Sliding window convergence and lacunary statistical convergence for measurable functions via modulus function

abstract: In this paper we study the concepts of sliding window convergence for real valued measurable functions defined on [0,∞) via modulus function. We also establish some inclusions and consistency theorems for sequential methods along with examples. Finally, we give a Cauchy convergence criterion.


Introduction and Preliminaries
A function M : [0, ∞) → [0, ∞) is said to be a modulus function if it satisfy the following conditions: 1. M (x) = 0 if and only if x = 0, 2. M (x + y) ≤ M (x) + M (y), for all x, y ≥ 0, 3. M is increasing, 4. M is continuous from the right at 0.
It follows that M must be continuous everywhere on [0, ∞).The modulus function may be bounded or unbounded.For example, if we take M (x) = x x+1 , then M (x) is bounded.If M (x) = x p , 0 < p < 1 then the modulus function M (x) is unbounded.For more details about modulus function and sequence spaces one may refer to ( [4], [6], [10], [22], [25], [26]) and references therein.The concept of statistical convergence was introduced by Steinhaus [28] and Fast [10] and later reintroduced by Schoenberg [27] independently.In recent years, statistical convergence was discussed in the theory of Fourier analysis, ergodic theory, 162 M. Mursaleen and Kuldip Raj number theory, measure theory, trigonometric series and Banach spaces, e.g. ( [1]- [3], [16]- [18]) .The corresponding notion of convergence for function of a real variable was established in ( [7], [8]) and recently investigated by Mòrlicz [23] .Fridy and Orhan ( [14], [15]) introduced lacunary statistical convergence, with some of their result constructing on the work of Freedman et al. [12].For latest work on the related topic can be found in [5], [19], [20], [21].In this paper we encompassed Fridy and Orhan's work into more general settings of functions of a real variable by using an modulus function.
Let M be a modulus function, p be positive real number then we define the following definitions: (2) The function g is statistically (γ, η, M, p) convergent to L and we write S(γ, η, M, p) − lim g = L(or g → LS(γ, η, M, p)) if and only if for all ǫ > 0. In this case we write that g is S(γ, η, M, p) convergent.We call either of the methods defined above a Sliding window method.
Note that the averages are taken over the disjoint intervals (k n−1 , k n ], the preceding definition for statistical (γ, η, M, p) convergence does not require the intervals (γ(r), η(r)] to be disjoint.For instant, if γ(r) = 0 and η(r) = r, we have that N (γ, η, M, p) and S(γ, η, M, p) are strong Cesàro summability and statistical convergence for measurable functions as consider in [23] and [8].
Let N (γ, η, M, p) and S(γ, η, M, p) are strong Cesàro summability and statistical convergence for measurable functions by means of an modulus function.A function g is statistical convergent to L provided for all ǫ > 0. Throughout this paper by S − lim g denote the statistical limit of g.If (γ, η) is a sliding window pair such that there is a function θ : N → (0, ∞) such that θ(n + 1) − θ(n) tends to infinity and a sequence (r n ) of real numbers for which, given s ∈ (r n , r n+1 ], γ(s) = θ(n) and η(s) = θ(n + 1), then N (γ, η, M, p) and S(γ, η, M, p) will be denoted by S θ (M, p) and N θ (M, p).Let I n = (θ(n), θ(n + 1)] and observe that g is S θ (M, p)− statistically convergent to L if and only if for all ǫ > 0. Note that the intervals I n are pairwise disjoint in this special case.In keeping with the sequential method, the method S θ (M, p) will be called lacunary statistical convergence.A similar construction of a pair (γ, η) can be used to show that λ−statistical convergence and λ−strong summability, as defined in [24], can be also be viewed as sliding window methods.The objective of this paper is to introduce a class of summability methods that can be applied to measurable functions defined on [0, ∞) by means of modulus function.These methods are known as sliding window methods are demonstrated on the methods of statistical convergence and lacunary statistical convergence by means of modulus function.We also establish some results for sequential summability to the setting of real valued functions defined on [0, ∞).

Correlation between Strong summability and Statistical convergence
In this section we establish relationship between strong summability and statistical convergence.
Theorem 2.1.Let M be a modulus function and p be a positive real numbers.Let (γ, η) be a sliding window pair, g be a measurable function and L be a real number.
Proof: Firstly we show that N (γ, η, M, p) − lim g = L implies S(γ, η, M, p) − lim g = L.For ǫ > 0, we have If g is bounded by B, then we have As the first term of the right-hand side tends to 0 as r tends to infinity, it follows that S(γ, η, M, p) Let M be a modulus function and p be a positive real numbers.If g is S(γ, η, M, p) convergent to L, then there is a measurable set A such that S(γ, η, M, p) Theorem 2.3.Let M be a modulus function and p be a positive real numbers.Let (γ, η) be a sliding window pair.Then there is a measurable function g that is not S(γ, η, M, p) convergent.
Proof: We construct a set A such that its characteristic function, i.e., a function that only takes the value 0 or 1, is not S(γ, η, M, p) convergent.Let r 1 be given and set A 1 = (γ(r 1 ), η(r 1 )].As A 1 is bounded, there is an r 2 > r 1 such that for all s ≥ r 2 we have that Define g by g(t) = χ A (t) and select ǫ such that 0 < ǫ < 1.The above calculation shows that lim and thus, g is not S(γ, η, M, p) convergent.✷ Next we show that in general S(γ, η, M, p) convergence is stronger than ordinary convergence and that S(γ, η, M, p) convergence does not imply N (γ, η, M, p) summability.
Theorem 2.4.Let M be a modulus function and p be a positive real numbers.Let (γ, η) be a sliding window pair.Then there is a function g such that lim t g(t) does not exist and S(γ, η, M, p) − lim g = 0 and an unbounded function h such that S(γ, η, M, p) − lim h = 0 and N (γ, η, M, p) − lim h does not exist.
Similarly we can show that S(γ, η, M, p) − lim h = 0 for any function h that has its support contained in j H j .Now define a function h by Observe that since for any L there is a J such that j > J implies which tends to infinity as j tends to infinity.Hence N (γ, η, M, p) − lim h = 0 does not exist and S(γ, η, M, p) − lim h = 0. Next we consider the case lim inf s (η(s) − γ(s)) = ∞.Select a sequence (s j ) increasing to infinity such that η(s 1 ) > 1, η(s j+1 ) > 2η(s j ) and r ≥ s j implies that η(r)−γ(r) ≥ 2 j .Set K j = (η(s j )−1, η(s j )] and define the function g by g(t) = 1 for t ∈ j K j and 0 otherwise.Note that lim t g(t) does not exist.We now demonstrate that S(γ, η, M, p) − lim g = 0. Let r ∈ [s j , s j+1 ) and ǫ > 0. Note that η(r) ≤ η(s j+1 ) implies that which tends to 0 as j tends to infinity.Hence S(γ, η, M, p) − lim g = 0.As before, S(γ, η, M, p) − lim h = 0 for any function that has its support contained in Define the function h by h(t) = j(η(s j ) − γ(s j )) when t ∈ K j for some j and 0 otherwise.Observe that for any L there is a J such that j > J implies Hence N (γ, η, M, p) − lim h does not exist and S(γ, η, M, p) − lim h = 0. ✷ Next we compare sliding window methods to statistical convergence.The next examples establishes that in general statistical sliding window convergence is not equivalent to statistical convergence.
Example 2.5.Let M be a modulus function and p be a positive real numbers.Let (γ, η) be a sliding window pair and a bounded function g with the property that g is S(γ, η, M, p) convergent but neither strongly Cesàro summable nor statistical convergence.Let M = I(identity) and p = 1.By using an inductive argument we constructed sliding window pair to generate a sequence (a n ).Let a 1 = 1 and select Proceeding inductively: if n − 1 is odd, select a n such that Define the sliding window pair (γ, η) using the function γ(t) = a 2n and η(t) = a 2n+3 for t ∈ (n, n + 1] and the function g by g(t) = 1 for t ∈ (a n−1 , a n ] when n is even and 0 otherwise.We establish that the function g is S(γ, η, M, p) convergent to 0 by observing, that for s ∈ (p, p + 1] and ǫ > 0, we have that . which tends to 0 as s hence p tends to infinity.Recall that strong Cesàro summability and statistical convergence correspond respectively to N (γ ′ , η ′ , M ′ , p ′ ) and S(γ ′ , η ′ , M ′ , p ′ ) convergence when M ′ (r) = I(identity), p ′ = 1, γ ′ (r) = 0, and η ′ (r) = r for all positive real r.First we establish g is not strongly Cesàro summable.
Observe that which tends to 1 as p tends to infinity.Next note that which tends to 0 as p tends to infinity.Hence g is not strongly Cesàro summable.As g is bounded, it follows from Theorem 2.1 that g is not statistical convergent.Lemma 2.6.[9] Let V = (0, v] be an interval and let U = {U z : z ∈ Z} be a collection of half-open, half-closed intervals such that V = {U z : z ∈ Z} and there is a b > 0 such that b < µ(U z ) for all z ∈ Z.Then, for any ǫ > 0, there is a finite, disjoint subcollection {U 1 , ....., U n } of U such that Lemma 2.7.[9] Let (γ, η) be a sliding window pair.
2. If Q is sufficiently large and ǫ > 0, then there is a finite disjoint subcollection Note that example 2.5 shows that it is necessary to assume that the function is convergent with respect to both methods.
Theorem 2.8.Let M be a modulus function and p be a positive real numbers.If g is statistically (γ, η) convergent and statistically convergent, then S(γ, η, M, p) − lim g = S − lim g.

Proof:
We assume without loss of generality, that g is a function such that S(γ, η, M, p) − lim g = 1 and S − lim g = 0 and derive a contradiction.First select 12 , be given and using Lemma 2.7 select a finite disjoint collection of intervals {I s : Since ǫ < 1 12 , we have .
Consequently, the statistical limit of g is not equal to 0 which contradicts the hypothesis on g. ✷ Theorem 2.9.Let M be a modulus function and p be a positive real numbers.Let (γ, η) be a sliding window pair.The following are equivalent: (1) If a function g is statistically convergent, then g is statistically (γ, η, M, p) convergent.
In this case Now, since γ n = η n 1+ǫn and r ≤ η n , we have that which also tends to 0 as n tends to infinity.As for all ǫ > 0, we have that S − lim g = 0. Now note that, by construction for all n, hence the S(γ, η, M, p)−lim g is either not equal to 0 or does not exist.By Theorem(2.8),since S − lim g = 0, if S(γ, η, M, p) − lim g exists then S(γ, η, M, p) − lim g = 0. Hence S(γ, η, M, p) − lim g does not exist.✷ Theorem 2.10.Let M be a modulus function and p be a positive real numbers.Let (γ, η) be a lacunary sliding window pair.The following are equivalent: (1) If a function g is lacunary statistically convergent, then g is statistically convergent.
Next observe that for 0 < ǫ < 1, and hence S θ − lim g = 0 or does not exist.As before, Theorem 2.8 yield that S − lim g does not exist.✷

Cauchy Criterion for S(γ, η, M, p)−convergence
In this section we make an effort to establish Cauchy criterion for S(γ, η, M, p)convergence and a criterion for two sliding window methods to be equivalent for bounded functions.
Definition 3.1.Let (γ, η) be a sliding window pair.The function g is said to be S(γ, η, M, p)-Cauchy if for every r > 0 there is an element t r ∈ I r such that lim r g(t r ) exists and for all ǫ > 0.
Theorem 3.2.Let M be a modulus function and p be a positive real numbers.Let (γ, η) be a sliding window pair and g be a measurable function.Then S(γ, η, M, p)− lim g exists if and only if g is S(γ, η, M, p)-Cauchy.
Proof: First we establish that S(γ, η, M, p)-convergent functions are S(γ, η, M, p)-Cauchy.Let g be a function such that S(γ, η, M, p) − lim g = L and j ∈ N. Set as r tends to infinity for each j ∈ N. Hence there is an increasing sequence of indices .