Nuray, Fatih

Savaş, Ekrem

English

Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaEkrem Savaş; Fatih NurayOn sigma-statistically convergence and lacunary sigma-statistically convergenceMathematica Slovaca, Vol. 43 (1993), No. 3, 309--315Persistent URL: http://dml.cz/dmlcz/132701Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1993Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMalheiTciticxi Slovaca ©1993 . . i í-i _.<% / . . . „ % . , ^ .-...x.-. .•.,«- Mathematical Institute Math. SlOVaCa, 4 3 (1993), NO. 3, 309-315 Slovák Academy of Sciences ON a-STATISTICALLY CONVERGENCE AND LACUNARY a-STATISTICALLY CONVERGENCE EKREM SAVA§ — FATIH NURAY (Communicated by Ladislav Mišík) ABSTRACT. In this note we introduce the concepts of ^--statistically conver-gence and lacunary ^-statistically convergence and give some inclusion relations. 1. In t roduct ion and background A complex number sequence x = (xk) is said to be statistically convergent to the number L if for every e > 0, l iml/n |{A; < n : \xk - L\ > e}\ = 0 , where the vertical bars indicate the number of elements in the enclosed set. In this case we write S- lim x = L or xk —• L (S). The idea of the statistical convergence of sequence of real numbers was in-troduced by F a s t [2]. S c h o n b e r g [12] studied statistical convergence as a summability method and listed some of the elementary properties of statis-tical convergence. Both of these authors noted that if a bounded sequence is statistically convergent to L, then it is Cesaro summable to L. Subsequently, statistical convergent sequences have been discussed in S a 1 a t [9], F r i d y [4], M a d d o x [6] and others independently. Most recently, in [1] it is shown that if a sequence is strongly p-Cesaro summable or wp-convergent to L , 0 < p < oo, then the sequence must be statistically convergent to L and that a bounded statistically convergent sequence must be ^-convergent. It is also shown that the statistically convergent sequences do not form a locally convex Fif-space. AMS Sub jec t C l a s s i f i c a t i o n (1991): Primary 40A05, 40C05, 40D05. Key words : Lacunary sequences, Strongly invariant convergence, Statistically conver-gence, Lacunary statistically convergence. 309 EKREM SAVA§ — FATIH NURAY Let a be a mapping of the set of positive integers into itself. A continuous linear functional <fi on l^ , the space of real bounded sequences, is said to be an invariant mean or cr-mean if and only if (1) 4>(x) > 0 when the sequence x = (xn) has xn > 0 for all n , (2) 4>(e) = 1, where e = ( 1 , 1 , . . . ) , and (3) (j)((xa(n))) = (j)(x) for all x e l^ . The mappings a are one-to-one and such that crm(n) ^ n for all positive integers n and m , where crm(n) denotes the m t h iterate of the mapping a at n. Thus (j) extends the limit functional on c, the space of convergent sequences, in the sense that </>(#) = limx for all x E c. In the case a is the translation mapping n —> n + 1, a a-mean is often called a Banach limit and Va , the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences [5]. If x = (xn), the set Tx = (Txn) = (xa^). It can be shown [10] that Va = {x = (xn) : limtmn(x) = Le uniformly i n n , L = a- lima;} , where *m n (x) = (xn + Txn + • • • + Tm : r n ) / (m + 1). Several authors including S c h a e f e r [12], M u r s a l e e n [7], S a v a § [11] and others have studied invariant convergent sequences. Recently, M u r s a l e e n [8] defined strongly cr-convergent sequences by saying that Xk —• L [Va] if and only if n - l lim 1/n 7 |£<-fc(m) — L| —> 0 uniformly in m . fc=0 By [Va] , we denote the set of all strongly a-convergent sequences. It is known ([8]) that c C [Va] C Va C loo • For a(m) = m-f 1 the space [V ]̂ is the space of strongly almost convergent sequences. By a lacunary sequence we mean an increasing integer sequence 9 = (kr) such that ko = 0 and hr = kr — fcr_i —* oo as r —* oo. Throughout this paper the intervals determined by 0 will be denoted by Jr — (fcr_i, A v ] . F r e e d m a n , S e m b e r and R a p h a e l [3] defined the space N$ in the following way: For any lacunary sequence 0 = (kr), N$ = < x = (xk) : for some L , lim \/hr \ J \xk — L| = 0 \. keir Quite recently, lacunary strong cr-convergent sequences were introduced by S a v a § [10] as below: LQ = ix = (xk) : lim \/hr ^ |z<Tfc(m) —--'1 = 0- uniformly in m l . keir 310 ON cr-STATISTICALLY CONVERGENCE . . . In [10], it is also shown that there is a strong connection between [V ]̂ and Lg as in Lemma 1. LEMMA 1. ([10]) LQ <=> [Va] for every lacunary sequence 9. The purpose of this paper is to introduce two concepts of convergence, S$ and Sae , and to give some inclusion relations between Sa- and 5<-0-convergence and also between LQ and ^^-convergence in the same way as LQ is related to Now we are ready to begin. 2. Definitions and theorems Before giving the promised inclusion relations we will give two new definitions. DEFINITION 1. A complex number sequence x = (xk) is said to be a-statisti-cally convergent to the number L if for every e > 0 lim 1/n | {0 < k < n : |#<-fc(m) — L\ > e} | = 0 uniformly in m = 1,2, . . . . In this case we write Sa-limx = L or Xk —• L (Sa) and we define Sa = {x = (xk) : for some L, Sa-limx = L} . DEFINITION 2. Let 9 = (kr) be a lacunary sequence; the number sequence x = (xk) is Sao-convergent to L provided that for every e > 0, lim l/hr \{k E Ir : \xak(m) - L\ > e}| = 0 uniformly in m = 1, 2 , . . . . In this case we write Sae-\imx = L or xk —• L (Sao) and we define Sae = [x = (xk) : for some L, SaQ-lim x = L} . We now give some inclusion relations between L^-convergence and S^-convergence and show that these are equivalent for bounded sequences. We also study relation between ^-convergence and Sae-convergence. THEOREM 1. Let 9 = (kr) be a lacunary sequence; then (i) xk -> L (Le) implies xk - • L (Sad), (ii) xEloo and Xk —> L (Sae) imply Xk —»• L (LQ) , (iii) Sae D /oo = Le . 311 EKREM SAVA$ — FATIH NURAY P r o o f . (i) If e > 0 and Xk —• L (L$), we can write 5 3 \x*k(m)-L\ > 5 3 \x*Hm)-L\ >e\{kelr : | - -k ( m ) - L| > e } | kGIr kGIr l*-fc(m)-I<l>e which yields the result. (ii) Suppose that Xk —» L(S(TQ) and x G/oo? say |x-fc(m) — L| < M for all fc and m. Given e > 0, we get 1/K 53 ko-fc(m) ~ i | fcGIr = l/ftr 53 l̂ fc(m) ~L\ + \/hr 5 ] l-Ĉ Cm) ~ L\ keir keir Kkim)-L\>e \Xak(m)-L\<£ <M/hr\{keIr: \xak{m)-L\>e}\+e from which the result follows. Let 0 be given and define Xk to be 1,2, . . . , [\Z7ir"] for k = a n ( m ) , n = fcr-i + 1, fcr-i + 2, . . . , fc r_i + [\/ft7] ; m > 1, and Xk = 0 otherwise (where [ ] denotes the greatest integer function). Note that x is not bounded. Further, for 0 < e < 1 we have l/hr\{kelr: \xak(m)-0\>e}\ = [y/hr']/hr-+0 as r - > o o , i.e. xk —• 0 ( 5 ^ ) . But, 1/fcr 5 Z K f c ( m ) - 0 | = l/hr([Jhr] {[y/hr]+l)/2) ^ 1 / 2 ^ 0 as V - OO , kGIr hence :c^ -*> 0(Le). Thus, inclusion (i) is proper and this example shows that the boundedness condition cannot be omitted from the hypothesis (ii). (iii) This is an immediate consequence of (i), (ii), Lemma 1 and [V ]̂ C /QO • This completes the proof. We now give a lemma which will be used in the proof of Theorem 2. 312 ON a -STATISTICALLY CONVERGENCE . . . LEMMA 2. Suppose for given S\ > 0 and every e > 0 , there exists n0 and m 0 such that l / n | { 0 < f c < n - l : \xak{m) - L\ > e}\ < ex for all n>n0 and m > m 0 , then x = (#&) G Sa . P r o o f . Let e\ > 0 be given. For every e > 0, choose n 0 , m 0 such that l / n | { 0 < f c < n - l : \xaHm) - L\ > e}\ < e1/2 (1) for all n > nfQ and m > m 0 . It is enough to prove that there exists n0' such that for n > n^ , 0 < m < m 0 , l / n | { 0 < f c < n - l : \xak{m) - L\ > e}\ < ex. (2) Since taking no = max(no ,no) , (2) will hold for n > no and for all m , which gives the result. Once mo has been chosen, 0 < m < mo, mo is fixed. So put K = |{0 < k < m 0 - 1 : \xak(m) -L\> e}\. Now taking 0 < m < mo and n > mo, by (1) we have l / n | { 0 < f c < n - l : | x ,* ( m ) - L\ > e}\ < l / n | { 0 < f e < m o - l : \xak{m) - L\ > e}\ -F 1/n | {m0 < k < n - 1 : \xak(m) - L\>e}\ <K/n + 1/n |{m 0 < k < n - 1 : \xak{mo) - L\>e}\ <K/n + sx/2, and taking n , sufficiently large, we can write <K/n + £ i /2 < e i , which gives (2), and hence the result follows. 313 EKREM SAVA$ — FATIH NURAY THEOREM 2. SGQ = Sa for every lacunary sequence 0. P r o o f . Let x 6 SaB • Then, from Definition 2, given s\ > 0, there exist r0 and L such that l/hr \{0 < k < hr - 1 : \xak(m) - L\ > e}\ < ex for r > 7'o and m = kr-i + l + u, u>0. Let n > hr, write n = i/ir + £, where 0 < t < hr, z is an integer. Since n > hr, i > 1. Now l / n | { 0 < f c < n - l : | x ^ ( m ) - L\ > e}\ < 1/n |{0 < fc < (i + l) / i r - 1 : \xaHm) - L\ > e}\ i = ^/n^2\{jhr<k< (j + l)hr-l: \xak{m)-L\ > e}\ j=o < l/n(i + l)hr E\ < 2ihr e\/n (i > 1) for hr/n < 1, and since ihr/n < 1, 1/n \{0 < k < n - 1 : \xak{m) - L\ > e}\ < 2ex. Then by Lemma 2, SGQ C Sa. It is easy to see that 5 a C Sae • This completes the proof. When a(m) = m + 1, from Definitions 1 and 2 we have the definitions of almost statistically convergence and lacunary almost statistically convergence of a sequence. So, similar inclusions to Theorems 1 and 2 hold between strongly almost convergent sequences and almost statistical convergent sequences, which have not appeared anywhere by this time. Acknowledgemen t Finally, the authors are grateful to Prof. Tibor Salat for his careful reading of this paper and several valuable suggestions which improved the presentation of the paper. REFERENCES [1] CONNOR, J . : The statistical and strong p-Cesdro convergence of sequences, Analysis 8 (1988), 47-63. [2] FAST, H . : Sur la convergence statistique, Colloq. Math. 2 (1951), 241-244. 314 ON a -STATISTICALLY CONVERGENCE . . . [3] FREEDMAN, A—SEMBER, J .—RAPHAEL, M. : Some Cesdro-type summability spaces, Proc. London Math. Soc. (3) 3 7 (1978), 508-520. [4] FRIDY, J . : On statistical convergence, Analysis 5 (1985), 301-310. [5] LORENTZ, G. G.: A contribution to the theory of divergent sequences, Acta Math . 80 (1948), 167-190. [6] MADDOX, I. J . : Statistical convergence in a locally convex space, Math . Proc. Cambridge Philos. Soc. 104 (1988), 141-145. [7] MURSALEEN : Invariant means and some matrix transformations, Tamkang J. Math. 10 (1979), 183-188. [8] MURSALEEN : Matrix transformations between some new sequence spaces, Houston J. Math. 9 (1983), 505-509. [9] SALAT, T . : On statistically convergent sequences of real numbers, Math . Slovaca 30 (1980), 139-150. [10] SAVA§, E .: On lacunary strong a-convergence, Indian J. Pure Appl. Math . 21 (1990), 359-365. [11] SAVA§, E .: Some sequence spaces involving invariant means, Indian J. Math. 31 (1989), 1-8. [12] SCHAEFER, P . : Infinite matrices and invariant means, Proc. Amer. Math. Soc. 36 (1972), 104-110. [13] SCHOENBERG, I. J . : The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361-365. Received December 3, 1991 Firat University Department of Mathematics Elazig Turkey 315 

On $sigma$-statistically convergence and lacunary $sigma$-statistically convergence

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On $sigma$ -statistically convergence and lacunary $sigma$ -statistically convergence