Mathematica Slovaca. λ-statistical convergence. Mohammad Mursaleen. Terms of use: Persistent URL:

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1 Mathematica Slovaca Mohammad Mursalee λ-statistical covergece Mathematica Slovaca, Vol. 50 (2000), No. 1, Persistet URL: Terms of use: Mathematical Istitute of the Slovak Academy of Scieces, 2000 Istitute of Mathematics of the Academy of Scieces of the Czech Republic provides access to digitized documets strictly for persoal use. Each copy of ay part of this documet must cotai these Terms of use. This paper has bee digitized, optimized for electroic delivery ad stamped with digital sigature withi the project DML-CZ: The Czech Digital Mathematics Library

2 Mathematica Slovaca 2000 KÍI.LL <-I m /o/-»i-.^\ M lir- Mathematical Istitute Math. SlOVaCa, 50 (2000), NO. 1, Slovák Academy of Scieces A-STATISTICAL CONVERGENCE MURSALEEN (Commuicated by L'ubica Hold) ABSTRACT. I this paper, we use the otio of (V, A)-summability to geeralize the cocept of statistical covergece. We call this ew method a X-statistical covergece ad deote by S x the set of sequeces which are A-statistically coverget. We fid its relatio to statistical covergece, (C, 1)-summability ad strog (V, A)-summability 1. Itroductio Let A = (A ) be a o-decreasig sequece of positive umbers tedig to oo such that A +1 <A + 1, A 1 = l. The geeralized de la Valee-Pousi mea is defied by *(*) : = x~ 71 ^2 X^ where I = [ - X + 1, ]. A sequece x = (x k ) is said to be (V, X) -summable to a umber L (see [8]) if ^( X as ) ~* L ri " yo ' If A =, the (V, A)-summability reduces to (C, 1)-summability. We write ad [C,l]:={x = (x ): 3LGR, lim 1 _ \x k - L\ = o} L -+oo k_^ J 1991 Mathematics Subject Classificatio: Primary 40A05, 40C05. Key words: Statistical covergece, A-statistical covergece, (V, A)-summability, strog (V, A)-summability. The preset research was supported by UGC (New Delhi) uder grat No. F. 8-14/94 111

3 MURSALEEN [y,a]:={x = (xj: 3FER, lim f Xfc - F = o} for the sets of sequeces x = (x k ) which are strogly Cesaro summable ad strogly (V, X)-summable to L, i.e. x k -> L [C, 1] ad x k -> L [V, A] respectively. The idea of statistical covergece was itroduced by F a s t [3] ad studied by various authors (see [1], [5] ad [9]). A sequece x = (x k ) is said to be statistically coverget to the umber L if for every e > 0 lim \{k < : \x, - L\ > e}\ =0, ->oo U ' L ' ^ ' J I ' where the vertical bars idicate the umber of elemets i the eclosed set. I this case, we write S-\imx = L or x k > L (S) ad 5 deotes the set of all statistically coverget sequeces. I this paper, we itroduce ad study the cocept of A-statistical covergece ad determie how it is related to [V, A] ad S. DEFINITION. A sequece x = (x ) is said to be X-statistically S x -coverget to L if for every e > 0 J^Lx^keI " : I**- >- }.= o. I this case we write S x - limx = L or x k -> L (S' A ), ad Remark. 5 A := {x : 3LGR, S A -lima; = L}. coverget or (i) If A =, the S x is the same as S. (ii) A-statistical covergece is a special case of A-statistical covergece (see [2], [7]) if the matrix A = (a k ) is take as [ikei > a = / k l o if k i i. 2. I this sectio, we fid the relatioship of S x with [V, A] ad (C, 1) methods. Let A deote the set of all o-decreasig sequeces A (A of positiv umbers tedig to oo such that A +1 < A ad X 1 1. The followig theort is the aalogue of [6; Theorem 1]. 112

4 THEOREM 2.1. Let \e A, the A-STATISTICAL CONVERGENCE (i) x k ->L[V,\] = x k^l(s x ) ad the iclusio [V, A] ^ S x is proper, (ii) if x G l^ ad x k > L (S x ), the x k -> L [V, A] ad hece x k > F(C, 1) provided x = (x k ) is ot evetually costat, (iii) 5 A ^0O = [Y,A]*? 0o, where i^ deotes the set of bouded sequeces. Proof. (i) Let e > 0 ad x k -> L [V, L]. We have X> fe - > l x *- L l>. e l{* e/» : K-i > }. /cgi kel \x k-l\>e Therefore x k -> L [17, A] =-[> a; fc -> F (5 A ). The followig example shows that 5 A ^ [V, A]. Defie x = (x k ) by x * = {o k for - [^/\ ~] +1 < fc<, otherwise. The x < i^ ad for every (0 < e < 1) ^\{kel : \x k -0\>e}\ = ^ ^ - - ^ 0 as -> oo, A A i.e. x k > 0 (5 A ). O the other had, Y^ 1^ 0 -> oo ( -> oo ), i.e. x k -/>0[V,\]. (ii) Suppose that x k > F (5 A ) ad xg^, say l.^ - F < M for all k. Give > 0, we have j-j2\x k -L\ = ^i^-li-f J2\ x k- L \ \xk L\>e \xk L\<e <^\{kel : which implies that x k -> L [V, A]. \x k -L\>e}\+e, 113

5 MURSALEEN Further, we have li:^-l)= l - f:(x k -L) + i -Y.^-L) k=l k=l kel -- T E K - z l + y- I** - L l fe=i Hece x k -» L (C, 1), sice x k -> L [V, A]. (iii) This immediately follows from (i) ad (ii). 2 v^, < r i**-j fee/ D It is easily see that S x ^ S for all A, sice X / sectio, we prove the followig relatio. THEOREM 3.1. S C S x if ad oly if is bouded by 1. I this Proof. For give e > 0 we have Therefore limif-^ >0. (3.1.1) oo {k < : \x k -L\>e}D{keI : \x k -L\>e}. ±\{k<: \ Xk -L\>e)\>l\{keI : \x k -L\>e}\ >^-i {fce/ : x fc -L >e}. Takig the limit as > oc ad usig (3.LI), we get x k -*L(S) => x k ->L (S x ). Coversely, suppose that limif ^ = 0. As i [4; p. 510], we ca choose a -»co subsequece ((j)) such that /ry < \. Defie a sequece x = (x-) by 1 if iel (j), j = l,2,..., x A, 0 otherwise. The x G [C, 1], ad hece, by [1; Theorem 2.1], x G S. But o the other had, x [V, A] ad Theorem 2.1 (ii) implies that x < S x. Hece (3.LI) is ecessary. D 114

6 A-STATISTICAL CONVERGENCE Ackowledgemet The author is sicerely grateful to Prof. J. Fridy for his kid help ad ecouragemet durig the preparatio of this paper. The author also thaks the referee for his useful remarks. REFERENCES [1] CONNOR, J. S. : The statistical ad strog p-cesaro covergece of sequeces, Aalysis 8 (1988), [2] CONNOR, J. S.: O strog matrix summability with respect to a modulus ad statistical covergece, Caad. Math. Bull. 32 (1989), [3] FAST, H.: Sur la covergece statistique, Colloq. Math. 2 (1951), [4] FREEDMAN, A. R. SEMBER, J. J. RAPHAEL, M.: Some Cesaro type summability spaces, Proc. Lodo Math. Soc. 37 (1978), [5] FRIDY, J. A.: O statistical covergece, Aalysis 5 (1985), [6] FRIDY, J. A. ORHAN, C: Lacuary statistical covergece, Pacific. J. Math. 160 (1993), [7] KOLK, E.: The statistical covergece i Baach spaces, Acta Commet. Uiv. Tartu 928 (1991), [8] LEINDLER, L.: Uber die de la Vallee-Pousische Summierbarkeit allgemeier Orthogoalreihe, Acta Math. Acad. Sci. Hugar. 16 (1965), [9] SALAT, T.: O statistically coverget sequeces of real umbers, Math. Slovaca 30 (1980), Received Jue 19, 1996 Departmet of Mathematics Revised August 27, 1997 Faculty of Sciece Aligarh Muslim Uiversity Aligarh INDIA 115