# λ-Statistical Convergence in Paranormed Space

Authors: Mohammed A. Alghamdi; Mohammad Mursaleen
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

The concept ofλ-statistical convergence for sequences of real numbers was introduced in Mursaleen (2000). In this paper, we prove decomposition theorem for λ-statistical convergence. We also define and study λ-statistical convergence, λ-statistically Cauchy, and strongly λp-summability in Paranormed Space.


---

## Body

## 1. Introduction

The notion of statistical convergence was first introduced by Fast [1]. In the recent years, statistical summability became one of the most active areas of research in summability theory, which was further generalized as lacunary statistical convergence [2], λ-statical convergence [3], statistical A-summability [4], and statistical σ-convergence [5]. Maddox [6] studied this notion in locally convex Hausdorff topological spaces and Kolk [7] defined and studied this notion in Banach spaces while Çakalli [8] extended it to topological Hausdorff groups. The concept of statistical convergence is studied in probabilistic normed space and in intuitionistic fuzzy normed spaces in [9, 10]. Recently, the statistical convergence has been studied in Paranormed Space and locally solid Riesz spaces in [11, 12], respectively. Therefore, one can choose either some different setup to study these concepts or generalizing the existing concepts through different means. In this paper, we will study the concept of λ-statistical convergence, λ-statistical Cauchy, and strongly  λp-summability in Paranormed Space.Aparanorm is a function g:X→ℝ defined on a linear space X such that for all x,y,z∈X(P1)
g(x)=0 if x=θ,(P2)
g(-x)=g(x),(P3)
g(x+y)≤g(x)+g(y),(P4)
if(αn) is a sequence of scalars with αn→α0  (n→∞) and xn, a∈X with xn→a  (n→∞) in the sense that g(xn-a)→0  (n→∞), then αnxn→α0a  (n→∞), in the sense that g(αnxn-α0a)→0  (n→∞).A paranormg for which g(x)=0 implies that x=θ is called a total paranorm on X, and the pair (X,g) is called a total Paranormed Space.


## 2.λ-Statistical Convergence

Letλ=(λn) be a nondecreasing sequence of positive numbers tending to ∞ such that
(1)λn+1≤λn+1,λ1=0.
The generalized de la Vallée-Poussin mean is defined by
(2)tn(x)=:1λn∑j∈In‍xj,
where In=[n-λn+1,n].A sequencex=(xj) is said to be (V,λ)-summable to a number L if
(3)tn(x)⟶Lasn⟶∞.LetK be a subset of the set of natural numbers ℕ. Then, the λ-density of K is defined as
(4)δλ(K)=limn1λn|{n-λn+1≤j≤n:j∈K}|.The number sequencex=(xj) is said to be λ-statistically convergent to the number L (c.f. [3, 13, 14]) if δλ(K(ϵ))=0; that is, if for each ϵ>0,
(5)limn1λn|{k∈In:|xk-L|≥ϵ}|=0.
In this case we write stλ-limkxk=L and we denote the set of all λ-statistically convergent sequences by Sλ. In case λn=n, λ-density reduces to the natural density and λ-statistical convergence reduces to statistical convergence. This notion for double sequences has been studied in [15].A sequencex=(xk) is said to be strongly  λp-summable  (0<p<∞) to the limit  L [14] if
(6)limn1λn∑k∈In‍|xk-L|p=0,
and we write it as xk→L[Vλ]p. In this case L is called the [Vλ]p-limit of x.The following relation was established in [14].Theorem 1.
If0<p<∞and a sequence x=(xk) is strongly  λp-summable to L, then it is λ-statistically convergent to L. If a bounded sequence is λ-statistically convergent to L, then it is strongly  λp-summable to L.
The following theorem isλ-statistical version of Connor’s Decomposition Theorem [16].Theorem 2.
Ifx=(xk) is strongly  λp-summable or statistically λ-convergent to L, then there is a convergent sequence y and a λ-statistically null sequence z such that y is convergent to L,x=y+z and
(7)limn1λn|{k∈In:zk≠0}|=0.
Moreover, if x is bounded, then y and z both are bounded.Proof.
By Theorem1, it follows that x is λ-statistically convergent to L if x is strongly  λp-summable to L. Set N0=0 and choose a strictly increasing sequence of positive integers N1<N2<N3<⋯ such that
(8)1λn|{k∈In:|xk-L|≥j-1}|<j-1
for n>Nj. Define y and z as follows.
IfN0<k<N1 set zk=0 and yk=xk. Let j≥1 and Nj<k≤Nj+1. Now we set
(9)yk={xk,zk=0,if|xk-L|<j-1;L,zk=xk-L,if|xk-L|≥j-1.
Clearly, x=y+z and y and z are bounded, if x is bounded. Also, we observe that for k>Nj, we have
(10)|yk-L|<ϵsince|yk-L|=|xk-L|<ϵ|yk-L||xk-Lhl|if|xk-L|<j-1,|yk-L|=|L-L|=0if|xk-L|>j-1.
Hence, limkyk=L, since ϵ was arbitrary.
Next we observe that(11)|{k∈In:zk≠0}|≥|{k∈In:|zk|≥ϵ}|
for any natural number n and ϵ>0. Hence, limn(1/λn)|{k∈In:zk≠0}|=0; that is, z is λ-statistically null.
We now show that ifδ>0 and j∈N such that j-1<δ, then |{k∈In:zk≠0}|<δ for all n>Nj. Recall from the construction that if Nj<k≤Nj+1, then zk≠0 only if |xk-L|>j-1. It follows that if Nℓ<k≤Nℓ+1, then
(12){k∈In:zk≠0}⊆{k∈In:|xk-L|>ℓ-1}.
Consequently,(13)1λn|{k∈In:zk≠0}|≤1λn|{k∈In:|xk-L|>ℓ-1}|<ℓ-1<j-1<δ,
if Nℓ<n≤Nℓ+1 and ℓ>j. That is,
(14)limn1λn|{k∈In:zk≠0}|=0.
This completes the proof of the theorem.


## 3. Application to Fourier Series

Letf:𝕋→ℂ be a Lebesgue integrable function on the torus 𝕋:=[-π,π); that is, f∈L1(𝕋). The Fourier series of f is defined by
(15)f(x)~∑j∈ℤ‍f^(j)eijx,x∈𝕋,
where the Fourier coefficients f^(j) are defined by
(16)f^(j)∶=12π∫𝕋‍f(t)e-ijtdt,j∈ℤ.
The symmetric partial sums of the series in (15) are defined by
(17)sk(f;x)∶=∑|j|≤k‍f^(j)eijx,x∈𝕋,k∈ℕ.
The conjugate series to the Fourier series in (15) is defined by [17, Vol. I, pp. 49]
(18)∑j∈ℤ‍(-isgnj)f^(j)eijx.
Clearly, it follows from (15) and (18) that
(19)∑j∈ℤ‍f^(j)eijx+i∑j∈ℤ‍(-isgnj)f^(j)eijx=1+2∑j=1∞‍f^(j)eijx,
and the power series
(20)1+2∑j=1∞‍f^(j)eijx,wherez∶=reix,0≤r<1,
is analytic on the open unit disk |z|<1, due to the fact that
(21)|f^(j)|≤12π∫π‍|f(t)|dt,j∈ℤ.
The conjugate function f^ of a function f∈L1(𝕋) is defined by
(22)f^(x)∶=-limɛ↓01π∫ɛ≤|t|≤πf(x+t)2tan(t/2)dt=limɛ↓01π∫ɛπf(x-t)-f(x+t)2tan(t/2)dt
in the “principal value” sense and that f^(x) exists at almost every x∈𝕋.The following isλ-statistical version of [18] (c.f. [19, Theorem 2.1 (ii)]).Theorem 3.
Iff∈L1(𝕋), then for any p>0 its Fourier series is strongly  λp-summable to f(x) at almost every x∈𝕋. Furthermore, its conjugate series (18) is strongly  λp-summable for any p>0 to the conjugate function f^(x) defined in (22) at almost every x∈𝕋.From Theorems1 and 3, we easily get the following useful result.Theorem 4.
Iff∈L1(𝕋), then its Fourier series is λ-statistically convergent to f(x) at almost every x∈𝕋. Furthermore, its conjugate series (18) is λ-statistically convergent to the conjugate function f^(x) defined in (22) at almost every x∈𝕋.


## 4.λ-Statistical Convergence in Paranormed Space

Recently, statistical convergence, statistical Cauchy, and strongly  Cesàro summability have been studied in Paranormed Space by Alotaibi and Alroqi [11].In this paper, we define and study the notion ofλ-summable, λ-statistical convergence, λ-statistical Cauchy, and strongly  λp-summability in Paranormed Space.Let(X,g) be a Paranormed Space.A sequencex=(xk) is said to be convergent to the number ξ in (X,g) if, for every ɛ>0, there exists a positive integer k0 such that g(xk-ξ)<ɛ whenever k≥k0. In this case, we write g-limx=ξ, and ξ is called the g-limit of x.We define the following.Definition 5.
A sequencex=(xk) is said to be λ-statistically convergent to the number ξ in (X,g) if, for each ϵ>0,
(23)limn1λn|{k∈In:g(xk-ξ)≥ϵ}|=0.
In this case we write stλ(g)-limx=ξ.Definition 6.
A sequencex=(xk) is said to be λ-statistically Cauchy sequence in (X,g) if for every ϵ>0 there exists a number N=N(ϵ) such that
(24)limn1λn|{j∈In:g(xj-xN)≥ϵ}|=0.Definition 7.
A sequencex=(xk) is said to be strongly  λp-summable  (0<p<∞) to the limit  ξ in (X,g) if
(25)limn1λn∑k∈In‍(g(xk-ξ))p=0,
and we write it as xk→ξ[Vλ,g]p. In this case ξ is called the [Vλ,g]p-limit of x.Now we define another type of convergence in Paranormed Space.Definition 8.
A sequence(xk) in a Paranormed Space (X,g) is said to stλ*(g)-convergent to ξ∈X if there exists an index set K={k1<k2<⋯<kn<⋯}⊆ℕ, n=1,2,…, with δλ(K)=1 such that g(xkn-ξ)→0(n→∞). In this case, we write ξ=stλ*(g)-limx.First we prove the following results onλ-statistical convergence in (X,g).Theorem 9.
Ifg-limx=ξ, then stλ(g)-limx=ξ but converse need not be true in general.Proof.
Letg-limx=ξ. Then, for every ɛ>0, there is a positive integer N such that
(26)g(xn-ξ)<ɛ
for all n≥N. Since the set A(ϵ)∶={k∈ℕ:g(xk-ξ)≥ɛ} is finite, δλ(A(ϵ))=0. Hence, stλ(g)-limx=ξ.
The following example shows that the converse need not be true.
Example10. Let X=ℓ(1/k)∶={x=(xk):∑k‍|xk|1/(k+1)<∞} with the paranorm g(x)=(∑k‍|xk|1/(k+1)). Define a sequence x=(xk) by
(27)xk∶={k,ifn-[λn]+1≤k≤n,n∈ℕ;0,otherwise,
and write
(28)K(ɛ)∶={k≤n:g(xk)≥ɛ},0<ɛ<1.
We see that(29)g(xk)∶={k1/(k+1),ifn-[λn]+1≤k≤n,n∈ℕ;0,otherwise,
and hence
(30)limkg(xk)∶={1,ifn-[λn]+1≤k≤n,n∈ℕ;0,otherwise.
Thereforeg-limx does not exist. On the other hand δλ(K(ɛ))=0; that is, stλ(g)-limx=0.
This completes the proof of the theorem.We can easily prove the following results onλ-statistical convergence in (X,g) similar to those of [11].Theorem 11.
If a sequencex=(xk) is λ-statistically convergent in (X,g), then stλ(g)-limit is unique.Theorem 12.
Letstλ(g)-limx=ξ1 and stλ(g)-limy=ξ2. Then,(i)
stλ(g)-lim(x±y)=ξ1±ξ2,(ii)
stλ(g)-lim αx=αξ1,α∈ℝ.Theorem 13.
Let(X,g) be a complete Paranormed Space. Then a sequence x=(xk) of points in (X,g) is λ-statistically convergent if and only if it is λ-statistically Cauchy.Theorem 14.
(a) If0<p<∞ and xk→ξ[Vλ,g]p, then x=(xk) is λ-statistically convergent to ξ in (X,g).
(b) Ifx=(xk) is bounded and λ-statistically convergent to ξ in (X,g), then xk→ξ[Vλ,g]p.Theorem 15.
Let(X,g) be a complete Paranormed Space. Then a sequence x=(xk) of points in (X,g) is λ-statistically convergent if and only if it is λ-statistically Cauchy.Note that the proof of Theorem 2.4 [11] is incorrect and the correct proof is given in the following theorem which is generalization of Theorem 2.4 [11]. Another form of this result is given in [20] for ideal convergence.Theorem 16.
A sequencex=(xk) in (X,g) is λ-statistically convergent to ξ if and only if it is stλ*(g)-convergent to ξ.Proof.
Suppose thatx=(xk) is λ-statistically convergent to ξ; that is, stλ(g)-limx=ξ. Now, write for r=1,2,….
(31)Kr∶={n∈ℕ:g(xkn-ξ)≥1r},Mr∶={n∈ℕ:g(xkn-ξ)<1r}(r=1,2,…).
Then δλ(Kr)=0,
(32)M1⊃M2⊃⋯⊃Mi⊃Mi+1⊃⋯,(33)δλ(Mr)=1,r=1,2,….
Now we have to show that, forn∈Mr, (xkn) is g-convergent to ξ. On contrary suppose that (xkn) is not g-convergent to ξ. Therefore, there is ɛ>0 such that g(xkn-ξ)≥ɛ for infinitely many terms. Let Mɛ:={n∈ℕ:g(xkn-ξ)<ɛ} and ɛ>1/r, r∈ℕ.
Then(34)δλ(Mɛ)=0,
and by (32), Mr⊂Mɛ. Hence δλ(Mr)=0, which contradicts (33) and we get that (xkn) is g-convergent to ξ. Hence, x is stλ*(g)-convergent to ξ.
Conversely, suppose thatx is stλ*(g)-convergent to ξ. Then there exists a set K={k1<k2<k3<⋯<kn<⋯} with δλ(K)=1 such that g-limn→∞xkn=ξ. Therefore, there is a positive integer N such that g(xn-ξ)<ɛ for n≥N. Put Kɛ∶={n∈ℕ:g(xn-ξ)≥ɛ} and K′:={kN+1,kN+2,…}. Then δλ(K′)=1 and Kɛ⊆ℕ-K′ which implies that δλ(Kɛ)=0. Hence x=(xk) is λ-statistically convergent to ξ; that is stλ(g)-limx=ξ.
This completes the proof of the theorem.



---

*Source: 264520-2013-12-23.xml*