𝜆-Statistical Convergence in Paranormed Space

Abstract

The concept of 𝜆-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 𝜆_𝑝-summability in Paranormed Space.

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  𝜆_𝑝-summability in Paranormed Space.

A paranorm 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 (𝛼_𝑛) is a sequence of scalars with 𝛼_𝑛→𝛼₀  (𝑛→∞) and 𝑥_𝑛, 𝑎∈𝑋 with 𝑥_𝑛→𝑎  (𝑛→∞) in the sense that 𝑔(𝑥_𝑛−𝑎)→0  (𝑛→∞), then 𝛼_𝑛𝑥_𝑛→𝛼₀𝑎  (𝑛→∞), in the sense that 𝑔(𝛼_𝑛𝑥_𝑛−𝛼₀𝑎)→0  (𝑛→∞).

A paranorm 𝑔 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

Let 𝜆=(𝜆_𝑛) be a nondecreasing sequence of positive numbers tending to ∞ such that 𝜆_𝑛+1≤𝜆_𝑛+1,𝜆₁=0. (1) The generalized de la Vallée-Poussin mean is defined by 𝑡_𝑛(𝑥)1=∶𝜆_{𝑛𝑗∈𝐼𝑛}𝑥_𝑗, (2) where 𝐼_𝑛=[𝑛−𝜆_𝑛+1,𝑛].

A sequence 𝑥=(𝑥_𝑗) is said to be (𝑉,𝜆)-summable to a number 𝐿 if 𝑡_𝑛(𝑥)⟶𝐿as𝑛⟶∞. (3)

Let 𝐾 be a subset of the set of natural numbers ℕ. Then, the 𝜆-density of 𝐾 is defined as 𝛿_𝜆(𝐾)=lim_𝑛1𝜆_𝑛𝑛−𝜆_𝑛+1≤𝑗≤𝑛∶𝑗∈𝐾. (4)

The number sequence 𝑥=(𝑥_𝑗) is said to be 𝜆-statistically convergent to the number 𝐿 (c.f. [3, 13, 14]) if 𝛿_𝜆(𝐾(𝜖))=0; that is, if for each 𝜖>0, lim_𝑛1𝜆_𝑛𝑘∈𝐼_𝑛∶𝑥_𝑘−𝐿≥𝜖=0. (5) In this case we write st_𝜆-lim_𝑘𝑥_𝑘=𝐿 and we denote the set of all 𝜆-statistically convergent sequences by 𝑆_𝜆. In case 𝜆_𝑛=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 sequence 𝑥=(𝑥_𝑘) is said to be strongly  𝜆_𝑝-summable  (0<𝑝<∞) to the limit  𝐿 [14] if lim_𝑛1𝜆_{𝑛𝑘∈𝐼𝑛}𝑥_𝑘−𝐿^𝑝=0, (6) and we write it as 𝑥_𝑘→𝐿[𝑉_𝜆]_𝑝. In this case 𝐿 is called the [𝑉_𝜆]_𝑝-limit of 𝑥.

The following relation was established in [14].

Theorem 1. If  0<𝑝<∞and a sequence 𝑥=(𝑥_𝑘) is strongly  𝜆_𝑝-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  𝜆_𝑝-summable to 𝐿.
The following theorem is 𝜆-statistical version of Connor’s Decomposition Theorem [16].

Theorem 2. If 𝑥=(𝑥_𝑘) is strongly  𝜆_𝑝-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 lim_𝑛1𝜆_𝑛𝑘∈𝐼_𝑛∶𝑧_𝑘≠0=0. (7) Moreover, if 𝑥 is bounded, then 𝑦 and 𝑧 both are bounded.

Proof. By Theorem 1, it follows that 𝑥 is 𝜆-statistically convergent to 𝐿 if 𝑥 is strongly  𝜆_𝑝-summable to 𝐿. Set 𝑁₀=0 and choose a strictly increasing sequence of positive integers 𝑁₁<𝑁₂<𝑁₃<⋯ such that 1𝜆_𝑛𝑘∈𝐼_𝑛∶𝑥_𝑘−𝐿≥𝑗⁻¹<𝑗⁻¹ (8) for 𝑛>𝑁_𝑗. Define 𝑦 and 𝑧 as follows.
If 𝑁₀<𝑘<𝑁₁ set 𝑧_𝑘=0 and 𝑦_𝑘=𝑥_𝑘. Let 𝑗≥1 and 𝑁_𝑗<𝑘≤𝑁_𝑗+1. Now we set 𝑦_𝑘=𝑥_𝑘,𝑧_𝑘=0,if𝑥_𝑘−𝐿<𝑗⁻¹;𝐿,𝑧_𝑘=𝑥_𝑘−𝐿,if𝑥_𝑘−𝐿≥𝑗⁻¹. (9) Clearly, 𝑥=𝑦+𝑧 and 𝑦 and 𝑧 are bounded, if 𝑥 is bounded. Also, we observe that for 𝑘>𝑁_𝑗, we have 𝑦_𝑘−𝐿<𝜖since𝑦_𝑘−𝐿=𝑥_𝑘−𝐿<𝜖if𝑥_𝑘−𝐿<𝑗⁻¹,𝑦_𝑘−𝐿=|𝐿−𝐿|=0if𝑥_𝑘−𝐿>𝑗⁻¹. (10) Hence, lim_𝑘𝑦_𝑘=𝐿, since 𝜖 was arbitrary.
Next we observe that 𝑘∈𝐼_𝑛∶𝑧_𝑘≠0≥𝑘∈𝐼_𝑛∶𝑧_𝑘≥𝜖 (11) for any natural number 𝑛 and 𝜖>0. Hence, lim_𝑛(1/𝜆_𝑛)|{𝑘∈𝐼_𝑛∶𝑧_𝑘≠0}|=0; that is, 𝑧 is 𝜆-statistically null.
We now show that if 𝛿>0 and 𝑗∈𝑁 such that 𝑗⁻¹<𝛿, then |{𝑘∈𝐼_𝑛∶𝑧_𝑘≠0}|<𝛿 for all 𝑛>𝑁_𝑗. Recall from the construction that if 𝑁_𝑗<𝑘≤𝑁_𝑗+1, then 𝑧_𝑘≠0 only if |𝑥_𝑘−𝐿|>𝑗⁻¹. It follows that if 𝑁_ℓ<𝑘≤𝑁_ℓ+1, then 𝑘∈𝐼_𝑛∶𝑧_𝑘≠0⊆𝑘∈𝐼_𝑛∶𝑥_𝑘−𝐿>ℓ⁻¹. (12)
Consequently, 1𝜆_𝑛𝑘∈𝐼_𝑛∶𝑧_𝑘≤1≠0𝜆_𝑛𝑘∈𝐼_𝑛∶𝑥_𝑘−𝐿>ℓ⁻¹<ℓ⁻¹<𝑗⁻¹<𝛿, (13) if 𝑁_ℓ<𝑛≤𝑁_ℓ+1 and ℓ>𝑗. That is, lim_𝑛1𝜆_𝑛𝑘∈𝐼_𝑛∶𝑧_𝑘≠0=0. (14)
This completes the proof of the theorem.

3. Application to Fourier Series

Let 𝑓∶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 𝑓(𝑥)∼_𝑗∈ℤ𝑓𝑗𝑒^𝑖𝑗𝑥,𝑥∈𝕋, (15) where the Fourier coefficients 𝑓(𝑗) are defined by 𝑓𝑗1∶=2𝜋_𝕋𝑓(𝑡)𝑒^{−𝑖𝑗𝑡}𝑑𝑡,𝑗∈ℤ. (16) The symmetric partial sums of the series in (15) are defined by 𝑠_𝑘𝑓;𝑥∶=_𝑗≤𝑘𝑓𝑗𝑒^𝑖𝑗𝑥,𝑥∈𝕋,𝑘∈ℕ. (17) The conjugate series to the Fourier series in (15) is defined by [17, Vol. I, pp. 49] −𝑖sgn𝑗𝑓_𝑗∈ℤ𝑗𝑒^𝑖𝑗𝑥. (18) Clearly, it follows from (15) and (18) that _𝑗∈ℤ𝑓𝑗𝑒^𝑖𝑗𝑥+𝑖_𝑗∈ℤ−𝑖sgn𝑗𝑓𝑗𝑒^𝑖𝑗𝑥=1+2^∞_𝑗=1𝑓𝑗𝑒^𝑖𝑗𝑥, (19) and the power series 1+2^∞_𝑗=1𝑓𝑗𝑒^𝑖𝑗𝑥,where𝑧∶=𝑟𝑒^𝑖𝑥,0≤𝑟<1, (20) is analytic on the open unit disk |𝑧|<1, due to the fact that 𝑓𝑗≤12𝜋_𝜋𝑓(𝑡)𝑑𝑡,𝑗∈ℤ. (21) The conjugate function 𝑓 of a function 𝑓∈𝐿¹(𝕋) is defined by 𝑓(𝑥)∶=−lim_𝜀↓01𝜋_{|𝑡|𝜀≤≤𝜋}𝑓(𝑥+𝑡)2tan(𝑡/2)𝑑𝑡=lim_𝜀↓01𝜋∫^𝜋_𝜀𝑓(𝑥−𝑡)−𝑓(𝑥+𝑡)2tan(𝑡/2)𝑑𝑡 (22) in the “principal value” sense and that 𝑓(𝑥) exists at almost every 𝑥∈𝕋.

The following is 𝜆-statistical version of [18] (c.f. [19, Theorem 2.1 (ii)]).

Theorem 3. If 𝑓∈𝐿¹(𝕋), then for any 𝑝>0 its Fourier series is strongly  𝜆_𝑝-summable to 𝑓(𝑥) at almost every 𝑥∈𝕋. Furthermore, its conjugate series (18) is strongly  𝜆_𝑝-summable for any 𝑝>0 to the conjugate function 𝑓(𝑥) defined in (22) at almost every 𝑥∈𝕋.

From Theorems 1 and 3, we easily get the following useful result.

Theorem 4. If 𝑓∈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 of 𝜆-summable, 𝜆-statistical convergence, 𝜆-statistical Cauchy, and strongly  𝜆_𝑝-summability in Paranormed Space.

Let (𝑋,𝑔) be a Paranormed Space.

A sequence 𝑥=(𝑥_𝑘) is said to be convergent to the number 𝜉 in (𝑋,𝑔) if, for every 𝜀>0, there exists a positive integer 𝑘₀ such that 𝑔(𝑥_𝑘−𝜉)<𝜀 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 sequence 𝑥=(𝑥_𝑘) is said to be 𝜆-statistically convergent to the number 𝜉 in (𝑋,𝑔) if, for each 𝜖>0, lim_𝑛1𝜆_𝑛𝑘∈𝐼_𝑛∶𝑔𝑥_𝑘−𝜉≥𝜖=0. (23) In this case we write st_𝜆(𝑔)-lim𝑥=𝜉.

Definition 6. A sequence 𝑥=(𝑥_𝑘) 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 lim_𝑛1𝜆_𝑛𝑗∈𝐼_𝑛∶𝑔𝑥_𝑗−𝑥_𝑁≥𝜖=0. (24)

Definition 7. A sequence 𝑥=(𝑥_𝑘) is said to be strongly  𝜆_𝑝-summable  (0<𝑝<∞) to the limit  𝜉 in (𝑋,𝑔) if lim_𝑛1𝜆_{𝑛𝑘∈𝐼𝑛}𝑔𝑥_𝑘−𝜉^𝑝=0, (25) and we write it as 𝑥_𝑘→𝜉[𝑉_𝜆,𝑔]_𝑝. In this case 𝜉 is called the [𝑉_𝜆,𝑔]_𝑝-limit of 𝑥.

Now we define another type of convergence in Paranormed Space.

Definition 8. A sequence (𝑥_𝑘) in a Paranormed Space (𝑋,𝑔) is said to st^∗_𝜆(𝑔)-convergent to 𝜉∈𝑋 if there exists an index set 𝐾={𝑘₁<𝑘₂<⋯<𝑘_𝑛<⋯}⊆ℕ, 𝑛=1,2,…, with 𝛿_𝜆(𝐾)=1 such that 𝑔(𝑥_𝑘𝑛−𝜉)→0(𝑛→∞). In this case, we write 𝜉=st^∗_𝜆(𝑔)-lim𝑥.

First we prove the following results on 𝜆-statistical convergence in (𝑋,𝑔).

Theorem 9. If 𝑔-lim𝑥=𝜉, then st_𝜆(𝑔)-lim𝑥=𝜉 but converse need not be true in general.

Proof. Let 𝑔-lim𝑥=𝜉. Then, for every 𝜀>0, there is a positive integer 𝑁 such that 𝑔𝑥_𝑛−𝜉<𝜀 (26) for all 𝑛≥𝑁. Since the set 𝐴(𝜖)∶={𝑘∈ℕ∶𝑔(𝑥_𝑘−𝜉)≥𝜀} is finite, 𝛿_𝜆(𝐴(𝜖))=0. Hence, st_𝜆(𝑔)-lim𝑥=𝜉.
The following example shows that the converse need not be true.
Example 10. Let 𝑋=ℓ(1/𝑘)∶={𝑥=(𝑥_𝑘)∶∑_𝑘|𝑥_𝑘|^1/(𝑘+1)<∞} with the paranorm 𝑔(𝑥)=(∑_𝑘|𝑥_𝑘|^1/(𝑘+1)). Define a sequence 𝑥=(𝑥_𝑘) by 𝑥_𝑘∶=𝑘,if𝑛−𝜆_𝑛+1≤𝑘≤𝑛,𝑛∈ℕ;0,otherwise, (27) and write 𝐾(𝜀)∶=𝑘≤𝑛∶𝑔𝑥_𝑘≥𝜀,0<𝜀<1. (28)
We see that 𝑔𝑥_𝑘𝑘∶=^1/(𝑘+1),if𝑛−𝜆_𝑛+1≤𝑘≤𝑛,𝑛∈ℕ;0,otherwise, (29) and hence lim_𝑘𝑔𝑥_𝑘∶=1,if𝑛−𝜆_𝑛+1≤𝑘≤𝑛,𝑛∈ℕ;0,otherwise. (30)
Therefore 𝑔-lim𝑥 does not exist. On the other hand 𝛿_𝜆(𝐾(𝜀))=0; that is, st_𝜆(𝑔)-lim𝑥=0.
This completes the proof of the theorem.

We can easily prove the following results on 𝜆-statistical convergence in (𝑋,𝑔) similar to those of [11].

Theorem 11. If a sequence 𝑥=(𝑥_𝑘) is 𝜆-statistically convergent in (𝑋,𝑔), then 𝑠𝑡_𝜆(𝑔)-limit is unique.

Theorem 12. Let 𝑠𝑡_𝜆(𝑔)-lim𝑥=𝜉₁ and 𝑠𝑡_𝜆(𝑔)-lim𝑦=𝜉₂. Then,(i)𝑠𝑡_𝜆(𝑔)-lim(𝑥±𝑦)=𝜉₁±𝜉₂,(ii)𝑠𝑡_𝜆(𝑔)-lim 𝛼𝑥=𝛼𝜉₁,𝛼∈ℝ.

Theorem 13. Let (𝑋,𝑔) be a complete Paranormed Space. Then a sequence 𝑥=(𝑥_𝑘) of points in (𝑋,𝑔) is 𝜆-statistically convergent if and only if it is 𝜆-statistically Cauchy.

Theorem 14. (a) If 0<𝑝<∞ and 𝑥_𝑘→𝜉[𝑉_𝜆,𝑔]_𝑝, then 𝑥=(𝑥_𝑘) is 𝜆-statistically convergent to 𝜉 in (𝑋,𝑔).
(b) If 𝑥=(𝑥_𝑘) is bounded and 𝜆-statistically convergent to 𝜉 in (𝑋,𝑔), then 𝑥_𝑘→𝜉[𝑉_𝜆,𝑔]_𝑝.

Theorem 15. Let (𝑋,𝑔) be a complete Paranormed Space. Then a sequence 𝑥=(𝑥_𝑘) 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 sequence 𝑥=(𝑥_𝑘) in (𝑋,𝑔) is 𝜆-statistically convergent to 𝜉 if and only if it is 𝑠𝑡^∗_𝜆(𝑔)-convergent to 𝜉.

Proof. Suppose that 𝑥=(𝑥_𝑘) is 𝜆-statistically convergent to 𝜉; that is, 𝑠𝑡_𝜆(𝑔)-lim𝑥=𝜉. Now, write for 𝑟=1,2,…. 𝐾_𝑟∶=𝑛∈ℕ∶𝑔𝑥_𝑘𝑛≥1−𝜉𝑟,𝑀_𝑟∶=𝑛∈ℕ∶𝑔𝑥_𝑘𝑛<1−𝜉𝑟(𝑟=1,2,…). (31) Then 𝛿_𝜆(𝐾_𝑟)=0, 𝑀₁⊃𝑀₂⊃⋯⊃𝑀_𝑖⊃𝑀_𝑖+1⊃⋯,𝛿_𝜆𝑀_𝑟=1,𝑟=1,2,…. (32)(33)
Now we have to show that, for 𝑛∈𝑀_𝑟, (𝑥_𝑘𝑛) is 𝑔-convergent to 𝜉. On contrary suppose that (𝑥_𝑘𝑛) is not 𝑔-convergent to 𝜉. Therefore, there is 𝜀>0 such that 𝑔(𝑥_𝑘𝑛−𝜉)≥𝜀 for infinitely many terms. Let 𝑀_𝜀∶={𝑛∈ℕ∶𝑔(𝑥_𝑘𝑛−𝜉)<𝜀} and 𝜀>1/𝑟, 𝑟∈ℕ.
Then 𝛿_𝜆𝑀_𝜀=0, (34) and by (32), 𝑀_𝑟⊂𝑀_𝜀. Hence 𝛿_𝜆(𝑀_𝑟)=0, which contradicts (33) and we get that (𝑥_𝑘𝑛) is 𝑔-convergent to 𝜉. Hence, 𝑥 is st^∗_𝜆(𝑔)-convergent to 𝜉.
Conversely, suppose that 𝑥 is st^∗_𝜆(𝑔)-convergent to 𝜉. Then there exists a set 𝐾={𝑘₁<𝑘₂<𝑘₃<⋯<𝑘_𝑛<⋯} with 𝛿_𝜆(𝐾)=1 such that 𝑔-lim_𝑛→∞𝑥_𝑘𝑛=𝜉. Therefore, there is a positive integer 𝑁 such that 𝑔(𝑥_𝑛−𝜉)<𝜀 for 𝑛≥𝑁. Put 𝐾_𝜀∶={𝑛∈ℕ∶𝑔(𝑥_𝑛−𝜉)≥𝜀} and 𝐾^′∶={𝑘_𝑁+1,𝑘_𝑁+2,…}. Then 𝛿_𝜆(𝐾^′)=1 and 𝐾_𝜀⊆ℕ−𝐾^′ which implies that 𝛿_𝜆(𝐾_𝜀)=0. Hence 𝑥=(𝑥_𝑘) is 𝜆-statistically convergent to 𝜉; that is st_𝜆(𝑔)-lim𝑥=𝜉.
This completes the proof of the theorem.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (130-073-D1434). The authors, therefore, acknowledge with thanks DSR technical and financial support.