Vietnam Journal of Mathematics 34:2 (2006) 129–138
9LHWQDP -RXUQDO
RI
0$7+(0$7,&6
9$67
Lacunary St rongly Sum m able Sequences and
q- Lacunary A lm ost St at ist ical Convergence
R ifat ¸Colak1,B.C.Tripathy
2,andMikˆail E t 1
1Department of Mathematics, Firat University, 23119, Elazı˘g-Turkey
2Mathematical Sciences Division
Institute of Advanced Study in Science and Technology,
Paschim Baragoan, Garchuk, Guwahati 781035, Assam, I ndia
Received J anuary 28, 2005
Revised February 28, 2006
A bst ract . A lacunary sequence is an increasing sequence θ= ( kr)of p osit ive int egers
such t hat k0= 0 and kr−kr−1→ ∞ as r→ ∞ .A sequence x= ( xk)is called q−lacunary almost
st at ist ical convergent t o ξprovided t hat for each ε > 0,lim r(kr−kr−1)−1{t he nu mb er of
k:kr−1< k kr:q(tk m (x)−ξ)ε}= 0.T h e p urp ose of t his pap er is t o int rod uce t he con cept
of q−lacunary st rongly alm ost convergence wit h resp ect t o an Orlicz funct ion and
q−lacu nary almost st at ist ical convergence, and exam ine som e p rop ert ies of t hese se-
quence spaces. We est ab lish some connect ions b et ween q−lacun ary st rongly almost
convergence and q−lacunary almost st at ist ical convergence. It is also shown t hat if a
sequence is q−lacunary st rongly almost convergent wit h resp ect t o an Orlicz funct ion
t hen it is q−lacunary almost st at ist ical convergent .
2000 Mat hem at ics Su b ject Classificat ion: 40A05, 40C05, 46A45.
K eywords: St at ist ical convergence, lacunary sequence, Orlicz funct ion, almost conver-
gence.
1. Int ro duction
Let wdenot e t he set of all real sequen ces x= (xn).By ℓ∞and c, we d enot e
t he Bana ch spaces of b ounded an d convergent sequences x= (xn)normedby
x= sup
n|xn|,respect ively. A linear funct ion al Lon ℓ∞is said t o b e a Ba nach
lim it [1] if it ha s t h e pr op ert ies:
130 Rifat ¸Colak, B. C. Tripathy, and Mikˆail Et
i) L(x)0ifx0 (i.e. xn0 for all n),
ii) L(e)= 1,where e= (1,1,...),
iii) L(D x )= L(x),
where Dis t he shift op er at or defin ed by (D x n)= (xn+ 1 ).
Let Bbe t he set of all Banach limit s on ℓ∞.A sequence xis said t o b e almost
convergent t o a numb er ξif L(x)= ξfor all L ∈B . Lor ent z [12] has shown t hat
xis almost convergent t o ξif and only if
tk m =tk m (x)= xm+xm+ 1 +...+xm+k
k+ 1 →ξas k→∞,uniformly in m .
Let fdenot e t he set of all alm ost convergent sequ ences. We writ e f−lim x=ξ
if xis a lmost convergent t o ξ. Ma ddox [13] and (in dep endent ly) Freedm an et a l.
[7] have defined xt o b e st rongly a lmost convergent t o a numb er ξif
1
k+ 1
k
i= 0
|xi+m−ξ|→0ask→∞,uniform ly in m .
Let [f] d enot e t he set of all st rongly alm ost convergent sequ ences. If xis
st r ongly alm ost convergent t o ξ, we writ e [f]−lim x=ξ. It is ea sy t o see t hat
[f]⊂f⊂ℓ∞.Das and Sa hoo [4] defined t h e sequ ence spa ce
[w(p)] =
x∈w:1
n+ 1
n
k= 0
|tk m (x)−ξ|pk→0asn→∞,uniformly in m
an d invest igat ed som e of it s prop ert ies.
T h e definit ion of st a t ist ical conver gen ce was int rod uced by Fast [6] in a sh ort
not e. Schoenb erg [20] st u died st at ist ical conver gen ce as a sum ma bility m et hod
and list ed some of t he elem ent ary p rop ert ies of st at ist ical convergence. Recent ly,
st a t ist ical convergence h as b een st ud ied by variou s aut hors (cf. [3, 8, 9, 14, 17,
18]).
T he st at ist ical convergence dep ends on t he den sit y of t he subset s of N,the
set of nat u ral numb ers. A su bset Eof Nis said t o have densit y δ(E)if
δ(E) = lim
n→ ∞
1
n
n
k= 1
χE(k) exist s,
where χEis t he char act erist ic fun ct ion of E .
A sequence (xn) is said t o b e st a t ist ica lly convergent t o ξif for every ε>
0,δ
{k∈N:|xk−ξ|ε}
= 0.In t his ca se we wr it e st at -lim xk=ξ.
Let θ= (kr) b e t h e sequ ence of p osit ive int egers su ch t ha t k0= 0,0< k
r<
kr+ 1 and hr=kr−kr−1→∞ as r→∞.T hen θis called a lacunary sequence.
T h e int ervals det er min ed by θwill b e denot ed by Ir= (kr−1,kr]andtheratio
kr/ kr−1will b e d enot ed by ηr.
Lacunary sequences have b een st udied in [2, 5, 7, 9, 19].
An O rlicz funct ion is a fun ct ion M:[0,∞)→[0,∞), which is cont inu ous,
non-decreasing and convex wit h M(0) = 0, M(x)>0forx > 0andM(x)→∞
as x→∞.
Lacunary Sequences and Almost Stati sti cal Convergence 131
Lin denst r auss a nd T zafriri [11] used t he idea of O rlicz funct ion t o con st ru ct
t he sequence space
ℓM=
x∈w:
∞
k= 1
M
|xk|
ρ
<∞for som e ρ> 0
.
T he space ℓMwit h t he n orm
x= inf
ρ> 0:
∞
k= 1
M
|xk|
ρ
1
becomes a Banach space, called an Orlicz sequence space. T he space ℓMis closely
relat ed t o t he space ℓpwhich is an O rlicz sequence space wit h M(x)= |x|pfor
1p< ∞.
R ecent ly O rlicz sequ ence spa ces have b een st ud ied by Mursaleen et al. [15],
Bhardwa j and Singh [2], Sava¸s and Rhoades [19] and many ot hers.
A sequence space Eis said t o b e solid (or n ormal) if (αkxk)∈Ewhenever
(xk)∈Efor all sequences (αk)ofscalarswith|αk|1 for all k∈N.
A sequence space Eis sa id t o be m onot one if it cont a ins t he canonical preim -
ages of it s st ep sp aces [10].
Remar k. T wo O rlicz funct ion s M1and M2ar e said t o b e equ ivalent if t her e are
p osit ive const ant s αand β,andx0such t ha t M1(α x )M2(x)M1(βx )for
all xwit h 0 xx0[10].
It is well known t hat if Mis a convex funct ion an d M(0) = 0, t hen M(λ x )
λ M (x) for all λwit h 0 < λ < 1.
2. M ain R esults
Let Mb e an O rlicz fun ct ion,p= (pk) b e a sequence of p osit ive real numb ers
and Xb e a seminorm ed space over t he field Cof complex nu mb ers wit h t h e
seminorm q.w(X) den ot es t he space of all sequen ces x= (xk),where xk∈X.
We define t he following sequence spaces:
(W , θ, M , p, q)=
x∈w(X) : lim
r
1
hr
k∈Ir
[M(q(tk m (x)−ξ
ρ))]pk= 0,
uniformly in mfor some ξand for some ρ> 0
,
(W , θ, M , p, q)0=
x∈w(X) : lim
r
1
hr
k∈Ir
[M(q(tk m (x)
ρ))]pk= 0,
uniformly in mfor some ρ> 0
,
(W , θ, M , p, q)∞=
x∈w(X):sup
r , m
1
hr
k∈Ir
[M(q(tk m (x)
ρ))]pk<∞,
for som e ρ> 0
.
132 Rifat ¸Colak, B. C. Tripathy, and Mikˆail Et
We get t he following sequ ence spaces from t he a bove sequence spaces on
giving p art icular values t o θ, M and p.
i) If pk= 1forallk∈N, t h en we shall denot e (W , θ, M , p, q),(W , θ, M , p, q)0and
(W , θ, M , p, q)∞by (W , θ, M , q),(W , θ, M , q)0and (W , θ, M , q)∞,resp ect ively.
If x∈(W , θ, M , q)wesaythat xis q−lacun ary st rongly almost convergent
wit h r esp ect t o t he Orlicz fun ct ion M.
ii) Taking pk= 1forallk∈Nand M(x)= x , we denot e t he a bove sequ ence
spaces by (W , θ, q), (W , θ, q)0and (W , θ, q)∞,resp ect ively.
iii) In t he case θ= (2
r) we shall denot e t he a bove sequen ce spaces by (W , M , p, q),
(W , M , p, q)0and (W , M , p, q)∞,resp ect ively.
T heore m 2.1. Let Mbe an Orlicz function. Then (W , θ, M , p, q)0⊂(W , θ, M , p, q)
⊂(W , θ, M , p, q)∞.
Proof. Let x∈(W , θ, M , p, q).T hen we have
1
hr
k∈Ir
M
tk m (x)
ρ
pk
D
hr
k∈Ir
M
q
tk m (x)–ξ
ρ
pk
+D
hr
k∈Ir
M
q(ξ)
ρ
pk
D
hr
k∈Ir
M
q
tk m (x)−ξ
ρ
pk
+Dmax
1,sup
M
q(ξ)
ρ
H
,
where supkpk=G,H= max(1,G)andD= max(1,2G−1).
T hus we get x∈(W , θ, M , p, q)∞.T he inclusion (W , θ, M , p, q)0⊂(W , θ, M , p, q)
is obvious.
T heore m 2.2. Let the sequence (pk)be bounded, t hen (W , θ, M , p, q)0,(W , θ,
M , p, q)and (W , θ, M , p, q)∞are li near spaces over t he set of complex number s.
Proof. O mit t ed .
T heore m 2.3. T he spaces (W , θ, M , p, q)0,(W , θ, M , p, q)and (W , θ, M , p, q)∞
are paranormed spaces (not totally paranormed), paranormed by
g(x)= inf
ρpr/ H :sup
k
M
q
tk m (x)
ρ
1,ρ>0,uniform ly in m
,
Proof. Clearly g(x)= g(−x),and q
tk m (¯
θ)
ρ
=q(¯
θ)= 0where¯
θis t h e zero
sequence. Not hin g t h at M(0) = 0, from the ab ove one get s, g(¯
θ)= 0.Next let
(xk),(yk)∈(W , θ, M , p, q)0.Let ρ1>0andρ2>0 b e such t h at
sup
k
M
q
tk m (x)
ρ1
1,uniformly in m(1)
and
sup
k
M
q
tk m (y)
ρ2
1,uniformly in m . (2)
Let ρ=ρ1+ρ2.T hen we have
Lacunary Sequences and Almost Statistical Convergence 133
sup
k
M
q
tk m (x+y)
ρ
ρ1
ρ1+ρ2
sup
k
M
q
tk m (x)
ρ1
+
ρ2
ρ1+ρ2
sup
k
M
q
tk m (y)
ρ2
1,uniformly in m
by (1) and (2). Hence g(x+y)g(x)+ g(y).
T he cont inu ity of sca lar mult iplicat ion follows from t he following equalit y:
g(λ x )= inf
ρpr/ H :sup
k
M
q
tk m (λ x )
ρ
1,ρ>0,uniform ly in m
= inf
(|λ|s)pr/ H :sup
k
M
q
tk m (x)
ρ
1,ρ>0,uniformly in m
,
where s=ρ
|λ|.
T heore m 2.4. Let M1and M2be Orlicz functions. Then we have
i) (W , θ, M 1,p,q)0∩(W , θ, M 2,p,q)0⊂(W , θ, M 1+M2,p,q)0,
ii) (W , θ, M 1,p,q)∩(W , θ, M 2,p,q)⊂(W , θ, M 1+M2,p,q),
iii) (W , θ, M 1,p,q)∞∩(W , θ, M 2,p,q)∞⊂(W , θ, M 1+M2,p,q)∞.
Proof. It is st ra ight forward and h ence om it t ed.
T heore m 2.5. Let 0< p
ktkand
tk
pk
be bounded. T hen (W , θ, M , t , q)⊂
(W , θ, M , p, q).
Proof. If we t a ke wk , m =
M
q
tk m (x)
ρ
tkfor all k, m and use t he sam e t ech-
niqu e of T heor em 2 of Nan da [16], t he t h eorem is easily t o b e proved .
T heore m 2.6. T he sequence spaces (W , θ, M , p, q)0and (W , θ, M , p, q)∞are
neither solid nor monotone.
Proof. We give t he proof only for (W , θ, M , p, q)0.F o r t h i s l e t pk= 1,for k∈N,
θ= (2
r)M(x)= xand q(x)= |x|.Consider two sequences xk= (−1)kand αk=
(−1)kfor all k∈N.Then(xk)∈(W , θ, M , p, q)0but (αkxk)/∈(W , θ, M , p, q)0.
Hence (W , θ, M , p, q)0is not solid.
Consider t he J−st ep spa ce [(W , θ, M , p, q)0]Jof (W , θ, M , p, q)0.G iven a se-
quence x= (xk)∈(W , θ, M , p, q)0let us define y= (yk)∈[(W , θ, M , p, q)0]Jas
yk=xkfor odd kand yk= 0,ot herwise. T hen (yk)/∈(W , θ, M , p, q)0.Hence
(W , θ, M , p, q)0is not m onot one.
T he ot her ca ses can b e proved on consid ering simila r exam ples.
T h e followin g t h eor em ca n b e proved u sing t he sa me t ech niqu es of T h eorem
2.5 and T heorem 2.6 of Savas a nd R hoades [19], t herefore we give wit h out proof.
T heore m 2.7. Let θ= (kr)be a lacunary sequence with 1<lim infrηr
lim suprηr<∞.Then for any Orlicz function M , we have (W , M , p, q)=
(W , θ, M , p, q).
