Statistical Tauberian theorems for Cesàro integrability mean based on post-quantum calculus

The notion of statistical convergence is more general than the classical convergence. Tauberian theorems via different ordinary summability means have been established by many researchers. In the present work, we have established some new Tauberian theorems based on post-quantum calculus via statistical Cesàro summability mean of real-valued continuous function of one variable under oscillating behavior and De la vallée Poussin mean of a single integral. Moreover, some remarks and corollaries are provided here to support our theorems.

Furthermore, Çanak and Totur [1] proved a Tauberian theorem for Cesàro summability of single integral and also, they established the alternative proofs of some classical type Tauberian theorems for the Cesàro summability of single integral. Moreover, in the year 2017, Jena et al. [11] proved the Tauberian theorems for Harmonic summability of double-integrable real-valued function over R 2 and also established the inclusion relation between the statistical convergence and classical convergence. Very recently, Çanak et al. [3] introduced and studied the concept of Tauberian theorem for Cesàro integrability mean based upon quantum calculus via usual convergence.
Motivated essentially by the above-mentioned investigations and results, here we prove the statistical versions of Tauberian theorems via Cesàro integrability mean based upon post-quantum calculus of a realvalued continuous function of one variable under slow oscillation and De la Vallée Poussin mean of the integral. In fact, we extend here the result of Çanak [3] using the idea of Tauberian theorems for Cesàro integrability mean based on quantum calculus via usual summability.
Let f (x) be a function in R with the partial sum The (C, 1) mean of f (x) is (see [20]), The integral The integral x 0 s(ζ )dζ is statistically (C, 1)-summable to a finite number if, for each > 0, In this case, we write stat lim x→∞ σ (s(x)) = . (2.2) If lim x→∞ s(x) = exists, then the relations (2.1) and (2.2) hold. However, in general, the converse is not true. To prove the sufficient part, we use the oscillatory behavior and De la vallée Poussin mean of the above integral over R. Such a condition is called a Tauberian condition and the resulting theorem is called a Tauberian theorem.
Quantum calculus (or q-calculus) is the modern name for the investigation of calculus which is focused on the idea of deriving q-analogues of results belonging to standard calculus without using limits. The q-calculus served as a bridge between mathematics and physics. Furthermore, there is a possibility of extension of the q-calculus to post-quantum calculus (or ( p, q)-calculus). It has gained noticeable importance and popularity during the past 3 decades due mainly to its applications in different mathematical areas such as number theory, combinatorics, orthogonal polynomials, basic hypergeometric series and other sciences such as statistical mechanics, quantum theory, and the theory of relativity.
We recall some basic concepts of ( p, q)-calculus and some associated properties of ( p, q)-Cesàro integrability mean method. For any n ∈ N, the ( p, q)-integer [n] p,q is defined by for all n, k ∈ N and n ≥ k.
We also recall that, suppose 0 < q < p 1 and the ( p, q)-derivative of the function f is defined as As a special case, when p = 1, ( p, q)-derivative reduces to q-derivative and also, the ( p, q)-derivative fulfills the following product derivative properties: and .
Next, let f be a real-valued continuous function and let a be a real number.
provided the sums converge absolutely.
converges absolutely. Moreover, we assume that, f (t) is a function defined on [0, ∞) and satisfying x 0 | f (t)d p,q t| < ∞ and also, let the partial sum of f (t) be defined as Next, we define the ( p, q) Cesàro mean of f (x) as follows: Furthermore, the functions(x) is ( p, q)-statistically Cesàro summable to if, for every > 0, In this case, we write stat lim x→∞ t (s(x)) = . (2.5) If lim x→∞s (x) = exists, then the relations (2.4) and (2.5) hold. However, in general, the converse is not true. To prove the oscillatory behavior and De la vallée Poussin mean based onsufficient part, we use the ( p, q)-calculus of the above integral over R. Such a condition is called a Tauberian condition and the resulting theorem is called a Tauberian theorem under post-quantum calculus.
For each non-negative integer k we define Remark 2.1 If k = 1, then (C, k)-summabllity mean is same as the (C, 1)-summability mean.
Next we have, the partial sum of the function is where Notice that x .
Now, let us define for each non-negative integer k, Here, the integral Similarly, for post-quantum calculus, the partial sum of the function is Notice that x .
Now, we define for each non negative integer k, Here, the integral Now recalling the De la Vallée Poussin mean (see [3]) of the integral In the similar way, we define the De la Vallée Poussin mean for the post-quantum calculus of the integral Note that, an integral We have If we choose β = x/2 i and β/α ≤ 2, we obtain That impliesv( f (x)) is oscillating slowly by Kronecker identity.

Now we have to show that t (s(x)) is oscillating slowly. Since
To prove the converse part, suppose thatv( f (x)) is bounded and oscillating slowly. The boundedness of v( f (x)) implies that t (s(x)) is oscillating slowly. Sincev( f (x)) is oscillating slowly, sos(x) is oscillating slowly by Kronecker identity. This establishes Lemma 3.2.
Proof We have by De la Vallée Poussin mean ofs(x), Subtracting t (s(x)) from the identity, alsō This establish Lemma 2.
Proof The proof of this theorem is similar to the proof of Theorem 3.4.

Conclusion
Tauberian theorems for single sequences as well as for functions of single variable have achieved a high degree of development; however, it is still in its infancy for double sequences and functions of two or more variables. The result established here that the statistical versions of Tauberian theorems for a real-valued continuous function of one variable under the post-quantum calculus (or ( p, q)-calculus) of integrals via statistical Cesàro summability mean generalizes some earlier existing Tauberian theorems for the function of a single variable in quantum calculus (or (q)-calculus) via classical Cesàro summability mean. Further, it will be encouraging if one can extend the result for functions of two or more variables using the post-quantum calculus (or ( p, q)calculus) of integrals via the statistical Cesàro mean and so also for other different statistical versions of summability means.
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