R E S E A R C H A R T I C L E
On Pointwise Lacunary Statistical Convergence of Order
a
of Sequences of Function
Mikail Et1,2•Hacer S¸engu¨l2Received: 5 March 2014 / Revised: 13 January 2015 / Accepted: 22 January 2015 / Published online: 15 May 2015 Ó The National Academy of Sciences, India 2015
Abstract In this paper we introduce the concepts of pointwise lacunary statistical convergence of order a and pointwise wpðf ; hÞ—summability of order a of sequences
of real valued functions. Also some relations between pointwise Sahðf Þ—statistical convergence and pointwise wapðf ; hÞ—summability are given.
Keywords Statistical convergence Sequences of function Cesa`ro summability
Mathematics Subject Classification 40A05 40C05 46A45
1 Introduction
The concept of statistical convergence was introduced by Steinhaus [1] and Fast [2] and later reintroduced by Schoenberg [3] independently. Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory, number theory, measure theory, trigonometric series, turnpike theory and Banach spaces. Later on it was further inves-tigated from the sequence space point of view and linked
with summability theory by Caserta et al. [4], C¸ akallı [5, 6], C¸ akallı and Khan [7], Connor [8], Et et al. [9,10], Fridy [11], Gu¨ngo¨r et al. [12], Kolk [13], Mursaleen [14], Salat [15], Tripathy et al. [16–18] and many others.
The definition of pointwise statistical convergence of sequences of real valued functions was given by Go¨khan and Gu¨ngo¨r [19] and independently by Duman and Orhan [20].
In this paper we introduce and examine the concepts of pointwise lacunary statistical convergence of order a and pointwise wpðf ; hÞ—summability of order a of sequences
of real valued functions.
2 Definition and Preliminaries
The idea of statistical convergence depends on the density of subsets of the set N of natural numbers. The asymptotic density of a subset E of N is defined by
dðEÞ ¼ lim n!1 1 n Xn k¼1
vEðkÞ provided the limit exists;
where vE is the characteristic function of E. It is clear that any finite subset of N has zero natural density and d Eð cÞ ¼ 1 d Eð Þ.
The order of statistical convergence of a sequence of numbers was given by Gadjiev and Orhan in [21] and after that statistical convergence of order a and strong p-Cesa`ro summability of order a was studied by C¸ olak [22].
By a lacunary sequence we mean an increasing integer sequence h¼ ðkrÞ such that hr¼ ðkr kr1Þ ! 1 as
r! 1. Throught this paper the intervals determined by h is denoted by Ir¼ ðkr1; kr and the ratiokkr1r is abbreviated
by qr. Recently lacunary sequence have been studied by
several authors [23–33]. & Mikail Et mikailet@yahoo.com; mikailet68@gmail.com; met@firat.edu.tr; met@siirt.edu.tr Hacer S¸engu¨l hacer.sengul@hotmail.com
1 Department of Mathematics, Faculty of Science, Firat
University, 23119 Elazig, Turkey
2 Department of Mathematics, Faculty of Science and Arts,
Siirt University, Siirt, Turkey DOI 10.1007/s40010-015-0199-z
Definition 2.1 Let h¼ ðkrÞ be a lacunary sequence and
a2 ð0; 1 be any real number. A sequence of functions ffkg
is said to be pointwise Sahðf Þ—statistically convergent (or pointwise lacunary statistical convergence of order a) to the function f on a set A, if for every e [ 0,
lim r 1 ha r k2 Ir : fjkðxÞ f xð Þj e; for every x2 A f g j j ¼ 0 where Ir ¼ ðkr1; kr and ha¼ ðhraÞ ¼ ðha1; ha2; . . .; har; . . .Þ.
In this case we write Sah lim fkðxÞ ¼ f ðxÞ on A. Sah
lim fkðxÞ ¼ f ðxÞ means that for every d [ 0 and 0\a 1,
there is an integer n0 such that
1 ha r k2 Ir: fjkðxÞ f xð Þj e; for every x2 A f g j j\d;
for all n [ n0ð¼ n0ðe; d; xÞÞ and for each e [ 0. The set of
all pointwise lacunary statistical convergence of order a will be denoted by Sahðf Þ. For h ¼ ð2rÞ, we shall write Saðf Þ
instead of Sa
hðf Þ which were defined and studied by C¸ınar
et al. [34] and in the special case a¼ 1, we write Shðf Þ
instead of Sahðf Þ.
Pointwise lacunary statistical convergence of order a of sequence of functions is well defined for 0\a 1, but is not well defined for a [ 1. For this let ff g be defined ask
follows: fkð Þ ¼x 1 k¼ 2r kx 1þ k2x2 k6¼ 2r r¼ 1; 2; 3. . .; x 2 0;1 2 h i ( Then, both lim r!1 1 ha r k2 Ir: fjkðxÞ 1j e; for every x2 A f g j j lim r!1 kr kr1 2ha r ¼ lim r!1 hr 2ha r ¼ 0 and lim r!1 1 ha r k2 Ir: fjkðxÞ 0j e; for every x2 A f g j j lim r!1 kr kr1 2ha r ¼ lim r!1 hr 2ha r ¼ 0 for a [ 1, and so Sa h lim fkðxÞ ¼ 1 and Sah lim fkðxÞ ¼ 0.
It is easy to see that every convergent sequence of functions is statistically convergent of order að0\a 1Þ. The following example shows that the converse of this does not hold. The sequenceffkg defined by
fkð Þ ¼x 1 k¼ n3 2kx 1þ k2x2 k6¼ n 3 (
is statistically convergent of order a with Sa lim fkð Þ ¼x
0 for a [1, but it is not convergent.
Definition 2.2 Let h¼ kð Þ be a lacunary sequence, a 2r
0; 1
ð and p be a positive real number. A sequence of functionsffkg is said to be pointwise wapðf ; hÞ—summable
(or pointwise wpðf ; hÞ—summable of order a), if there is a
function f such that lim r!1 1 ha r P k2Ir;x2A fkðxÞ f xð Þ j jp¼ 0:
In this case we write wapðf ; hÞ lim fkðxÞ ¼ f ðxÞ on A. The
set of all pointwise wpðf ; hÞ—summable sequence of
functions order a will be denoted by wapðf ; hÞ.
Let A be any non empty set, by BðAÞ we denote the set of all bounded real valued functions defined on A.
3 Main Results
Theorem 3.1 Leth¼ ðkrÞ be a lacunary sequence, a 2
ð0; 1 be any real number and ffkg; fgkg be sequences of
real valued functions defined on a set A:
ðiÞ If Sah lim fkðxÞ ¼ f ðxÞ and c2 R, then
Sah lim cfkðxÞ ¼ cf ðxÞ,
ðiiÞ If Sa
h lim fkðxÞ ¼ f ðxÞ and Sah lim gkðxÞ ¼ gðxÞ,
then Sah lim ðfkðxÞ þ gkðxÞÞ ¼ f ðxÞ þ gðxÞ.
Theorem 3.2 Leth¼ ðkrÞ be a lacunary sequence and
a; b2 ð0; 1 ða bÞ: Then Sa hðf Þ S
b
hðf Þ and the inclusion
is strict for the case a\b:
Proof To show the strictness of the inclusion Sa
hðf Þ S b
hðf Þ, let us define a sequence ffkg by
fkð Þ ¼x 1 k¼ r2 kxþ 2 1þ k2x2 k6¼ r 2; ( x2 ½2; 3: Then x2 Sbhðf Þ for1 2\b 1, but x 62 S a hðf Þ for 0\a 1 2. h
Theorem 3.3 Let h¼ ðkrÞ be a lacunary sequence,
0\a b 1 and p be a positive real number. Then wa
pðf ; hÞ wbpðf ; hÞ and the inclusion is strict for the case
a\b:
Proof Taking h¼ ð2rÞ we show the strictness of the
inclusion wapðf ; hÞ wb
pðf ; hÞ for a special case. For this
consider the sequenceffkg defined by
fkð Þ ¼x k2x2 1þ k2x2 k¼ n 2 0 k6¼ n2 ; 8 < : x2 ½1; 2: Then 1 nb X k¼1 n fkðxÞ 0 j jp ffiffiffi n p nb ¼ 1 nb12 ! 0 as n ! 1
and 1 na X k¼1 n fkðxÞ 0 j jp ffiffiffi n p 2na! 1 as n ! 1
and so the sequenceffkg is pointwise wpðf ; hÞ—summable
of order b for 1
2\b 1; but is not pointwise wpðf ; hÞ—
summable of order a for 0\a\12. h The following result is established using standard tech-niques, so we state the result without proof.
Theorem 3.4 Let h¼ ðkrÞ be a lacunary sequence and
leta and b be fixed real numbers such that 0\a b 1 and 0\p\1: If a sequence of functions ffkg is pointwise
wpðf ; hÞ—summable of order a, to the function f, then it is
pointwise lacunary statistical convergence of order b, to the function f.
Theorem 3.5 Let h¼ ðkrÞ be a lacunary sequence and
a2 ð0; 1: If lim infrqr[ 1 then Saðf Þ Sahðf Þ:
Proof Suppose that lim infrqr[ 1, then there exists a
d [ 0 such that qr 1 þ d for sufficiently large r, which
implies that hr kr d 1þ d¼) hr kr a d 1þ d a ¼) 1 ka r d a 1þ d ð Þa 1 ha r : If Sa lim f
kðxÞ ¼ f ðxÞ on A, then for every e [ 0 and for
sufficiently large r, we have 1
ka r
kkr: fjkðxÞ f xð Þje; for every x 2 A
f g
j j
1 ka r
k2 Ir: fjkðxÞ f xð Þje; for every x 2 A
f g j j d a 1þ d ð Þa 1 ha r
k2 Ir: fjkðxÞ f xð Þje; for every x 2 A
f g
j j;
this proves the proof. h
Theorem 3.6 Let h¼ ðkrÞ be a lacunary sequence and
a2 ð0; 1: If lim suprqr\1 then Sahðf Þ Sðf Þ:
Proof If lim suprqr\1, then there is an H [ 0 such that
qr\H for all r. Suppose that Sah lim fkðxÞ ¼ f ðxÞ on A
and let Nr¼jfk2 Ir : fjkðxÞ f xð Þj e; for every x 2 Agj.
By the definition for a given e [ 0; there is an r02 N such
that for 0\a 1, Nr
ha r
\e¼)Nr hr
\e; for all r [ r0:
The rest of proof follows from Lemma 3 in [26]. h Theorem 3.7 Let h¼ ðkrÞ be a lacunary sequence and
a2 ð0; 1: If lim infr
har kr
[ 0 then Sðf Þ Sa hðf Þ:
Proof For a given e [ 0, we have 1 kr k kr : fjkðxÞ f xð Þj e; for every x2 A f g j j 1 kr k2 Ir: fjkðxÞ f xð Þj e; for every x2 A f g j j ¼h a r kr 1 ha r k2 Ir: fjkðxÞ f xð Þj e; for every x2 A f g j j:
Taking limit as r! 1, we get Sa
h lim fkðxÞ ¼ f ðxÞ on A.
h The proofs of the following theorems are obtained by using the standard techniques.
Theorem 3.8 Let 0\a 1 and h ¼ ðkrÞ be a lacunary
sequence. If lim infrqr[ 1 then wapðf Þ wapðf ; hÞ:
Theorem 3.9 Let h¼ ðkrÞ be a lacunary sequence. If
lim suprqr\1 then wpðf ; hÞ wpðf Þ:
Theorem 3.10 Let 0\a 1 and 0\p\q\1
then waqðf ; hÞ wa pðf ; hÞ:
Let h¼ ðkrÞ and h0¼ ðsrÞ be two lacunary sequences
such that Ir Jr for all r2 N; a and b be fixed real
num-bers such that 0\a b 1. Now we give some inclusion relations between the sets of Sahðf Þ—statistically conver-gent sequences and pointwise waðf ; hÞ—summable
se-quences for different a0s and h0s which also include Theorems3.2,3.3and3.4as a special case.
Theorem 3.11 Let h¼ ðkrÞ and h0¼ ðsrÞ be two
lacu-nary sequences such that Ir Jrfor all r2 N and let a and
b be fixed real numbers such that 0\a b 1 (i) If lim r!1inf har ‘br [ 0 ð1Þ then Sbh0ðf Þ Sahðf Þ, (ii) If lim r!1 ‘r hbr ¼ 1 ð2Þ then Sahðf Þ Sbh0ðf Þ; where Ir ¼ ðkr1; kr, Jr¼ ðsr1; sr; hr¼ kr kr1 and‘r ¼ sr sr1:
Proof (i) Easy, so omitted.
(ii) Let fðkð Þx Þ 2 Sahð Þ and Eq. (2) be satisfied. Sincef
1 ‘br k2 Jr: fjkðxÞ f xð Þj e; for every x2 A f g j j ¼1 ‘br sr1\k kr1: fjkðxÞ f xð Þj e; for every x2 A f g j j þ1 ‘br kr\k sr: fjkðxÞ f xð Þj e; for every x2 A f g j j þ1 ‘br kr1\k kr: fjkðxÞ f xð Þj e; for every x2 A f g j j kr1 sr1 ‘br þsr kr ‘br þ1 ‘br k2 Ir: fjkðxÞ f xð Þj e; f j
for every x2 Agj ¼‘r hr ‘br þ1 ‘br k2 Ir: fjkðxÞ f xð Þj e; for every x2 A f g j j ‘r h b r hbr þ 1 hbr k2 Ir: fjkðxÞ f xð Þj e; for every x2 A f g j j ‘r hbr 1 þ1 ha r k2 Ir: fjkðxÞ f xð Þj e; for every x2 A f g j j
for all r2 N. This implies that Sa hð Þ Sf
b
h0ð Þ.f h
From Theorem3.11we have the following results. Corollary 3.11.1 Let h¼ kð Þ and hr 0¼ sð Þ be two la-r
cunary sequences such that Ir Jr for all r2 N:
If Eq. (1) holds then
(i) Sah0ð Þ Sf ahð Þ for each a 2 0; 1f ð and for all x 2 A,
(ii) Sh0ð Þ Sf a
hð Þ for each a 2 0; 1f ð and for all
x2 A,
(iii) Sh0ð Þ Sf hð Þ and for all x 2 A:f
If Eq. (2) holds then (i) Sahð Þ Sf a
h0ð Þ for each a 2 0; 1f ð and for all x 2 A,
(ii) Sahð Þ Sf h0ð Þ for each a 2 0; 1f ð and for all x 2 A;
(iii) Shð Þ Sf h0ð Þ for all x 2 A:f
Theorem 3.12 Let h¼ kð Þ and hr 0¼ sð Þ be two lacu-r
nary sequences such that Ir Jr for all r2 N; a and b be
fixed real numbers such that 0\a b 1 and 0\p\1. Then we have
(i) If Eq. (1) holds then wbpðf;h0Þ wa
pðf;hÞ for all
x2 A,
(ii) Let Eq. (2) holds, f xð Þ 2 B Að Þ and ff g be a sequencek
of bounded real valued functions defined on a set A then wa
pðf;hÞ wbp f;h 0
ð Þ for all x 2 A:
Proof Suppose that Eq. (2) holds and ff g be a sequencek
of bounded real valued functions defined on a set A. Since f xð Þ 2 B Að Þ then there exists some M [ 0 such that
fkð Þ f xx ð Þ
j j M for all k 2 N and for all x 2 A. Now, we may write 1 ‘br X k2Jr fkð Þ f xx ð Þ j jp ¼ 1 ‘br X k2JrIr;x2A fkð Þ f xx ð Þ j jpþ1 ‘br X k2Ir;x2A fkð Þ f xx ð Þ j jp ‘r hr ‘br Mpþ 1 ‘br X k2Ir;x2A fkð Þ f xx ð Þ j jp ‘r h b r hbr Mpþ 1 hbr X k2Ir;x2A fkð Þ f xx ð Þ j jp ‘r hbr 1 Mpþ 1 ha r X k2Ir;x2A fkð Þ f xx ð Þ j jp
for every r2 N. Therefore wa
pðf;hÞ wbpðf;h0Þ. h
From Theorem3.12we have the following results. Corollary 3.12.1 Let h¼ kð Þ and hr 0¼ sð Þ be two la-r
cunary sequences such that Ir Jr for all r2 N:
If Eq. (1) holds then (i) wapðf;h0Þ wa
pðf;hÞ for each a 2 0; 1ð and for all
x2 A;
(ii) wpðf;h0Þ wapðf;hÞ for each a 2 0; 1ð and for all
x2 A,
(iii) wpðf;h0Þ wpðf;hÞ for all x 2 A,
Let Eq. (2) holds, f xð Þ 2 B Að Þ and ff g be a sequence ofk
bounded real valued functions defined on a set A, then (i) wapðf;hÞ wa
pðf;h0Þ for each a 2 0; 1ð and for all
x2 A, (ii) wa
pðf;hÞ wpðf;h0Þ for each a 2 0; 1ð and for all
x2 A,
(iii) wpðf;hÞ wpðf;h0Þ for all x 2 A.
Theorem 3.13 Leth¼ kð Þ and hr 0¼ sð Þ be two lacunaryr
sequences such that Ir Jr for all r2 N; a and b be fixed
real numbers such that 0\a b 1 and 0\p\1. Then (i) Let Eq. (1) holds, if a sequence of real valued
functions defined on a set A is pointwise wbpðf;h0Þ—
summable to f; then it is pointwise lacunary statistical convergence of ordera to the function f on a set A, (ii) Let Eq. (2) holds, f xð Þ 2 B Að Þ and ff g be a sequencek
of bounded real valued functions defined on a set A, if a sequence is pointwise lacunary statistical conver-gence of ordera to the function f then it is pointwise wbpðf;h0Þ—summable to f.
Proof (i) Let wapðf;hÞ lim fkð Þ ¼ f xx ð Þ on A and e [ 0;
1 ‘br X k2Jr;x2A xk L j jph a r ‘br 1 ha r k2 Ir : fjkð Þ f xx ð Þj e f j
for every x2 Agjep:
Since Eq. (1) holds, the sequence ffkg is a pointwise
la-cunary statistically convergent sequence of order a to the function f on a set A:
(ii) Suppose that the sequence ffkg is a pointwise
lacunary statistically convergent sequence of order a to the function f on a set A: Since fðxÞ 2 BðAÞ and ffkg is a
bounded sequence of real valued functions defined on a set A; there exists a M [ 0 such that fjkð Þ f xx ð Þj M for all
k: Then for every e [ 0 we may write
1 ‘br X k2Jr;x2 A fkð Þ f xx ð Þ j jp ¼ 1 ‘br X k2JrIr;x2 A fkð Þ f xx ð Þ j jpþ1 ‘br X k2Ir;x2 A fkð Þ f xx ð Þ j jp ‘r hr ‘br Mpþ 1 ‘br X k2Ir;x2 A fkð Þ f xx ð Þ j jp ‘r h b r ‘br Mpþ 1 ‘br X k2 Ir;x2 A fkð Þ f xx ð Þ j jp ‘r hbr 1 Mpþ 1 hbr X k2 Ir fkð Þf xx ð Þ j j e;x 2 A fkð Þ f xx ð Þ j jp þ 1 hbr X k2 Ir fkð Þf xx ð Þ j j\e;x 2 A fkð Þ f xx ð Þ j jp ‘r hbr 1 Mp þM p hbr k 2 Ir: fjkð Þ f xx ð Þj e for every x 2 A þ hr hbr ep ‘r hbr 1 MpþM p ha r k 2 Ir: fjkð Þ f xx ð Þj e for every x 2 A þ ‘r hbr ep
for all r2 N: Using Eq. (2) we obtain that wb
pðf;h0Þ
lim fkð Þ ¼ f xx ð Þ; whenever Sahð Þ lim ff kð Þ ¼ f xx ð Þ: h
From Theorem3.13we have the following result. Corollary 3.13.1 Let a be any fixed real number such that 0\a 1; 0\p\1 and let h ¼ kð Þ and hr 0¼ sð Þ ber
two lacunary sequences such that Ir Jr for all r2 N:
If Eq. (1) holds then
(i) If a sequence of real valued functions defined on a set A is pointwise wapðf;h0Þ—summable to f, then it is
pointwise lacunary statistically convergent sequence of ordera to the function f on a set A,
(ii) If a sequence of real valued functions defined on a set A is pointwise wpðf;h0Þ—summable to f, then it is
pointwise lacunary statistically convergent sequence of ordera to the function f on a set A,
(iii) If a sequence of real valued functions defined on a set A is pointwise wpðf;h0Þ—summable to f, then it is
pointwise lacunary statistically convergent sequence to the function f on a set A,
Let Eq. (2) holds, f xð Þ 2 B Að Þ and ff g be a sequence ofk
bounded real valued functions defined on a set A, then (i) If a sequence is pointwise lacunary statistical
conver-gence of ordera to the function f then it is pointwise wapðf;h0Þ—summable to f,
(ii) If a sequence is pointwise lacunary statistical convergence of order a to the function f then it is pointwise wpðf;h0Þ—summable to f,
(iii) If a sequence is pointwise lacunary statistical convergence to the function f then it is pointwise wpðf;h0Þ—summable to f.
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