Some Tauberian Conditions for the Weighted Mean Method of Summability

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1 A. Ştiiţ. Uiv. Al. I. Cuza Iaşi. Mat. N.S. Tomul LXIII, 207, f. 3 Some Tauberia Coditios for the Weighted Mea Method of Summability Ümit Totur İbrahim Çaak Received: 2.VIII.204 / Accepted: 6.III.205 Abstract Let p = be a seuece of oegative umbers ad P := k=0 as. Let the weighted geeral cotrol modulo of the oscillatory behavior of iteger order m of a seuece u be deoted by ω,p m u. We prove that if the weighted geerator seuece of a seuece u = u of real umbers is summable to a fiite umber by the weighted mea method, σ,pω m u is icreasig, the coditios λ α lim sup P P k= k,p u ω m = o, λ, >, α > ad λ α lim sup P P k= k,p u ω m = o, λ, >, α > are satisfied, ad certai coditios o are hold, the u is slowly oscillatig. Keywords Tauberia theorems Weighted meas Weighted geeral cotrol modulo Slowly decreasig seuece Mathematics Subject Classificatio E05 Ümit Totur Departmet of Mathematics Ada Mederes Uiversity Aydi 0900, Turkey utotur@adu.edu.tr; utotur@yahoo.com İbrahim Çaak Departmet of Mathematics Ege Uiversity Izmir 3500, Turkey ibrahim.caak@ege.edu.tr; ibrahimcaak@yahoo.com

2 2 Ümit Totur, İbrahim Çaak Itroductio Let p = be a seuece of oegative umbers with p 0 > 0 ad P := as. k=0 The -th weighted mea of a seuece u = u is defied [] by σ,pu := P Let u be a seuece of real umbers. If u = v k= u k. k=0 P k v k u 0, for some v = v, we say that the seuece u is regularly weighted geerated by the seuece v ad v is called a weighted geerator of u. For a seuece u, u σ,pu = V,p 0 u,. where V 0,p u = P P k u k. Note that u = u u ad u = 0. k= Sice σ,pu = u 0 k= P k V 0 k,p u = V 0,p u u,. ca be rewritte as k= P k V 0 k,p u u 0..2 Therefore, we say that the seuece V,p 0 u is called a weighted geerator of u. The weighted classical cotrol modulo of the oscillatory behavior of u is deoted by ω,pu 0 = P u. The weighted geeral cotrol modulo of the oscillatory behavior of iteger order m of a seuece u is defied i [2] by For a seuece u, we defie P where P ω m,p u = ω m,p u σ,pω m u. u = m = P u = u, ad 0 P P P P m u, m u = P u. u

3 Some Tauberia Coditios for the Weighted Mea Method of Summability 3 It is proved i [3] that for ay iteger m 2, ω,p m P u = m V,p m u, where V,p m u = σ,pv m 2 u. Throughout this work, the symbol deotes the itegral part of the product λ. A seuece u is called slowly oscillatig [4] if k lim lim sup max λ k u j = 0. If ω 0,pu = O with j= P = O.3 holds for a seuece u, the u is slowly oscillatig. The weighted de la Vallée Poussi meas of u are defied by τ >,,p u = P P for λ > ad sufficietly large ad τ <,,p u = P P k= k= u k, u k, for 0 < λ < ad sufficietly large. A seuece u is said to be summable by the weighted mea method determied by the seuece p, i short, N, p summable to a fiite umber s if lim σ,pu = s..4 If the limit lim u = s.5 exists, the.4 is also exists. The coverse is ot ecessarily true. However,.4 may imply.5 by addig some suitable coditio o the seuece u. Such a coditio is called a Tauberia coditio ad the resultig theorem is called a Tauberia theorem. Hardy [] proved that the coditio ω,pu 0 = O is a Tauberia coditio for the N, p summability method. Çaak ad Totur [2] replaced the Hardy s Tauberia coditio by ω,pu H,.6

4 4 Ümit Totur, İbrahim Çaak for some H > 0, ad showed that if u is N, p summable to s ad the coditio.6 is satisfied with certai coditios o, the u coverges to s. Recetly, Totur ad Çaak [3] have itroduced the coditio ω m,p u = O,.7 where m is some oegative iteger, as a geeral Tauberia coditio for N, p summability method. Istead of recoverig covergece of a seuece from the existece of.4 ad some additioal coditio, we ca obtai more geeral iformatio o the seuece u by replacig the N, p summability of u by the N, p summability of the geerator seuece of u. Çaak ad Totur [5] ivestigated coditios, which are give i terms of the weighted geeral cotrol modulo of the oscillatory behavior of order 2 of u, uder which N, p summability of V,p 0 u implies the slow oscillatio of u. I the light of this iformatio, Çaak ad Totur [5] proved the followig theorem. Theorem. Let the coditio.3 be satisfied. For a real seuece u let there exist a oegative seuece M = M with slow oscilatio of such that ad lim sup lim sup ω 2,pu M, P P P P P P k= Mk k lim sup τ >,,p M = o, λ, lim sup τ <,,p M = o, λ. If V 0,p u is N, p summable to s, the u is slowly oscillatig. I this paper we obtai the slow oscillatio of u from N, p summability of V,p 0 u by addig some coditios o the weighted de la Vallée Poussi meas of the th power, >, of the geeral cotrol modulo of the oscillatory behavior of iteger order m of u. 2 Mai Result The mai theorem of this paper ivolves the cocepts of a regularly varyig seuece of idex α >. Defiitio 2. [6] A positive seuece R is said to be regularly varyig of idex α > if R lim = λ α, λ > 0, α >. 2. R Throughout this sectio m is ay oegative iteger ad is ay positive real umber greater tha.

5 Some Tauberia Coditios for the Weighted Mea Method of Summability 5 Theorem 2.2 Let the coditio.3 be satisfied ad P be regularly varyig seuece of idex α >. For a real seuece u, let σ,pω m u be icreasig ad the coditios λ α lim sup τ >,,p ωm u = o, λ, >, 2.2 ad λ α lim sup τ <,,p ωm u = o, λ, >, 2.3 be hold. If V 0,p u is N, p summable to s, the u is slowly oscillatig. I the complex case, oe of the coditios λ α lim sup τ >,,p ωm u = o, λ, >, 2.4 ad λ α lim sup τ <,,p ωm u = o, λ, >, 2.5 is sufficiet to recover the slow oscillatio of u out of N, p summability of its geerator seuece. Cotrary to the mai theorem, i the complex case we do ot reuire the coditio that σ,pω m u is icreasig. We eed the followig lemmas for the proof of our theorem. Lemma 2.3 [2] Let u be a seuece of real umbers. i For λ > ad sufficietly large, u σ,pu = P σ P P,pu σ,pu P P ii For 0 < λ < ad sufficietly large, u σ,pu = k= u k u. P σ P P,pu σ,p u P P k= u u k.

6 6 Ümit Totur, İbrahim Çaak Lemma 2.4 For each iteger m, m m ω,p m u = j j where m j = m m 2...m j j!. P Proof. We do the proof by iductio. For m =, we have V j,p u, ω,pu = P u V,p 0 u = P V,p 0 u 0 0 = j P V,p j u. j Assume the observatio is true for m = k. That is, assume that k k ω,pu k = j j P V j,p u. 2.6 We must show that the observatio is true for m = k. That is, we must show that ω k,p u = k k j j P V j,p u. By defiitio, By 2.6, ω,p k u = ω,pu k σ,pω k u. k k ω, u = j j k k j j P P V j,p u V,p j u. Lettig j = i i the secod sum. Usig this substitutio k ω, k u = j j k i= i k i P P V j,p u V i,p u. 2.7

7 Some Tauberia Coditios for the Weighted Mea Method of Summability 7 I the secod sum of 2.7, we reame the idex of summatio j, split the first term off i the first sum ad the last term i the secod sum of 2.7, we have k ω,p k u= 0 P V 0 0 Thus, we have k j j k j= k ω, u = 0 0 k j Sice k j P P k,p u k j j j= V j,p u k k k P P k [ k V,p 0 u j j ] P j= V j,p u k k k k j = k j, the last idetity ca be writte k ω, u = 0 0 k k j j j= P P V 0,p u V j,p u P V j,p u V k,p u. V k,p u. k k k k k = j j j= P P V k,p u V j,p u. Thus, we coclude that Lemma 2.4 is true for every positive iteger m. 3 Proof of Theorem 2.2 Sice V 0,p u is N, p summable to s, we have It follows from the idetity V,p u s,. 3. ad 3. that V,p u V 2,p u = P V 2,p u = σ 2,pω u σ 2,pω u = o. 3.2

8 8 Ümit Totur, İbrahim Çaak Hece, we get, by the idetity., for ay iteger m 2. Applyig Lemma 2.3 i to σ,pω m u we have σ 2,pω m u = o, 3.3 σ,pω m u σ 2,pω m u = σ,pω 2 m u P P P σ 2 P P,p ωm u k= k j= σ j,p ωm u 3.4 for λ > ad sufficietly large. For the secod term o the right-had side of 3.4, we have P P k σ j,p ωm u max σ j,p k ωm u k= j= j= j= p j k ω m j u. P j Let > ad p =. Applyig Hölder ieuality to the right-had side of the last ieuality, we obtai P P k= k j= σ j,p ωm u j= pj p P j P P P p = P P P = P P P p j= P P j= ω m j,p u ω m j,p u j= ω m j τ,,p > ω m u. u

9 Coseuetly, we get Some Tauberia Coditios for the Weighted Mea Method of Summability 9 P P k= P P P k j= σ j ω m u τ,,p > ω m u. 3.5 Takig the lim if of both sides of 3.4, we see that the first term o the right-had side vaishes. Therefore we deduce lim if σ,pω m u σ 2 lim if P P P,pω m u τ,,p > ω m u P P lim sup P lim su τ,,p > ω m u by takig 3.5 ito cosideratio. Sice P is regularly varyig of idex α >, we have lim if σ,pω m u σ,pω 2 m u λ α lim su τ,,p > ω m u. It the follows by the coditio 2.2 that lim ifσ,pω m u σ,pω 2 m u Similarly, applyig Lemma 2.3 ii to σ ω m u, we have σ,pω m u σ,pω 2 m u = P σ,pω 2 m u σ 2 P P P P k= j=k,p ωm u σ j,p ωm u 3.7 for 0 < λ < ad sufficietly large.

10 0 Ümit Totur, İbrahim Çaak For the secod term o the right-had side of 3.7, we have P P k= max j=k k j=k σ j,p ωm u σ j,p ωm u j= p j P j ω m j,p u. Let > ad p =. Applyig Hölder ieuality to the right-had side of the last ieuality, we obtai P P k= j=k j= σ j,p ωm u pj p P j P P P p = P P P = P P P p j= j= ω m j,p u ω m j,p u ω m j,p P P u j= τ [,λ],p < ω m u. Coseuetly, we get P P k= j=k P P P σ j,p ωm u τ,,p < ω m u. 3.8

11 Some Tauberia Coditios for the Weighted Mea Method of Summability Takig the lim sup of both sides of 3.7 we see that the first term o the right-had side vaishes. Therefore we deduce lim su σ,pω m u σ 2 lim su,pω m u P P P τ,,p < ω m u P P lim sup P lim su τ[,λ],p < ω m u by takig 3.8 ito cosideratio. Sice P is regularly varyig of idex α >, we have lim su σ,pω m u σ,pω 2 m u λ α lim su τ[,λ],p < ω m u λ α It the follows by the coditio 2.3 that From 3.6 ad 3.9 we have lim supσ,pω m u σ,pω 2 m u σ,pω m u σ 2,pω m u = o Usig 3.3, we obtai By Lemma 2.4, we get σ,pω m u = o. 3. m 2 m 2 σ,pω m u = j j P V V j,p u = o. 3.2 It easily follows from 3.2 ad 3.2 that P,p u = o. From the euality P V,p u = V,p 0 u V,p u, we get V,p 0 u s as by 3.. We coclude by.2 that u is slowly oscillatig. As a corollary, we have the followig Tauberia theorem for N, p summability. Corollary 3. For the real seuece u = u, let σ,pω m u be icreasig ad the coditios 2.2 ad 2.3 be hold. If u is N, p summable to s, the u is coverget to s. Proof. Assume that u is N, p summable to s. It follows by the idetity. that V,p 0 u is N, p summable to 0. By Theorem 2.2, V,p 0 u is coverget to 0. Sice u is N, p summable to s, u is coverget to s by..

12 2 Ümit Totur, İbrahim Çaak Corollary 3.2 [7] For the real seuece u = u, let σ ω m u be icreasig ad the coditios λ α lim sup τ >, ωm u = o, λ, >, α > ad λ α lim sup τ <, ωm u = o, λ, >, α > be hold. If V 0 u is Cesàro summable to s, the u is slowly oscillatig. Proof. Take = for every oegative iteger i Theorem 2.2. Refereces. Hardy, G.H. Diverget series, Oxford, at the Claredo Press 949, xvi396 pp. 2. Çaak, İ.; Totur, Ü. Some Tauberia theorems for the weighted mea methods of summability, Comput. Math. Appl , o. 6, Totur, Ü.; Çaak, İ. Some geeral Tauberia coditios for the weighted mea summability method, Comput. Math. Appl , o. 5, Schmidt, R. Über divergete Folge ud lieare Mittelbilduge Germa, Math. Z , o., Çaak, İ.; Totur, Ü. Exteded Tauberia theorem for the weighted mea method of summability. Ukraiia Math. J , o. 7, Karamata, J. Sur certais Tauberia theorems de M. M. Hardy et Littlewood, Mathematica, 3 930, Çaak, İ.; Totur, Ü. Some Tauberia coditios for Cesàro summability method, Math. Slovaca , o. 2,