On μ -strong Cesaro summability at inﬁnity and its application to the Fourier–Stieltjes transforms

The concept of μ -strong Cesaro summability at inﬁnity for a locally integrable function is introduced in this work. The concept of μ -statistical convergence at inﬁnity is also considered and the relationship between these two concepts is established. The concept of μ [ p ]-strong convergence at inﬁnity point, generated by the measure μ ( · ) is also considered. Similar results are obtained in this case too. This approach is applied to the study of the convergence of the Fourier–Stieltjes transforms


Introduction
The concept of Cesaro summability of integrals is long known. We refer the readers to the monograph by Hardy [17] (see pp. [10][11][12][13] where this concept is referred to as the (C; 1) summability, and to E.C. Titchmarsh [30] (see pp. [26][27][28][29][30]. See also Moricz [22]. In this work, we consider the similar concepts in a Borel measurable space and obtain results at a point. We introduce the concept of μ-strong Cesaro summability of integrals, which is related to the one of μ-statistical convergence introduced and studied by Bilalov and Sadigova [3].
They introduced the concepts of μ-stat convergence and μ-stat fundamentality, proved their equivalence and studied some of their properties. They also introduced the concept of μ-stat continuity. μ-stat convergence is a direct generalization of the statistical convergence in continuous case, as it turns out from this concept as a special case.
The concept of μ-strong Cesaro summability at infinity for a locally integrable function is introduced in this work. The concept of μ-statistical convergence at infinity is also considered and the relationship between these two concepts is established. The concept of μ [ p]-strong convergence at infinity point, generated by the measure μ (·) is also considered. Similar results are obtained in this case too. This approach is applied to the study of the convergence of the Fourier-Stieltjes transforms. It should be noted that the results obtained in this paper generalize the results of work Morics [22].

Necessary notations and information
We will use the standard notation. R is the set of all real numbers; ∃ will mean "there exist(s)"; ∃! will mean "there exists a unique"; ⇒ will mean "it follows"; ⇔ will mean equivalence. I +∞ a ≡ [a ; +∞); I −∞ a ≡ (−∞; a]. Let I ∞ a ; B; μ be a measurable space with measure μ : B → I ∞ a , where B σ -algebra of Borel subsets in I ∞ a . We will assume that the measure μ is a σ -finite measure and μ I ∞ a = +∞. The measure of the set M ∈ B will be denoted by |M|, i.e. |M| = μ (M).
We will need some concepts and facts from the work [3].
The set of all subsets of I ∞ a , for which the infinity (∞) is the point of μ-stat density, will be denoted by I st (∞).
We will need the concept of μ-stat-fundamentality which introduced in [3].

Definition 2.3
We say that the B-measurable function f : We will also use the following information. Let J ⊂ R be some segment, B J be σ -algebra of all Borel subsets J and μ : Assume that the measure μ (·) satisfies the following condition: We say that the point x 0 ∈ J is a point μ-stat density for M, if measurable function and ε > 0 be some number. For given number l ∈ R assume Denote by J st (x 0 ) the family of all sets of B J , which x 0 is the point of μ-stat density. Definition 2. 4 We say that l is μ-stat limit of the function f at a point This limit will be denoted as μ-st lim Thus, (J ; B J ) measurable functions with μ-stat limit at the point x 0 ∈ J form a linear space over a field K , and we denote this space by B st (x 0 ) .
Similarly we define the concepts of one-sided μ-stat limits at the point x 0 . So, assume

Definition 2.5
We say that is a right-hand μ-stat limit of the function f at the point We say that x 0 ∈ J is a point of right-hand μ-stat density for the set By J + st (x 0 ) we denote the family of all subsets of B J , for which the point x 0 is a point of right-hand μ-stat density.
Similarly, we define the concept of the point of left-hand and right-hand μ-stat density and the family Proceeding from these concepts μ-stat one-sided limits of the function f (·) at the point x 0 are defined. Namely, we say that the function f (·) has a μ-stat right-hand (left-hand) limit equal to the l at the point x 0 , if and this fact will be denoted as Let us note that these concepts are introduced and studied in [24,25].
This limit will be denoted by Let us show that this definition is well-define. Let We have Consider the particular cases.
It is not difficult to see that this operation of taking the limit is linear, i.e.
In case of 1 ≤ p < +∞, the above assertion follows from the Minkowski inequality, and in case of 0 < p < 1 it follows from the inequality Let 0 < p _1 < p 2 < +∞. Applying Hölder's inequality with the exponents p 2 p 1 and p 2 p 2 − p 1 , we have Note that in the discrete case, p-Cesaro summability has the following form (see, e.g., [17, p. 147 The theorem below is the μ-analogue of the discrete case.
From the arbitrariness of ε it follows that We have From the arbitrariness of ε > 0 it follows Theorem is proved.
Note that the obtained results are the generalizations of the results of [22] to the μ-case.

μ [ p]-strong Cesaro convergence at finite point
Let J ⊂ R be some interval, B J be a σ -algebra of all Borel subsets of J and μ : Let f : J → R be some (J ; B J )-measurable function, and μ : B J → R + be a Borel measure satisfying the condition α). As before, we assume μ (M) = |M| , ∀M ∈ B J . Let p ∈ (0, +∞) be some number. Introduce the following definition.

Definition 4.1 We will say that the function f is μ [ p]-strongly Cesaro convergent to
Using the well-known inequality quite similar to the previous case we can prove

x), and this limit is equal to A.
In accordance with the μ-statistical convergence, we introduce the following definition. Definition 4. 3 We will say that the function f : J → R has a μ [ p]-strong Cesaro limit (or is Cesaro convergent) at the point a ∈ R from the right (from the left) if there exists the limit and this limit is denoted by μ Hence it follows that μ Hence it follows that if The following theorem is true.
If the measure μ (·) satisfies the condition β) and
where p ∈ [1, +∞) is some number. In case where the measure μ (·) is defined by dμ (t) = ρ (t) dt, the μ [ p]-statistical convergence at a point will be called a ρ [ p]-statistical convergence.
Introduce the following definition.
Denote the set of all p th order ρ-Lebesgue points of the function f by L ρ ( f ; p). Applying Hölder's inequality, we obtain that L ρ ( f ; In what follows we'll need some facts of the theory of measurable spaces with a measure. Recall the definition of a regular Borel measure. (X ; τ ) be a Hausdorff topological space and B τ be a σ -algebra of its Borel sets. A measure μ : B τ → [0, +∞] is called a regular Borel measure if the following conditions are satisfied:

Definition 5.2 Let
The following theorem is true.

Theorem 5.3
Let (X ; τ ) be a locally compact Hausdorff space with a regular Borel measure μ (·) on X . Let f be a p th order integrable function, p ∈ [1, +∞), with respect to the measurable space (X ; B τ ; μ). Then for ∀ε > 0 there exists a function g (·) continuous on X with compact support such that For more details see, e.g., [1]. Let f ∈ L p,ρ L 1 .Consider and define the partial integralsS τ conjugate to S τ as follows The expressions are well known. In case ρ ≡ 1, denote the set L ρ ( f ; p) (i.e. the set of p th order Lebesgue points) by E ( f ; p), and let The theorem below was proved in [15].
Theorem 5.4 [15] i) Let f ∈ L (R), and p ∈ (0, +∞). Then m (R\E ( f ; p)) = 0, where m (·) is a Lebesgue measure. In other words, the set of points x ∈ R, for which the relation lim In the sequel, we will need some weighted versions of Hausdorff-Young inequality for the Fourier transform. Let's state some facts related to this matter.
Let u (·) and ϑ (·) be some weight functions on R. As usual, byf we denote the Fourier transform of the function f :f Denote by HY p;q the class of weights (u; ϑ) for which the following Hausdorff-Young type inequality holds where c is a constant independent of f . Numerous works have been dedicated to this matter. A simple version of this inequality is the Pitt's theorem, which states that if the conditions hold, then the inequality (5.1) is true for ∀ f ∈ L p;u (R). More details about this fact can be found in [28, p.489]. In [20], the sufficient conditions for the pair (u; ϑ) to belong to the class HY p;q are given (see also [16,20] and references therein). Let's state one sufficient condition for the validity of the inequality (5.1).
Let f ∈ L 1 (R) L p;u (R). Consider the partial integrals S ν ( f ; x) of the function f (·). Let ν ∈ (0, T ). Following [15], we have Consequently Assume that the weight function ϑ (·) satisfies the following condition: Then the previous relation implies Consider I 2 (T ; ν): We have Assume that the pair (u; ϑ) belongs to the class HY p;q , i.e. (u; ϑ) ∈ HY p;q . We have It's absolutely obvious that As a result, we obtain from these relations that Assume that the measure ϑ (·) additionally satisfies the condition Then it is clear that So the following main theorem is proved.
Introduce the following definition.
Definition 5. 7 We will say that the Fourier transform of the function f (·) at the point x ∈ R is H q (ϑ)summable to f (x) if the relation (5.2) holds.

(5.3)
Let's verify that the conditions of Theorem 5.6 are fulfilled. It is not difficult to see that the condition γ ) is true for the weight function ϑ(·). For γ = 1 p + 1 q −1, the condition μ) is also true. From the Pitt's theorem it follows that 1; |x| −γ q ∈ HY p;q . So the following corollary is valid. When γ = 0, from this corollary we get the result of [15]: Corollary 5.9 Let f ∈ L 1 (R) L p (R), 1 ≤ p < 2. If x ∈ E ( f ), then the Fourier transform of the function f (·) at the point x is H q -summable to f (x), 1 p + 1 q = 1. Remark 5.10 In work [18] it is considered some questions of convergence of a sequences in a strong and statistical sense. We consider so called μ-strong Cesaro summability of local integrable functions (with Borel measure).

Remark 5.11
As respected Reviewer noted, in particular, if we take the atomic measure μ we can get appropriate results can be obtained for sequences.