r k= t km (x ξe)z...z n =0 uniformly in m for some ξ and for every nonzero z...z n X x : lim r r ( R p r ) n = P r k= t km (x ξe) z...z n =0 uniformly in m for some ξ and for every nonzero z...z n X x : lim r r R p r n = P r k= t km ( x ξe z...z n )=0 uniformly. in m for some ξ and for every nonzero z...z n X (4) If we select θ =(k r )=2 r for r >0and the base space as (X ) then these sequence spaces above are reduced to the sequence spaces which can be seen in [5]. (5) If we choose =for all k N and θ =(k r )=2 r for r >0 then these sequence spaces above are reduced to the sequence spaces [C ] n (C ) n C n respectively. Now we give the following theorem to demonstrate some inclusion relations among the sequence spaces R p r θ n ( R p r θ) n [ R p r θ] n C θ n (C θ) n [C θ] n with the spaces F and [F]. Theorem 3. The following statements are true: () [F] F ( R p r θ) n [ R p r θ] n [C θ] n. (2) [F] R p r θ n ( R p r θ) n [ R p r θ] n [C θ] n. (3) [F] R p r θ n C θ (C θ) n [C θ] n.

5 Koncaand Başarır Journal of Inequalities and Applications :8 Page 5 of Proof We give the proof only for (2). The proofs of () and (3) can be done similarly. So we omit them. Let x [F] and[f] lim x = ξ. Thent km ( x ξe z...z n ) 0as k uniformlyinm for every nonzero z...z n X.Since as r then its weighted lacunary mean also converges to ξ as r uniformly in m. This proves that x R p r θ n and [F] lim x = R p r θ n lim x = ξ.alsosince t km (x ξe) z...z n H r tkm (x ξe) z...z n ( t km x ξe z...z n ) then it follows that [F] R p r θ n ( R p r θ) n [ R p r θ] n and [F] lim x = R p r θ n lim x =( R p r θ) n lim x =[ R p r θ] n lim x = ξ. Since uniform convergence of t km (x ξe) z...z n witespect to m asr impliesconvergencefor m = 0 and for every nonzero z...z n X. It follows that [ R p r θ] n [C θ] n and [ R p r θ] n lim x =[C θ] n lim x = ξ. This completes the proof. Theorem 3.2 Let θ =(k r ) be a lacunary sequence and lim inf r Q r >.Then ( R p r ) n ( R p r θ) n with ( R p r ) n lim x =( R p r θ) n lim x = ξ. Proof Suppose that lim inf r Q r > then there exists a δ >0suchthatQ r +δ for sufficiently large values of r which implies that δ P kr +δ.ifx ( R p r ) n with ( R p r ) n lim x = ξ then for sufficiently large values of rwehave P kr k r k= = P kr t km (x ξe) z...z n ( kr t km (x ξe) z...z n + k= k r k=k r + H ( r ) tkm (x ξe) z...z n P kr δ +δ. tkm (x ξe) z...z n t km (x ξe) z...z n ) for each m 0 and for every nonzero z...z n X. Then it follows that x ( R p r θ) n with ( R p r θ) n lim x = ξ by taking the limit as r. This completes the proof. Theorem 3.3 Let θ =(k r ) be a lacunary sequence with lim sup r Q r <. Then ( R p r θ) n ( R p r ) n with ( R p r θ) n lim x =( R p r ) n lim x = ξ. Proof Let x ( R p r θ) n with ( R p r θ) n lim x = ξ.thenforε >0thereexistsq 0 such that for every q > q 0 L q = H q k I q tkm (x ξe) z...z n < ε (3.)

6 Koncaand Başarır Journal of Inequalities and Applications :8 Page 6 of for each m 0 and for every nonzero z...z n X thatiswecanfindsomepositive constant M such that L q M for all q. (3.2) lim sup r Q r < implies that there exists some positive number K such that Q r K for all r. (3.3) Therefore for k r < r k r wehaveby(3.) (3.2) and (3.3) P r r p tkm k (x ξe) z...z n k= P kr = P kr + k r k= ( k I q0 tkm (x ξe) z...z n k I t km (x ξe) z...z n + k I2 t km (x ξe) z...z n + tkm (x ξe) z...z n + + = P kr (L H + L 2 H L q0 H q0 + L q0 +H q L r ) M (H + H H q0 )+ ε (H q ) P kr P kr ) tkm (x ξe) z...z n = M (P k P k0 + + P kq0 P P )+ ε kq0 (P kq0 P kr P + + P kq0 k r P kr ) kr = M P k q0 P kr + ε P k r P kq0 P kr M P k q0 P kr + εk for each m 0 and for every nonzero z...z n X. SinceP kr as r weget x ( R p r ) n with ( R p r ) n lim x = ξ. This completes the proof. Corollary 3.4 Let <lim inf r Q r lim sup r Q r <. Then ( R p r θ) n =( R p r ) n ( R p r θ) n lim x =( R p r ) n lim x = ξ. Proof It follows from Theorem 3.2 and Theorem 3.3. and In the following theorem we give the relations between the sequence spaces (w θ ) n and ( R p r ) n. Theorem 3.5 () If <for all k N then (w θ ) n ( R p r ) n and (w θ ) n lim x =( R p r ) n lim x = ξ. (2) If >for all k N and ( ) is upper-bounded then ( R p r ) n (w θ ) n and ( R p r ) n lim x =(w θ ) n lim x = ξ.

7 Koncaand Başarır Journal of Inequalities and Applications :8 Page 7 of Proof () If <for all k Nthen < for all r N. So there exists an M aconstant such that 0<M <for all r N.Letx (w θ ) n with (w θ ) n lim x = ξthenfor an arbitrary ε >0we have tkm (x ξe) z...z n M tkm (x ξe) z...z n for each m 0 and for every nonzero z...z n X. Therefore we get the result by taking the limit as r. (2) Let >for all k Nthen > for all r N.Supposethat( ) is upper-bounded then there exists an M 2 aconstantsuchthat< M 2 < for all r N.Letx ( R p r ) n and ( R p r ) n lim x = ξ. So the result is obtained by taking the limit as r for each m 0 and for every nonzero z...z n Xfromthe following inequality: t km (x ξe) z...z n M 2 t km (x ξe) z...z n. Nowwe define a new concept of statistical convergence in n-normed space which will be called weighted almost lacunary statistical convergence: Definition 3.6 The weighted almost lacunary density of K N is denoted by δ ( Rθ) (K)= lim r K r (ε) if the limit exists. We say that the sequence x =(x j )isweightedalmost lacunary statistically convergent to ξ if for every ε >0thesetK r (ε) =k I r : t km (x ξe) z...z n ε has weighted lacunary density zero i.e. lim r k I r : tkm (x ξe) z...z n ε =0 (3.4) uniformly in m for every nonzero z...z n X.Inthiscasewewrite(S ( Rθ) n) lim k x k = ξ. By(S ( Rθ) n) we denote the set of all weighted almost lacunary statistically convergent sequences in n-normed space. () If we take =for all k N in (3.4) then we obtain the definition of almost lacunary statistical convergence in n normed space that is x is called almost lacunary statistically convergent to ξ if for every ε >0theset K θ (ε)=k I r : t km (x ξe) z...z n ε has lacunary density zero i.e. lim r k Ir : tkm (x ξe) z...z n ε =0 (3.5) uniformly in m for every nonzero z...z n X.Inthiscasewewrite (S θ n) lim j x j = ξ.by(s θ n) we denote the set of all weighted almost lacunary statistically convergent sequencesin n-normed space. (2) Let us choose θ =(k r ) for r >0then the definition of weighted almost lacunary statistical convergence which is given in (3.4) is reduced to the definition of weighted almost statistically convergence that is x is called weighted almost

8 Koncaand Başarır Journal of Inequalities and Applications :8 Page 8 of statistically convergent to ξ if for every ε >0theset K Pr (ε)=k P r : t km (x ξe) z...z n ε has weighted density zero i.e. lim r P r k Pr : t km (x ξe) z...z n ε =0 (3.6) uniformly in m for every nonzero z...z n X.Inthiscasewewrite (S R n) lim j x j = ξ.by(s R n) we denote the set of all weighted almost lacunary statistically convergent sequencesin n-normed space. (3) Let us choose θ =(k r ) for r >0and =for all k N then the definition of weighted almost lacunary statistical convergence which is given in (3.4) is reduced to the definition of almost statistical convergence. Theorem 3.7 If the sequence x is ( R p r θ) n -convergent to ξ then the sequence x is weighted almost lacunary statistically convergent to ξ. Proof Let the sequence x be ( R p r θ) n -convergent to ξ and K rm (ε) =k I r : t km (x ξe) z...z n ε.thenforagivenε >0wehave tkm (x ξe) z...z n k K rm (ε) ε K rm (ε) tkm (x ξe) z...z n for each m 0 and for every nonzero z X. Hence we see that the sequence x is weighted almost statistically convergent to ξ by taking the limit as r. Theorem 3.8 Let t km (x ξe) z...z n Mforallk N for each m 0 and for every nonzero z...z n X. Then (S ( Rθ) n) ( R p r θ) n with (S ( Rθ) n) lim x =( R p r θ) n lim x = ξ. Proof Let x be convergent to ξ in (S ( Rθ) n)andletustake K rm (ε)= k I r : p k tkm (x ξe) z...z n ε. Since t km (x ξe) z...z n M for all k N for each m 0 for every nonzero z...z n X and as r thenforagivenε >0wehave tkm (x ξe) z...z n = k K rm(ε) + k / K rm(ε) tkm (x ξe) z...z n tkm (x ξe) z...z n M K rm(ε) + ε M K rm(ε) + ε

9 Koncaand Başarır Journal of Inequalities and Applications :8 Page 9 of for each m 0 and for every nonzero z...z n X. Sinceε is arbitrary we have x ( R p r θ) n by taking the limit as r. Theorem 3.9 The following statements are true. () If for all k N then (S θ n) (S ( Rθ) n). (2) Let for all k N and ( ) be upper-bounded then (S ( Rθ) n) (S θ n). Proof () If for all k Nthen for all r N.SothereexistM and M 2 constants such that 0<M M 2 for all r N.Letx (S θ n) with (S θ n) lim x = ξ then for an arbitrary ε >0we have k I r : t km (x ξe) z...z n ε = Pkr < k P kr : tkm (x ξe) z...z n ε M Pkr k r < k P kr k r : tkm (x ξe) z...z n ε = M kr < k k r : t km (x ξe) z...z n ε = M k Ir : tkm (x ξe) z...z n ε for each m 0 and for every nonzero z...z n X. Hence we obtain the result by taking the limit as r. (2) Let ( ) be upper-bounded then there exist M and M 2 constants such that M M 2 < for all r N.Supposethat for all k Nthen for all r N.Letx ( R p r ) n and ( R p r ) n lim x = ξ then for an arbitrary ε >0we have k Ir : tkm (x ξe) z...z n ε = kr < k k r : t km (x ξe) z...z n ε M 2 kr P kr < k k r P kr : tkm (x ξe) z...z n ε = M 2 Pkr < k P kr : t km (x ξe) z...z n ε = M 2 k I r : tkm (x ξe) z...z n ε for each m 0 and for every nonzero z...z n X. Hencetheresultisobtainedby taking the limit as r. Theorem 3.0 For any lacunary sequence θ if lim inf r Q r >then (S R n) (S ( Rθ) n) and (S R n) lim x =(S ( Rθ) n) lim x = ξ. Proof Suppose that lim inf r Q r > then there exists a δ >0suchthatQ r +δ for sufficiently large values of r which implies that δ P kr +δ.ifx (S R n) with(s R n) lim x = ξ

10 Koncaand Başarır Journal of Inequalities and Applications :8 Page 0of then for every ε > 0 and for sufficiently large values of rwehave P kr k Pkr : t km (x ξe) z...z n ε Pkr < k P kr : t km (x ξe) z...z n ε P kr = H ( r Pkr < k P kr : t km (x ξe) z...z n ε ) P kr δ +δ ( k I r : t km (x ξe) z...z n ε for each m 0 and for every nonzero z...z n X. Hencewegettheresultbytaking the limit as r. Theorem 3. Let θ =(k r ) be a lacunary sequence with lim sup r Q r < then (S ( Rθ) n) (S R n) and (S R n) lim x =(S ( Rθ) n) lim x = ξ. Proof If lim sup r Q r < then there is a K >0suchthatQ r K for all r N.Supposethat x (S ( Rθ) n)with(s ( Rθ) n) lim x = ξ and let N r := k I r : p kt km (x ξe) z...z n ε. (3.7) ) By (3.7) given ε >0thereisar 0 N such that N r < ε for all r > r 0.NowletM := maxn r : r r 0 and let r be any integer satisfying k r < r k r thenforeachm 0 and for every nonzero z...z n X we can write P r k Pr : t km (x ξe) z...z n ε P kr Pkr < k P kr : tkm (x ξe) z...z n ε = P kr (N + N N r0 + N r N r ) M.r 0 P kr + P kr ε( ) = M.r 0 P kr + ε (P k r P kr0 ) P kr M.r 0 + εq r M.r 0 + εk P kr P kr which completes the proof by taking the limit as r. Corollary 3.2 Let <lim inf r Q r lim sup r Q r <. Then (S ( Rθ) n)=(s R n) and (S R n) lim x =(S ( Rθ) n) lim x = ξ. Proof It follows from Theorem 3.0 and Theorem 3..

11 Koncaand Başarır Journal of Inequalities and Applications :8 Page of Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally in the preparation of this article. Both authors read and approved the final manuscript. Acknowledgements This paper has been presented in 2nd International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-203) and it was supported by the Research Foundation of Sakarya University (Project Number: ). Received: 4 October 203 Accepted: 9 January 204 Published: 8 Feb 204 References. Gähler S: Lineare 2-normierte raume. Math. Nachr (965) 2. MisiakA: n-inner product spaces. Math. Nachr (989) 3. Gürdal M Pehlivan S: Statistical convergence in 2-normed spaces. Southeast Asian Bull. Math (2009) 4. Başarır M Konca Ş Kara EE: Some generalized difference statistically convergent sequence spaces in 2-normed space. J. Inequal. Appl. 203(77)-2 (203) 5. Başarır M Konca Ş: Some sequence spaces derived by Riesz mean in a real 2-normed space. Iran. J. Sci. Technol. Trans. A Sci. Accepted 6. Gunawan H Mashadi M: On n-normed spaces. Int. J. Math. Math. Sci. 27(0) (200) 7. Gunawan H: The space of p-summable sequences and its natural n-norm. Bull. Aust. Math. Soc (200) 8. Gunawan H Setya-Budhi W Mashadi M Gemawati S: On volumes of n-dimensional parallelepipeds in l p spaces. Publ. Elektroteh. Fak. Univ. Beogr. Mat (2005) 9. ReddyBS:Statistical convergence in n-normed spaces. Int. Math. Forum (200) 0. Konca ŞBaşarır M: Generalized difference sequence spaces associated with a multiplier sequence on a real n-normed space. J. Inequal. Appl. 203(335)-8 (203). Zygmund A: Trigonometric Spaces. Cambridge University Press Cambridge (979) 2. Fast H: Sur la convergence statistique. Colloq. Math. 2(3-4) (95) 3. Steinhaus H: Sur la convergence ordinate et la convergence asymptotique. Colloq. Math (95) 4. Schoenberg IJ: The integrability of certain functions and related summability methods. Am. Math. Mon (959) 5. Buck RC: Generalized asymptotic density. Am. J. Math (953) 6. Fridy JA Orhan C: Lacunary statistical convergence. Pac. J. Math (993) 7. Moricz F Orhan C: Tauberian conditions under which statistical convergence follows from statistical summability by weighted means. Studia Sci. Math. Hung (2004) 8. Karakaya V Chishti TA: Weighted statistical convergence. Iran. J. Sci. Technol. Trans. A Sci. 33(A3) (2009) 9. Mursaleen M Karakaya V Ertürk M Gürsoy F: Weighted statistical convergence and its application to Korovkin type approximation theorem. Appl. Math. Comput (202) 20. Başarır M Öztürk M: On the Riesz difference sequence space. Rend. Circ. Mat. Palermo (2008) 2. Başarır M Kara EE: On compact operators on the Riesz difference sequence spaces. Iran. J. Sci. Technol. Trans. A Sci. 35(A4) (20) 22. Başarır M Kara EE: On compact operators on the Riesz difference sequence spaces II. Iran. J. Sci. Technol. Trans. A Sci (202) 23. Başarır M Kayıkçı M: On the generalized Riesz difference sequence space and property. J. Inequal. Appl Article ID (2009). doi:0.55/2009/ Başarır M Konca Ş: On some spaces of lacunary convergent sequences derived by Norlund-type mean and weighted lacunary statistical convergence. Arab J. Math. Sci. (203 in press) 25. Lorentz GG: A contribution to the theory of divergent sequences. Acta Math. 80()67-90 (948) 26. Maddox IJ: A new type of convergence. Math. Proc. Camb. Philos. Soc (978) 0.86/ X Cite this article as: Konca and Başarır: On some spaces of almost lacunary convergent sequences derived by Riesz mean and weighted almost lacunary statistical convergence in a real n-normed space. Journal of Inequalities and Applications :8