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1 ABSTRACT ON ABEL CONVERGENT SERIES OF FUNCTIONS ERDAL GÜL AND MEHMET ALBAYRAK Yildiz Techical Uiversity, Departmet of Mathematics, Eseler, Istabul I this paper, we are cocered with Abel uiform covergece ad Abel poit-wise covergece of series of real fuctios where a series of fuctios f is called Abel uiformly coverget to a fuctio f if for each ε > 0 there is a δ > 0 such that f x t f t < ε For 1 δ < x < 1 ad t X, ad a series of fuctios f is called Abel poit-wise coverget to f if for each t X ad ε > 0 there is a δ(ε, t) such that for 1 δ < x < 1 f x t f t < ε Mathematics Subject Classifİcatio: 40A30. Key Words: Poitwise covergece; Uiform covergece; Abel covergece Coucil for Iovative Research Peer Review Research Publishig System Joural: JOURNAL OF ADVANCES IN MATHEMATICS Vol.11, No.9 editorjam@gmail.com 5639 P a g e J a u a r y 0 5,

2 1 INTRODUCTION Firstly, we give some otatios ad defiitios i the followig. Throughout this paper, N will deote the set of all positive itegers. We will use boldface p, r, w,... for sequeces p =(p ), r = (r ), w = (w ),... of terms i R, the set of all real umbers. Also, s ad c will deote the set of all sequeces of poits i R ad the set of all coverget sequeces of poits i R, respectively. A sequeces (p ) of real umbers is called Abel coverget (or Abel summable), (See [1,3]), to l if for 0 x < 1 the series k=0 p k x k is coverget ad Lim 1 x k=0 p k x k = l Abel proved that if lim p = l, the Abel lim p = l (Abel). A series the series p of real umbers is called Abel coverget series (See [1,3]), (or Abel summable) to l if for 0 x < 1 p k I this case we write Abel- k=0 x k is coverget ad lim (1 x) S k k=0 x k = l, where S = k=0 p k p = l. Abel proved that if lim = l k=0, the Abel- p = l (Abel), i.e. every coverget series is Abel summable. As we kow the coverse is false i geeral, e.g Abel- ( 1) = 1 2 (Abel), but ( 1) RESULTS We are cocered with Abel covergece of sequeces of fuctios defied o a subset X of the set of real umbers. Particularly, we itroduce the cocepts of Abel uiform covergece ad Abel poit-wise covergece of series of real fuctios ad observe that Abel uiform covergece iherits the basic properties of uiform covergece. Let (f ) be a sequeces of real fuctios o X ad for all t X let f x t = (1 x) S (t) x, where S t = k=0 f k t. Defiitio 2.1 A series of fuctios ε > 0 there is a δ(ε, t) such that for 1 δ < x < 1 f called Abel poit-wise coverget to a fuctio f if for each t X ad f x t f t < ε. I this case we write f f (Abel) o X. It is easy to see that ay poit-wise coverget sequece is also Abel poit-wise coverget. But the coverse is ot always true as beig see i the followig example. Example 2.1 Defie f : [0,1] R ad The, for every ε > 0, Hece by f t = 1 = S t = 1, N ad odd; 1, N ad eve 0, odd; 1, eve 1 x S t 1 2 lim x 1 (1 x) S x < ε. (t) x = 1 2 So f is Abel poit-wise coverget to 1 2 o [0,1]. But observe that f is ot poit-wise o [0,1] P a g e J a u a r y 0 5,

3 Defiitio 2.2 A series of fuctios a δ > 0 such that for 1 δ < x < 1 ad t X. f is called Abel uiform coverget to a fuctio f if for each ε > 0 there is f x t f t < ε I this case we write f f (Abel) o X. The sequece is equicotiuous if for every ε > 0 ad every x X, there exists a δ > 0, such that for all ad all x X with x x < δ we have f x f x < ε. The ext result is a Abel aalogue of a well-kow result. Theorem 2.1 Let f be equicotiuous o X. If a series of fuctios o X, the f is cotiuous o X. f coverges Abel uiform to a fuctio f Proof. Let t 0 be a arbitrary poit of X. By hypothesis f f (Abel) o X. The, for every ε > 0, there is a δ 1 > 0 such that 1 δ 1 < x < 1 implies f x t f t < ε ad f 3 x t 0 f t 0 < ε for each t X. Sice f is quicotiuous at t 0 X, there is a δ 2 > 0 ad N such that t t 0 < δ 2 implies f k t f k t 0 < ε for each t X, so f x t f x t 0 = 1 x S t x (1 x) =0 S (t 0 ) x 3 = (1 x) (S t S (t 0 ))x 1 x (S t S (t 0 ) x 1 x ε x = ε 3 3 Now for all 0 < x < 1, for δ = mi {δ 1, δ 2 } ad for all t X for which t t 0 < δ, we have f t f(t 0 ) = f t f x t + f x t f x t 0 + f x t 0 f(t 0 ) f t f x t + f x t f x t 0 + f x t 0 f t 0 < ε. Sice t 0 X is arbitrary, f is cotiuous o X. The ext example shows that either of the coverse of Theorem 2.1 is true. Example 2.2 Defie f : [0,1] R by f t = 2 t(1 t) The we have f 0,1 f = 0 (Abel) o [0,1]. Though all f ad f are cotiuous o 0,1, it follows from Defiitio that the Abel poit-wise covergece of (f ) is ot uiform, sice c = max 0 t 1 The followig result is a differet form of Dii s theorem. k=0 f k t f(t) = ad Abel-lim c = 0. Theorem 2.2 Let X be compact subset of R, (f ) be a sequece of cotiuous fuctios o X. Assume that f is +1 cotiuous ad f f (Abel) o X. Also let k=0 f k be mootoic decreasig o X ; k=0 f k (t) k=0 f k (t) 5641 P a g e J a u a r y 0 5,

4 ( = 1,2,3, ) for every t X. The f f (Abel) o X. Proof. Put t = k=0 (f k t f t ). By hypothesis, each is cotiuous ad 0 (Abel) o X, also is a mootoic decreasig sequece o X. Sice cotiuous fuctios 2 o set compact X, it is bouded o X. As all a series of fuctios is boud ad mootoic decreasig, it is poitwise covergece for all a t X. Sice is Abel poitwise to zero for all a t X, it fid poitwise covergece to zero for all a t X. Hece for every ε > 0 ad each t X there exists a umber t (ε, t) N such that 0 t < ε 2 for all t. Sice t is cotiuous a t X for every ε > 0, there is a ope set V(t) which cotais t such that t l t (t) < ε 2 for all l V(t). Hece for give ε > 0, by mootoicity we have for every l V(t) ad for all t. Sice X theorem it has a fiite ope coverig as 0 l t l = t l t t + t t < t l t t + t t < ε V(t) t X ad it is compact set, by the the Heie Borel X V t 1 V t 2 V t m. Now, let N = max t 1, t 2, t 3,, t m. The 0 l < ε for every t X ad for all N. So f f (Abel) o X. Usig Abel uiform covergece, we ca also get some applicatios. We merely state the followig theorems ad omit the proofs. Theorem 2.3 If a series fuctio sequece f coverges Abel uiformly o [a, b] to a fuctio f o [a, b]ad each f is a itegrable o [a, b] the, f is itegrable o [a, b]. Moreover, b Lim f x t dt = f(t) dt a a Theorem 2.4 Suppose that If f f o [a, b] ad f g (Abel) o [a, b], the f f (Abel) o [a, b], where f is differetiable ad f = g. 3 FUNCTIONS SERİES THAT PRESERVE ABEL CONVERGENCE b f is a fuctio series such that each (f ) has a cotiuous derivative o [a, b]. Recall that a fuctio sequece (f ) is called covergece-preservig (or coservative) o X R if the trasformed sequece (f (p )) coverges for each coverget sequece p = (p ) from X (see [4]). I this sectio, aalogously, we describe the fuctio sequeces which preserve the Abel covergece of sequeces. Our argumets also give a sequetial characterizatio of the cotiuity of Abel limit fuctios of Abel uiformly coverget fuctio series. First we itroduce the followig defiitio. Defiitio 3.1 Let X R ad let f be a series of real fuctios, ad f a real fuctio o X. The series of fuctios f is called Abel preservig Abel covergece (or Abel coservative) o X, if it trasforms Abel coverget sequeces to Abel coverget sequeces, i.e. series of fuctios f (p ) is Abel coverget to f(l) wheever (p ) is Abel coverget to l. If series of fuctios f is Abel coservative ad preserves the limits of all Abel coverget sequeces from X, the series of fuctios f is called Abel regular o X. Hece, if series of fuctios f is coservative o X, the series of fuctios f is Abel coservative o X. But the followig example shows that the coverse of this result is ot true. Example 3.1 Let f : 0,1 R defied by f t = 1 =, odd;, eve ad S t = 1, N ad odd; 2, N ad eve 2 Suppose that (w ) is a arbitrary sequece i [0,1] such that lim (1 x) w (t)x = L. The, for every ε > 0, 1 x (S w ( 1 4 ))x < ε. Hece Lim 1 x S w = 1. So f 4 is Abel coservative o [0,1]. But observe that f is ot coservative o [0,1]. The ext well-kow theorem plays a importet role i the proof of Theorem P a g e J a u a r y 0 5,

5 Theorem 3.1 If the series for all t X the f is Abel poitwise coverget to f o X ad f (t) 0 for sufficietly large f coverges to f for all t X. Proof. There exists 0 such that if > 0 the f t > 0 for all t X. Thus the (S ) 0+1 if S is bouded the f t = f(t) for all t X. So for all t X lim x 1 (1 x) f k=0 k(t) x k = k=0 f k (t) is a icreasig sequece If S is ot bouded LimS =, so f t is ot Abel poit-wise coverget for all t X (which cotradicts the hypothesis). Now we are ready to prove the followig theorem. Theorem 3.2 Let (f ) be a sequece of oegative fuctios defied o a closed iterval [a, b] R, a, b > 0. The a series of oegative fuctios f is Abel coservative o [a, b] if ad oly if a series of oegative fuctios f coverges Abel uiformly o [a, b] to a cotiuous fuctio. Proof. Necessity. Assume that a series of oegative fuctios f is Abel coservative o [a, b]. Choose the sequece (r ) = (r, r, ) for each r a, b. Sice A lim(r ) = r, A lims r exists, hece A lims r = f(r) for all r a, b : We claim that f is cotiuous o [a, b]. To prove this we suppose that f is ot cotiuous at a poit p 0 a, b.the there exists a sequece (p k ) i [a, b] such that lim k p k = p 0, but limf p k exists ad limf p k = L f(p 0 ). Sice a series of oegative fuctios f k is Abel poitwise coverget to f o [a, b], we obtai f f (Abel) o [a, b], from Theorem 3.1. Hece we write, for k = 1 lim 1 x =0 (S p 1 for k = 2 lim 1 x =0 (S p 2 for k = 3 lim 1 x =0 (S p for k = j lim 1 x (S p j Now, by the diagoal process as i [5] ad [6] So we have The, 1 x (S p S p x = (S p f p 1 x = 0 lims p 1 = f p 1 f p 2 x = 0 lims p 2 = f p 2 f p 3 x = 0 lims p 3 = f 3 f p j x = 0 lim S p j = f p j. f p x 1 x (S p j j =1 f p j x (S p f p )x = 0 (3.1) f p + f p )x = (S p f p )x + f p x ad hece from (3.1) oe obtais If limf p = L, So we fid that the S p x = lim 1 x f p f p x = L. x S p x = L. (3.2) Hece series of oegative fuctios f p is ot Abel coverget sice the series of fuctios f p has two differet limit value. So, the series of oegative fuctios f p is ot Abel coverget 5643 P a g e J a u a r y 0 5,

6 coverget, which cotradicts the hypothesis. Thus f must be cotiuous o [a, b]. It remais to prove that series of oegative fuctios f coverges Abel uiformly o [a, b] to f. Assume that a series of fuctios f is ot Abel uiformly coverget to f o [a, b]. Hece there exists a umber ε 0 > 0 ad umbers r a, b such that 1 x =0 (S r f r x 2ε 0. We obtai from Theorem 3.1 that S r f r 2ε 0. The bouded sequece r = (r ) cotais a coverget subsequece (r i ), r i=0 i x i = α, say. By the cotiuity of f, limf r i = f α. So there is a idex i 0 such that f r i f α < ε 0, i i 0. For the same i s, we have 1 x (S i r i i=0 f α )x i 1 x i=0 (S i r i f r i )x i 1 x i=0 (f r i f α )x i ε 0. Hece a series of oegative fuctios f i r i is ot Abel coverget, which cotradicts the hypothesis. Thus a series of oegative fuctios f must be Abel uiformly coverget to f o [a, b]. Sufficiecy. Assume that f f (Abel) o a, b ad f is cotiuous. Let p = (p ) be a Abel coverget Sequece i [a, b] with A limp = p 0. Sice Theorem 3.1 ad f f (Abel) o [a, b] ad, we obtai that limp = p 0. Sice limp = p 0 ad f is cotiuous, we obtai that there is A limf(p ) ad let A limf(p ) = f(p 0 ). Let ε > 0 be give. We write 1 x (f p [a, b], we have 1 x (f t f t )x < ε 2 1 x (f p f p 0 )x 1 x (f p for every t a, b. f p )x + 1 x f p 0 )x < ε 2. As f f(abel) o Hece takig t = (p ) we have (f p f p 0 )x < ε. This shows that f p f p 0 (Abel), whece the proof follows. Theorem 3. 2 cotais the followig ecessary ad sufficiet coditio for the cotiuity of Abel limit fuctios of fuctio series that coverge Abel uiformly o a closed iterval. Theorem 3.3 Let f k be a series of oegative fuctios that coverges Abel uiformly o a closed iterval [a, b], a, b > 0 to a fuctio f. The A-limit fuctio f is cotiuous o [a, b] if ad oly if the series of oegative fuctios f k is Abel coservative o [a, b]. Now, we study the Abel regularity of fuctio series. If series of oegative fuctios f k is Abel regular o [a, b], the obviously A-lim f t = t for all t [a, b], a, b > 0. So, takig f t = t i Theorem 3.2, we immediately get the followig result. Theorem 3.4 Let f k be a series of oegative fuctios o [a, b], a, b > 0. The series of oegative fuctios (f k ) is Abel regular o [a, b] if ad oly if series of oegative fuctios f k is Abel uiformly coverget o [a, b] to the fuctio fdefied by f t = t REFERENCES [1] N.H. Abel Resherches sur la srie N.H. Abel, Recherches sur la srie 1 + m 1 1 (1826) m (m 1) x + x 2 +, 1.2 [2] Tauber, A, Ei Satz aus der Theorie der Uedliche Reihe. Moatsh. Phys., VII (1897) [3] Hardy, G. H., Diverget series. Secod Editio, AMS Chelsea Publishig, 396s. USA. J. Fr. Math. [4] Kolk, E. Covergece-preservig fuctio sequeces ad uiform covergece. J. Math. Aal. Appl. 238 (1999), [5] Bartle, R. G. Elemts of Real Aalysis. Joh ad Sos Ic., New York, [6] Duma,O. ad Orha, C., μ statistacally coverget fuctio sequeces, 84 (129) (2004), P a g e J a u a r y 0 5,