ON ABEL CONVERGENT SERIES OF FUNCTIONS

In this paper, we are concerned with Abel uniform convergence and Abel point-wise convergence of series of real functions where a series of functions fn is called Abel uniformly convergent to a function f if for each ε > 0 there is a δ > 0 such that |fx t − f t | < ε For 1 − δ < x < 1 and ∀ t ∈ X, and a series of functions fn is called Abel point-wise convergent to f if for each t ∈ X and ∀ε > 0 there is a δ(ε, t) such that for 1 − δ < x < 1 |fx t − f t | < ε . 2010 Mathematics Subject Classifİcation: 40A30.


INTRODUCTION
Firstly, we give some notations and definitions in the following. Throughout this paper, N will denote the set of all positive integers. We will use boldface p, r, w, ... for sequences p =( ), r = ( ), w = ( ),... of terms in R, the set of all real numbers. Also, s and c will denote the set of all sequences of points in R and the set of all convergent sequences of points in R, respectively.
A series of real numbers is called Abel convergent series (See [1,3]), (or Abel summable) to ℓ if for 0 ≤ < 1 the series ∞ =0 is convergent and In this case we write Abel-∞ = = ℓ. Abel proved that if lim →∞ = ℓ =0 , then Abel- (Abel), i.e. every convergent series is Abel summable . As we know the converse is false in general, e.g Abel-

RESULTS
We are concerned with Abel convergence of sequences of functions defined on a subset of the set of real numbers. Particularly, we introduce the concepts of Abel uniform convergence and Abel point-wise convergence of series of real functions and observe that Abel uniform convergence inherits the basic properties of uniform convergence.
Let (f n ) be a sequences of real functions on and for all

Definition 2.1 A series of functions
called Abel point-wise convergent to a function if for each ∈ and ∀ > 0 there is a ( , ) such that for 1 − < < 1 In this case we write It is easy to see that any point-wise convergent sequence is also Abel point-wise convergent. But the converse is not always true as being seen in the following example. In this case we write The sequence is equicontinuous if for every > 0 and every ∈ , there exists a > 0 , such that for all and all * ∈ with * − < we have The next result is a Abel analogue of a well-known result. Proof. Let 0 be an arbitrary point of . By hypothesis ⇒ (Abel) on . Then, for every > 0, there is a Now for all 0 < < 1 , for = { 1 , 2 } and for all ∈ for which | − 0 | < , we have Since 0 ∈ is arbitrary, is continuous on .
The next example shows that neither of the converse of Theorem 2.1 is true.

Example 2.2 Define
also is a monotonic decreasing sequence on . Since continuous functions on set compact , it is bounded on . As all a series of functions is bound and monotonic decreasing, it is pointwise convergence for all a ∈ . Since is Abel pointwise to zero for all a ∈ , it find pointwise convergece to zero for all a ∈ . Hence for every > 0 and each ∈ there exists a number ≔ ( , ) ∈ such that 0 ≤ < 2 for all ≥ .
Since is continuous a ∈ for every > 0, there is an open set ( ) which contains such that | ℓ − ( )| < 2 for all ℓ ∈ ( ). Hence for given > 0, by monotonicity we have and it is compact set, by the the Heine Borel theorem it has a finite open covering as . Then 0 ≤ ℓ < for every ∈ and for all ≥ . So ⇒ (Abel) on .
Using Abel uniform convergence, we can also get some applications. We merely state the following theorems and omit the proofs.

FUNCTIONS SERİES THAT PRESERVE ABEL CONVERGENCE
Recall that a function sequence ( ) is called convergence-preserving (or conservative) on ⊂ if the transformed sequence ( ( )) converges for each convergent sequence p = ( ) from (see [4]). In this section, analogously, we describe the function sequences which preserve the Abel convergence of sequences. Our arguments also give a sequential characterization of the continuity of Abel limit functions of Abel uniformly convergent function series. First we introduce the following definition. Hence, if series of functions is conservative on , then series of functions is Abel conservative on . But the following example shows that the converse of this result is not true. The next well-known theorem plays an importent role in the proof of Theorem 3.2 . J a n u a r y 0 5 , 2 0 1 6 is not Abel point-wise convergent for all ∈ (which contradicts the hypothesis). Now we are ready to prove the following theorem.
Now, by the "diagonal process" as in [5] and [6] and hence from (3.1) one obtains So we find that Hence series of nonnegative functions ∞ =0 is not Abel convergent since the series of functions J a n u a r y 0 5 , 2 0 1 6 = , say. By the continuity of , So there is an index 0 such that − < 0 , ≥ 0 . For the same ′s, we have Hence a series of nonnegative functions is not Abel convergent, which contradicts the hypothesis. Hence taking This shows that → 0 (Abel), whence the proof follows. Theorem 3. 2 contains the following necessary and sufficient condition for the continuity of Abel limit functions of function series that converge Abel uniformly on a closed interval.