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 200 40E05 Ümit Totur Departmet of Mathematics Ada Mederes Uiversity Aydi 0900, Turkey E-mail: utotur@adu.edu.tr; utotur@yahoo.com İbrahim Çaak Departmet of Mathematics Ege Uiversity Izmir 3500, Turkey E-mail: ibrahim.caak@ege.edu.tr; ibrahimcaak@yahoo.com
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
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 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.
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 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
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 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
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 0. 3.6 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.
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
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 0. 3.9 σ,pω m u σ 2,pω m u = o. 3.0. 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..
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. 62 20, o. 6, 2609 265. 3. Totur, Ü.; Çaak, İ. Some geeral Tauberia coditios for the weighted mea summability method, Comput. Math. Appl. 63 202, o. 5, 999 006. 4. Schmidt, R. Über divergete Folge ud lieare Mittelbilduge Germa, Math. Z. 22 925, o., 89 52. 5. Çaak, İ.; Totur, Ü. Exteded Tauberia theorem for the weighted mea method of summability. Ukraiia Math. J. 65 203, o. 7, 032 04. 6. Karamata, J. Sur certais Tauberia theorems de M. M. Hardy et Littlewood, Mathematica, 3 930, 33 48. 7. Çaak, İ.; Totur, Ü. Some Tauberia coditios for Cesàro summability method, Math. Slovaca 62 202, o. 2, 27 280.