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On lacunary σ-statistical convergence
Information Sciences 166 (2004) 271–280 www.elsevier.com/locate/ins
On lacunary r-statistical convergence Vatan Karakaya Department of Mathematics, Education Faculty of Adiyaman, Gaziantep University, Adiyaman, Turkey Received 6 October 2002; received in revised form 20 November 2003; accepted 2 December 2003
Abstract The purpose of this paper is to introduce two new spaces and also to define str and sthr of strongly r-statistically convergent and lacunary strongly r-statistically convergent sequences, respectively. Also, we give some inclusion relations involving these spaces. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Lacunary sequence; Statistical convergence; Invariant means and modulus functions
1. Introduction and background A complex number sequence x ¼ ðxk Þ is said to be statistically convergent to the number ‘ if for every e > 0 lim
n!1
1 jfk 6 n : jxk ‘j P egj ¼ 0; n
where the vertical bars indicate the number of elements in the enclose set. In this case we write s lim x ¼ ‘ or x ¼ ‘ðstÞ. The idea of the statistical convergence of sequence of real numbers was introduced by Fast [1]. Schoenberg [16] studied statistical convergence as a summability method and listed some of the elementary properties of statistical
E-mail address: [email protected] (V. Karakaya). 0020-0255/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2003.12.005
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V. Karakaya / Information Sciences 166 (2004) 271–280
convergence. Recently, in [13], Savasß applied statistically convergent to Fuzzy numbers. at [10], Fridy Statistical convergence sequence have been discussed by Sal and Miller [3], Miller and Orhan [7], Savasß and Nuray [14], Savasß [11], Fridy [4] and others independently. Let r be a mapping of the set of positive integers into itself. A continuous linear functional / on ‘1 , the space of real bounded sequences, is said to be an invariant mean or r-mean if and only if (i) /ðxÞ P 0 when the sequence x ¼ ðxk Þ has xk P 0 for all n, (ii) /ðeÞ ¼ 1, where e ¼ ð1; 1; 1; . . .Þ and, (iii) /ððxrðnÞ ÞÞ ¼ /ðxÞ for all x 2 ‘1 . The mapping r is one to one such that rm ðnÞ 6¼ n for all positive integer n and m where rm ðnÞ denotes the mth iterate of the mapping r at n. Thus / extends the limit functional on c, the spaces of convergent sequence, in the sense that /ðxÞ ¼ lim x for all x 2 c. If x ¼ ðxn Þ, write Tx ¼ Txn ¼ ðxrðnÞ Þ. It can be shown [15] that Vr ¼ x 2 ‘1 : lim tkn ðxÞ ¼ Le; uniformly in n; L ¼ r lim x ; k
where xn þ xr1 ðnÞ þ þ xrk ðnÞ : kþ1 In the case r is the translation mapping n ! n þ 1, r-mean is often called a Banach limit and Vr , the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequence, (see [5]). Several authors including Schaefer [15], Mursaleen [8], and many others have studied invariant convergent sequences. By a lacunary sequence, we mean an increasing integer sequence h ¼ ðkr Þ such that k0 ¼ 0 and hr ¼ kr kr1 ! 1 as r ! 1. Throughout this paper, the intervals determined by h will be denoted by Ir ¼ ðkr1 ; kr . Freedman et al. [2] defined the space Nh in the following way: For any lacunary sequence h ¼ ðkr Þ ( ) 1 X Nh ¼ x ¼ ðxk Þ : lim jxk ‘j ¼ 0; for some ‘ : r!1 hr k2Ir tkn ðxÞ ¼
Again, in [2], it is defined that the lacunary sequence h0 ¼ ðkr0 Þ is called a lacunary refinement of the lacunary sequence h ¼ ðkr Þ if fkr g fkr0 g. A modulus function f is a function from ½0; 1Þ to ½0; 1Þ such that (i) (ii) (iii) (iv)
f ðxÞ ¼ 0 if and only if x ¼ 0, f ðx þ yÞ 6 f ðxÞ þ f ðyÞ, for all x, y > 0, f is increasing, f is continuous from the right at zero.
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273
Since jðxÞ f ðyÞj 6 f ðjx yjÞ, it follows from conditions (ii) and (iv) that f is continuous everywhere on ½0; 1Þ. A modulus function may be bounded and unbounded. For example, 1 f ðtÞ ¼ tþ1 is bounded but f ðtÞ ¼ tp ð0 < p 6 1Þ is unbounded. Ruckle [9] and Maddox [6] and other authors used modulus function to construct new sequence spaces. Recently, Savasß [12] defined and studied some sequence spaces by using a modulus f . Now we will introduce two new sequence spaces. Also we will investigate inclusion relations between these new spaces and strongly r-statistically convergent and lacunary strongly r-statistically convergent.
2. Definitions and theorems Before giving the promised inclusion relations, we will give some new definitions. Definition 1. Let f be a modulus and h ¼ ðkr Þ be lacunary sequence. We define following new spaces: ( ) 1 X wr ½f h ¼ x ¼ ðxk Þ : lim f ðjtkn ðx ‘eÞjÞ ¼ 0 uniformly in n ; r!1 hr k2Ir ( ) m 1 X f ðjtkn ðx ‘eÞjÞ ¼ 0 uniformly in n : wr ½f ¼ x ¼ ðxk Þ : lim m!1 m k¼1 Definition 2. A sequence x ¼ ðxk Þ is said to be strongly r-statistically convergent to the number ‘ if for every e > 0 lim sup
m!1
n
1 jf0 6 k 6 m : jtkn ðjx ‘ejÞj P egj ¼ 0: m
In this case we write str lim x ¼ ‘e. Definition 3. Let h ¼ ðkr Þ be a lacunary sequence. The number sequence x ¼ ðxk Þ is said to be lacunary strongly r-statistically convergent if for every e>0 lim sup
r!1
n
1 jfk 2 Ir : jtkn ðjx ‘ejÞj P egj ¼ 0: hr
In this case, we have sthr lim x ¼ ‘e. We now give our theorems and lemmas.
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Lemma 1. Let f be any modulus and 0 < d < 1. Then f ðjtkn ðxÞjÞ 6 2f ð1Þd1 jtkn ðxÞj; where jtkn ðxÞj P d for all n and k. Theorem 1. Let f be any modulus. Then wr ½f h sthr . Theorem 2. wr ½f h ¼ sthr if and only if f is bounded. Let us give a lemma without proof which will be used in the proof of Theorem 3. Lemma 2. Suppose that for given e1 and every e > 0, there exists m0 and n0 such that 1 jf0 6 k 6 m : jtkn ðjx ‘ejÞj P egj < e1 m 1
for all m P m0 and n P n0 , then ftkn ðjx ‘ejÞgk¼1 2 str for all n. Since the proofs of above Theorems and Lemmas are routine, we won’t give their proofs. Theorem 3. sthr ¼ str for every lacunary sequence h. Proof. Let x 2 sthr . Then, form Definition 3, for given e1 > 0, there exist r0 and ‘ such that 1 jf0 6 k 6 hr 1 : jtkn ðjx ‘ejÞj P egj < e1 hr for r P r0 and n ¼ kr1 þ u, u P 0. Let m P hr , write m ¼ phr þ t, where 0 6 t 6 hr , p is an integer number. Since m P hr , p P 1, we have 1 jf0 6 k 6 m 1 : jtkn ðjx ‘ejÞj P egj m p 1 X jfjhr 6 k 6 ðj þ 1Þhr 1 : jtkn ðjx ‘ejÞj P egj 6 m j¼0 6 For
hr m
1 2phr e1 ðp þ 1Þhr e1 6 ðp P 1Þ: m m
6 1 and since
phr m
6 1, we have
1 jf0 6 k 6 m 1 : jtkn ðjx ‘ejÞj P egj 6 2e1 : m
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So by the Lemma 2, we get that sthr str . We now show that str sthr . Let x 2 str . Then from Definition 2, we can write 1 jf0 6 k 6 m 1 : jtkn ðjx ‘ejÞj P egj 6 e1 : m
ð2:1Þ
Let m P hr . Taking m ¼ phr þ t, where 0 6 t 6 hr , p is an integer number. Since m P phr and m 6 ðp þ 1Þhr , we have 1 jf0 6 k 6 m 1 : jtkn ðjx ‘ejÞj P egj m 1 P jf0 6 k 6 phr 1 : jtkn ðjx ‘ejÞj P egj ðp þ 1Þhr P
1 jf0 6 k 6 phr 1 : jtkn ðjx ‘ejÞj P egj 2phr
¼
p 1 X jfðj 1Þhr 6 k 6 jhr 1 : jtkn ðjx ‘ejÞj P egj 2phr j¼1
¼
1 jf0 6 k 6 hr 1 : jtkn ðjx ‘ejÞj P egj: 2hr
By (2.1), we get 1 jf0 6 k 6 hr 1 : jtkn ðjx ‘ejÞj P egj 6 e1 : 2hr So it is founded that 1 jf0 6 k 6 hr 1 : jtkn ðjx ‘ejÞj P egj 6 2e1 : hr Consequently, it has been shown that str sthr . This completes the proof of the theorem. h Let h0 ¼ ðkr0 Þ be lacunary refinement sequence of lacunary sequence h ¼ ðkr Þ. Now it will be examined connection between sthr and sth0 r . Also we will give the theorems concerning inclusion relation between sthr and sth0 r where h and h0 are any two lacunary sequences. Theorem 4. Suppose that h0 ¼ ðkr0 Þ is a lacunary refinement of the lacunary se0 quence h ¼ ðkr Þ. Let Ir ¼ ðkr1 ; kr and Ir0 ¼ ðkr1 ; kr0 , r ¼ 1; 2; 3; . . .. If there exists d > 0 such that Ij0 Pd Ii
for every Ij0 Ii :
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Then sthr limk tkn ðxÞ ¼ ‘e implies sth0 r limk tkn ðxÞ ¼ ‘e uniformly in n. Proof. For any e > 0 and every Ij0 , we can find Ii such that Ij0 Ii ; then we have, for every n 1 jfk 2 Ij0 : jtkn ðjx ‘ejÞj P egj jIj0 j ! jIi j 1 ¼ jfk 2 Ij0 : jtkn ðjx ‘ejÞj P egj jIj0 j jIi j jIi j jIj0 j
6 1 d
6
!
1 jfk 2 Ii : jtkn ðjx ‘ejÞj P egj jIi j
1 jfk 2 Ii : jtkn ðjx ‘ejÞj P egj: jIi j
So it seen that sthr sth0 r . h Theorem 5. Suppose that h ¼ ðkr Þ, h0 ¼ ðkr0 Þ are two lacunary sequences. Let 0 Ir ¼ ðkr1 ; kr and Ir0 ¼ ðkr1 ; kr0 , r ¼ 1; 2; 3; . . .. There exists a sequence 1 ftkn ðjx ‘ejÞgk¼1 and a number ‘ such that sth0 r limk tkn ðjxjÞ ¼ ‘e and sthr limk tkn ðjxjÞ 6¼ ‘e uniformly in n, if and only if there exist fsi g, fti g N and d > 0 such that (i) Is0i \ Iti 6¼ £, (ii) limi (iii)
jIs0 j i
jIti j jIs0 \Iti j i
jIti j
¼ 1,
P d i ¼ 1; 2; 3; . . .. 1
Proof. If there exists a sequence ftkn ðjx ‘ejÞgk¼1 and a number ‘ such that sth0 r limk tkn ðjxjÞ ¼ ‘e and sthr limk tkn ðjxjÞ 6¼ ‘e uniformly in n, then there exist a subsequence fti g N , e > 0, d > 0 such that 1 ð2:2Þ jfk 2 Iti : jtkn ðjx ‘ejÞj P egj P d jIti j for each ti , there exist a positive integer si and a whole number ri such that Is0i þj \ Iti 6¼ £
for every j ¼ 1; 2; 3; . . .
and
i ¼ 1; 2; 3; . . . :
ð2:3Þ
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277
Then we can write 1 d6 jfk 2 Iti : jtkn ðjx ‘ejÞj P egj jIti j ) ! ( n [ 1 0 ¼ Isi þj \ Iti : jtkn ðjx ‘ejÞj P e k2 jIti j j¼0 X ri 1 ¼ jfk 2 Is0i þj \ Iti : jtkn ðjx ‘ejÞj P egj jIti j j¼0 1 ¼ fj ¼ 0; ri jfk 2 Is0i þj \ Iti : jtkn ðjx ‘ejÞj P egjg jIti j X 1 þ jfk 2 Is0i þj \ Iti : jtkn ðjx ‘ejÞj P egj jIti j 0
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This implies that n oo 1 n d j ¼ 0; ri : k 2 Is0i þj \ Iti : jtkn ðj x ‘ejÞj P e P 2 jIti j uniformly in n, which ensures that at least one of the following inequalities hold: o 1 n d ð2:5Þ k 2 Is0i þj \ Iti : jtkn ðj x ‘ejÞj P e P 4 jIti j or
1 d k 2 Is0i þri \ Iti : jtkn ðj x ‘ejÞj P e P : 4 jIti j
ð2:6Þ
Suppose that the (2.5) holds. From this inequality and since k 2 I 0 \ It : jtkn ðj x ‘ejÞj P e 6 I 0 \ It ; i i si si we conclude that d Is0i \ Iti 6 ; 4 jIti j which proves (iii). For such si and ti chosen in the proof of (iii), from (2.5), we have d 1 6 k 2 Is0i \ Iti : jtkn ðj x ‘ejÞj P e 4 jIti j ! 0 ! I 1 si 6 k 2 Is0i \ Iti : jtkn ðj x ‘ejÞj P e 0 jIti j I si for all n. Since ! 1 0 !0 \ I : t ð x k 2 I ‘ejÞj P e j j t kn i s i I 0
ð2:7Þ
as i ! 1
si
jI 0 j
uniformly in n, (2.7) implies jIsti j ! 1 as i ! 1, which proves (ii). It is clear i that the intervals Is0i and Iti chosen in the proof of (iii) satisfy (i). Conversely, suppose that h ¼ ðkr Þ and h0 ¼ ðkr0 Þ are two lacunary sequences. Then there exist sequences fsi g, fti g N and d > 0 which satisfy the conditions (i)–(iii) in the theorem. 1 Let us define ftkn ðjxjÞgk¼1 as follows, tkn ðjxjÞ ¼ 1 for all n if k 2 Is0i \ Iti and tkn ðjxjÞ ¼ 0 for all n if k 62 Is0i \ Iti . For any 0 < e < 1, if j 6¼ si for any i ¼ 1; 2; 3; . . ..
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! o j £j 1 n k 2 Ij0 : jtkn ðjx 0jÞj P e ¼ ¼ 0 Ij0 Ij0 uniformly in n; if j ¼ si , for some i ! Is0i \ Iti 1 jIt j 0 k 2 Isi \ Iti : jtkn ðjx 0jÞj P e ¼ 0 6 0i ! 0 I 0 I I si
si
si
as i ! 1 uniformly in n. Hence sth0 r limk tkn ðjxjÞ ¼ 0. But 0 I \ It 1 i si Pd jfk 2 Iti : jtkn ðjx 0jÞj P egj ¼ jIti j jIti j for all n and i ¼ 1; 2; 3; . . .. This implies that sthr limk tkn ðjxjÞ 6¼ 0. Hence this completes the proof. h
3. Conclusion In this work, by combining modulus function, r-mean and lacunary sequence, two new spaces wr ½f h and wr ½f , and two different concept of statistical convergence have been introduced. It has been shown that there is a connection between the sequence space wr ½f h and lacunary strongly r-statistically convergent respect to modulus function. In the same manner, it has been shown that sthr ¼ str for every lacunary sequence h. Furthermore, by using lacunary sequence and its lacunary refinement sequence, it has been shown that the spaces sthr and sth0 r have the same limit if there exists d > 0 such that Ij0 Pd Ii
for every Ij0 Ii :
Finally, the Theorem 5 involving the inclusion properties of different lacunary methods has been given and proved.
References [1] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241–244. [2] A.R. Freedman, J.J. Sember, M. Raphael, Some Cesaro-type summability spaces, Proc. Lond. Math. Soc. 37 (1978) 508–520. [3] J.A. Fridy, H.I. Miller, A matrix characterization of statistical convergence, Analysis 11 (1991) 59–66.
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[4] J.A. Fridy, On statistical convergence, Analysis 5 (1985) 301–313. [5] G.G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80 (1948) 167– 190. [6] I.J. Maddox, Sequence spaces defined by a modulus, Math. Proc. Camb. Phil. Soc. 100 (1986) 161–166. [7] H.I. Miller, C. Orhan, On almost convergent and statistically convergent subsequences, Acta Math. Hungar. 93 (1–2) (2001) 135–151. [8] Mursaleen, On infinite matrices and invariant means, Indian J. Pure Appl. Math. 10 (4) (1979) 457–460. [9] W.H. Ruckle, FK spaces in which the sequence of coordinate vectors in bounded, Canad. J. Math. 25 (1973) 973–978. at, On statistically convergence sequences of real numbers, Math. Slovaca 30 (1980) 139– [10] T. Sal 150. [11] E. Savasß, On sequence spaces and b S -convergence, Demost. Math. 33 (1) (2000) 165–170. [12] E. Savasß, On some generalized sequence spaces defined by a modulus, Indian J. Pure Appl. Math. 38 (1) (1999) 459–464. [13] E. Savasß, On statistically convergent sequences of fuzzy numbers, Inform. Sci. 137 (1-4) (2001) 277–282. [14] E. Savasß, F. Nuray, On r-statistically convergence and lacunary r-statistically convergence, Math. Slovaca 43 (3) (1993) 309–315. [15] P. Schaefer, Infinite matrices and invariant means, Proc. Am. Math. Soc. 36 (1) (1972) 104– 110. [16] I.J. Schoenberg, The integrability of certain functions and related summability methods, Am. Math. Monthly 66 (1959) 361–375.
On lacunary r-statistical convergence Vatan Karakaya Department of Mathematics, Education Faculty of Adiyaman, Gaziantep University, Adiyaman, Turkey Received 6 October 2002; received in revised form 20 November 2003; accepted 2 December 2003
Abstract The purpose of this paper is to introduce two new spaces and also to define str and sthr of strongly r-statistically convergent and lacunary strongly r-statistically convergent sequences, respectively. Also, we give some inclusion relations involving these spaces. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Lacunary sequence; Statistical convergence; Invariant means and modulus functions
1. Introduction and background A complex number sequence x ¼ ðxk Þ is said to be statistically convergent to the number ‘ if for every e > 0 lim
n!1
1 jfk 6 n : jxk ‘j P egj ¼ 0; n
where the vertical bars indicate the number of elements in the enclose set. In this case we write s lim x ¼ ‘ or x ¼ ‘ðstÞ. The idea of the statistical convergence of sequence of real numbers was introduced by Fast [1]. Schoenberg [16] studied statistical convergence as a summability method and listed some of the elementary properties of statistical
E-mail address: [email protected] (V. Karakaya). 0020-0255/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2003.12.005
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V. Karakaya / Information Sciences 166 (2004) 271–280
convergence. Recently, in [13], Savasß applied statistically convergent to Fuzzy numbers. at [10], Fridy Statistical convergence sequence have been discussed by Sal and Miller [3], Miller and Orhan [7], Savasß and Nuray [14], Savasß [11], Fridy [4] and others independently. Let r be a mapping of the set of positive integers into itself. A continuous linear functional / on ‘1 , the space of real bounded sequences, is said to be an invariant mean or r-mean if and only if (i) /ðxÞ P 0 when the sequence x ¼ ðxk Þ has xk P 0 for all n, (ii) /ðeÞ ¼ 1, where e ¼ ð1; 1; 1; . . .Þ and, (iii) /ððxrðnÞ ÞÞ ¼ /ðxÞ for all x 2 ‘1 . The mapping r is one to one such that rm ðnÞ 6¼ n for all positive integer n and m where rm ðnÞ denotes the mth iterate of the mapping r at n. Thus / extends the limit functional on c, the spaces of convergent sequence, in the sense that /ðxÞ ¼ lim x for all x 2 c. If x ¼ ðxn Þ, write Tx ¼ Txn ¼ ðxrðnÞ Þ. It can be shown [15] that Vr ¼ x 2 ‘1 : lim tkn ðxÞ ¼ Le; uniformly in n; L ¼ r lim x ; k
where xn þ xr1 ðnÞ þ þ xrk ðnÞ : kþ1 In the case r is the translation mapping n ! n þ 1, r-mean is often called a Banach limit and Vr , the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequence, (see [5]). Several authors including Schaefer [15], Mursaleen [8], and many others have studied invariant convergent sequences. By a lacunary sequence, we mean an increasing integer sequence h ¼ ðkr Þ such that k0 ¼ 0 and hr ¼ kr kr1 ! 1 as r ! 1. Throughout this paper, the intervals determined by h will be denoted by Ir ¼ ðkr1 ; kr . Freedman et al. [2] defined the space Nh in the following way: For any lacunary sequence h ¼ ðkr Þ ( ) 1 X Nh ¼ x ¼ ðxk Þ : lim jxk ‘j ¼ 0; for some ‘ : r!1 hr k2Ir tkn ðxÞ ¼
Again, in [2], it is defined that the lacunary sequence h0 ¼ ðkr0 Þ is called a lacunary refinement of the lacunary sequence h ¼ ðkr Þ if fkr g fkr0 g. A modulus function f is a function from ½0; 1Þ to ½0; 1Þ such that (i) (ii) (iii) (iv)
f ðxÞ ¼ 0 if and only if x ¼ 0, f ðx þ yÞ 6 f ðxÞ þ f ðyÞ, for all x, y > 0, f is increasing, f is continuous from the right at zero.
V. Karakaya / Information Sciences 166 (2004) 271–280
273
Since jðxÞ f ðyÞj 6 f ðjx yjÞ, it follows from conditions (ii) and (iv) that f is continuous everywhere on ½0; 1Þ. A modulus function may be bounded and unbounded. For example, 1 f ðtÞ ¼ tþ1 is bounded but f ðtÞ ¼ tp ð0 < p 6 1Þ is unbounded. Ruckle [9] and Maddox [6] and other authors used modulus function to construct new sequence spaces. Recently, Savasß [12] defined and studied some sequence spaces by using a modulus f . Now we will introduce two new sequence spaces. Also we will investigate inclusion relations between these new spaces and strongly r-statistically convergent and lacunary strongly r-statistically convergent.
2. Definitions and theorems Before giving the promised inclusion relations, we will give some new definitions. Definition 1. Let f be a modulus and h ¼ ðkr Þ be lacunary sequence. We define following new spaces: ( ) 1 X wr ½f h ¼ x ¼ ðxk Þ : lim f ðjtkn ðx ‘eÞjÞ ¼ 0 uniformly in n ; r!1 hr k2Ir ( ) m 1 X f ðjtkn ðx ‘eÞjÞ ¼ 0 uniformly in n : wr ½f ¼ x ¼ ðxk Þ : lim m!1 m k¼1 Definition 2. A sequence x ¼ ðxk Þ is said to be strongly r-statistically convergent to the number ‘ if for every e > 0 lim sup
m!1
n
1 jf0 6 k 6 m : jtkn ðjx ‘ejÞj P egj ¼ 0: m
In this case we write str lim x ¼ ‘e. Definition 3. Let h ¼ ðkr Þ be a lacunary sequence. The number sequence x ¼ ðxk Þ is said to be lacunary strongly r-statistically convergent if for every e>0 lim sup
r!1
n
1 jfk 2 Ir : jtkn ðjx ‘ejÞj P egj ¼ 0: hr
In this case, we have sthr lim x ¼ ‘e. We now give our theorems and lemmas.
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Lemma 1. Let f be any modulus and 0 < d < 1. Then f ðjtkn ðxÞjÞ 6 2f ð1Þd1 jtkn ðxÞj; where jtkn ðxÞj P d for all n and k. Theorem 1. Let f be any modulus. Then wr ½f h sthr . Theorem 2. wr ½f h ¼ sthr if and only if f is bounded. Let us give a lemma without proof which will be used in the proof of Theorem 3. Lemma 2. Suppose that for given e1 and every e > 0, there exists m0 and n0 such that 1 jf0 6 k 6 m : jtkn ðjx ‘ejÞj P egj < e1 m 1
for all m P m0 and n P n0 , then ftkn ðjx ‘ejÞgk¼1 2 str for all n. Since the proofs of above Theorems and Lemmas are routine, we won’t give their proofs. Theorem 3. sthr ¼ str for every lacunary sequence h. Proof. Let x 2 sthr . Then, form Definition 3, for given e1 > 0, there exist r0 and ‘ such that 1 jf0 6 k 6 hr 1 : jtkn ðjx ‘ejÞj P egj < e1 hr for r P r0 and n ¼ kr1 þ u, u P 0. Let m P hr , write m ¼ phr þ t, where 0 6 t 6 hr , p is an integer number. Since m P hr , p P 1, we have 1 jf0 6 k 6 m 1 : jtkn ðjx ‘ejÞj P egj m p 1 X jfjhr 6 k 6 ðj þ 1Þhr 1 : jtkn ðjx ‘ejÞj P egj 6 m j¼0 6 For
hr m
1 2phr e1 ðp þ 1Þhr e1 6 ðp P 1Þ: m m
6 1 and since
phr m
6 1, we have
1 jf0 6 k 6 m 1 : jtkn ðjx ‘ejÞj P egj 6 2e1 : m
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So by the Lemma 2, we get that sthr str . We now show that str sthr . Let x 2 str . Then from Definition 2, we can write 1 jf0 6 k 6 m 1 : jtkn ðjx ‘ejÞj P egj 6 e1 : m
ð2:1Þ
Let m P hr . Taking m ¼ phr þ t, where 0 6 t 6 hr , p is an integer number. Since m P phr and m 6 ðp þ 1Þhr , we have 1 jf0 6 k 6 m 1 : jtkn ðjx ‘ejÞj P egj m 1 P jf0 6 k 6 phr 1 : jtkn ðjx ‘ejÞj P egj ðp þ 1Þhr P
1 jf0 6 k 6 phr 1 : jtkn ðjx ‘ejÞj P egj 2phr
¼
p 1 X jfðj 1Þhr 6 k 6 jhr 1 : jtkn ðjx ‘ejÞj P egj 2phr j¼1
¼
1 jf0 6 k 6 hr 1 : jtkn ðjx ‘ejÞj P egj: 2hr
By (2.1), we get 1 jf0 6 k 6 hr 1 : jtkn ðjx ‘ejÞj P egj 6 e1 : 2hr So it is founded that 1 jf0 6 k 6 hr 1 : jtkn ðjx ‘ejÞj P egj 6 2e1 : hr Consequently, it has been shown that str sthr . This completes the proof of the theorem. h Let h0 ¼ ðkr0 Þ be lacunary refinement sequence of lacunary sequence h ¼ ðkr Þ. Now it will be examined connection between sthr and sth0 r . Also we will give the theorems concerning inclusion relation between sthr and sth0 r where h and h0 are any two lacunary sequences. Theorem 4. Suppose that h0 ¼ ðkr0 Þ is a lacunary refinement of the lacunary se0 quence h ¼ ðkr Þ. Let Ir ¼ ðkr1 ; kr and Ir0 ¼ ðkr1 ; kr0 , r ¼ 1; 2; 3; . . .. If there exists d > 0 such that Ij0 Pd Ii
for every Ij0 Ii :
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Then sthr limk tkn ðxÞ ¼ ‘e implies sth0 r limk tkn ðxÞ ¼ ‘e uniformly in n. Proof. For any e > 0 and every Ij0 , we can find Ii such that Ij0 Ii ; then we have, for every n 1 jfk 2 Ij0 : jtkn ðjx ‘ejÞj P egj jIj0 j ! jIi j 1 ¼ jfk 2 Ij0 : jtkn ðjx ‘ejÞj P egj jIj0 j jIi j jIi j jIj0 j
6 1 d
6
!
1 jfk 2 Ii : jtkn ðjx ‘ejÞj P egj jIi j
1 jfk 2 Ii : jtkn ðjx ‘ejÞj P egj: jIi j
So it seen that sthr sth0 r . h Theorem 5. Suppose that h ¼ ðkr Þ, h0 ¼ ðkr0 Þ are two lacunary sequences. Let 0 Ir ¼ ðkr1 ; kr and Ir0 ¼ ðkr1 ; kr0 , r ¼ 1; 2; 3; . . .. There exists a sequence 1 ftkn ðjx ‘ejÞgk¼1 and a number ‘ such that sth0 r limk tkn ðjxjÞ ¼ ‘e and sthr limk tkn ðjxjÞ 6¼ ‘e uniformly in n, if and only if there exist fsi g, fti g N and d > 0 such that (i) Is0i \ Iti 6¼ £, (ii) limi (iii)
jIs0 j i
jIti j jIs0 \Iti j i
jIti j
¼ 1,
P d i ¼ 1; 2; 3; . . .. 1
Proof. If there exists a sequence ftkn ðjx ‘ejÞgk¼1 and a number ‘ such that sth0 r limk tkn ðjxjÞ ¼ ‘e and sthr limk tkn ðjxjÞ 6¼ ‘e uniformly in n, then there exist a subsequence fti g N , e > 0, d > 0 such that 1 ð2:2Þ jfk 2 Iti : jtkn ðjx ‘ejÞj P egj P d jIti j for each ti , there exist a positive integer si and a whole number ri such that Is0i þj \ Iti 6¼ £
for every j ¼ 1; 2; 3; . . .
and
i ¼ 1; 2; 3; . . . :
ð2:3Þ
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277
Then we can write 1 d6 jfk 2 Iti : jtkn ðjx ‘ejÞj P egj jIti j ) ! ( n [ 1 0 ¼ Isi þj \ Iti : jtkn ðjx ‘ejÞj P e k2 jIti j j¼0 X ri 1 ¼ jfk 2 Is0i þj \ Iti : jtkn ðjx ‘ejÞj P egj jIti j j¼0 1 ¼ fj ¼ 0; ri jfk 2 Is0i þj \ Iti : jtkn ðjx ‘ejÞj P egjg jIti j X 1 þ jfk 2 Is0i þj \ Iti : jtkn ðjx ‘ejÞj P egj jIti j 0
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This implies that n oo 1 n d j ¼ 0; ri : k 2 Is0i þj \ Iti : jtkn ðj x ‘ejÞj P e P 2 jIti j uniformly in n, which ensures that at least one of the following inequalities hold: o 1 n d ð2:5Þ k 2 Is0i þj \ Iti : jtkn ðj x ‘ejÞj P e P 4 jIti j or
1 d k 2 Is0i þri \ Iti : jtkn ðj x ‘ejÞj P e P : 4 jIti j
ð2:6Þ
Suppose that the (2.5) holds. From this inequality and since k 2 I 0 \ It : jtkn ðj x ‘ejÞj P e 6 I 0 \ It ; i i si si we conclude that d Is0i \ Iti 6 ; 4 jIti j which proves (iii). For such si and ti chosen in the proof of (iii), from (2.5), we have d 1 6 k 2 Is0i \ Iti : jtkn ðj x ‘ejÞj P e 4 jIti j ! 0 ! I 1 si 6 k 2 Is0i \ Iti : jtkn ðj x ‘ejÞj P e 0 jIti j I si for all n. Since ! 1 0 !0 \ I : t ð x k 2 I ‘ejÞj P e j j t kn i s i I 0
ð2:7Þ
as i ! 1
si
jI 0 j
uniformly in n, (2.7) implies jIsti j ! 1 as i ! 1, which proves (ii). It is clear i that the intervals Is0i and Iti chosen in the proof of (iii) satisfy (i). Conversely, suppose that h ¼ ðkr Þ and h0 ¼ ðkr0 Þ are two lacunary sequences. Then there exist sequences fsi g, fti g N and d > 0 which satisfy the conditions (i)–(iii) in the theorem. 1 Let us define ftkn ðjxjÞgk¼1 as follows, tkn ðjxjÞ ¼ 1 for all n if k 2 Is0i \ Iti and tkn ðjxjÞ ¼ 0 for all n if k 62 Is0i \ Iti . For any 0 < e < 1, if j 6¼ si for any i ¼ 1; 2; 3; . . ..
V. Karakaya / Information Sciences 166 (2004) 271–280
279
! o j £j 1 n k 2 Ij0 : jtkn ðjx 0jÞj P e ¼ ¼ 0 Ij0 Ij0 uniformly in n; if j ¼ si , for some i ! Is0i \ Iti 1 jIt j 0 k 2 Isi \ Iti : jtkn ðjx 0jÞj P e ¼ 0 6 0i ! 0 I 0 I I si
si
si
as i ! 1 uniformly in n. Hence sth0 r limk tkn ðjxjÞ ¼ 0. But 0 I \ It 1 i si Pd jfk 2 Iti : jtkn ðjx 0jÞj P egj ¼ jIti j jIti j for all n and i ¼ 1; 2; 3; . . .. This implies that sthr limk tkn ðjxjÞ 6¼ 0. Hence this completes the proof. h
3. Conclusion In this work, by combining modulus function, r-mean and lacunary sequence, two new spaces wr ½f h and wr ½f , and two different concept of statistical convergence have been introduced. It has been shown that there is a connection between the sequence space wr ½f h and lacunary strongly r-statistically convergent respect to modulus function. In the same manner, it has been shown that sthr ¼ str for every lacunary sequence h. Furthermore, by using lacunary sequence and its lacunary refinement sequence, it has been shown that the spaces sthr and sth0 r have the same limit if there exists d > 0 such that Ij0 Pd Ii
for every Ij0 Ii :
Finally, the Theorem 5 involving the inclusion properties of different lacunary methods has been given and proved.
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