A. Ştiiţ. Uiv. Al. I. Cuza Iaşi. Mat. N.S. Tomul LXIII, 207, f. Some Tauberia theorems for weighted meas of bouded double sequeces Cemal Bele Received: 22.XII.202 / Revised: 24.VII.203/ Accepted: 3.VII.203 Abstract I this paper we defie double weighted geerator sequece ad prove some Tauberia theorems uder which the covergece of bouded double sequeces follows from its summability by weighted meas. Keywords Tauberia theorem summability weighted mea method Mathematics Subject Classificatio 200 40C05 40E05 Itroductio Recetly, the cocept of weighted geerator sequece of a sigle sequece, which is the differece betwee the sequece ad its weighted mea, has bee itroduced by Çaak ad Totur [6]. They proved that certai coditios i terms of weighted geerator sequeces are Tauberia coditios for the weighted mea method see also [4]. Tauberia theorems for double weighted mea summability method have bee examied by several authors such as Baro ad Stadtmüller [2], Che ad Hsu [3], Che ad Chag [4,5], Móricz ad Stadtmüller [], Stadtmüller [3]. The purpose of this paper is to itroduce the double aalogue of weighted geerator sequece ad show that some coditios based o the double weighted geerator sequece are Tauberia coditios that Prigsheim s covergece of a bouded complex real double sequece follows from its weighted summability. Let p = p j, q = q k be two sequeces of oegative umbers such that p 0 > 0, q 0 > 0 ad P m := m j=0 p j as m, Q := k=0 q k as. The weighted meas σm αβ of a give double sequece u jk are defied by σ m:= P m Q Cemal Bele Ordu Uiversity, Faculty of Educatio, 52200, Ordu, Turkey E-mail: cbele52@gmail.com p j q k u jk, σ 0 m := P m j=0 p j u j, σ 0 m := Q k=0 q k u mk,
2 Cemal Bele where m, 0. We say that u jk is N, p, q; α, β summable to L if lim m, σ αβ m = L, where α, β =,,, 0 or 0,. Here we mea the covergece i the sese of Prigsheim, i.e., m ad ted to ifiity idepedetly of each other. Uder the coditios P m m ad Q, we have by a theorem of Kojima-Robiso see, e.g. [7] that for bouded double sequeces the correspodig weighted mea method is bouded regular or RH-regular. Let SV A be the set of all oegative sequeces such that p 0 >0 ad lim if P m P m > 0, for all λ > 0 with λ, where := [λm] deotes the itegral part of λm. Lemma. [3] p SV A is equivalet to ay of the followig coditios: lim if m m > λ >, lim if P m m P m > 0 < λ <, P m < 0 < λ <, < λ >. P m m We shall write throughout for simplicity i otatio for all j, k that 0 u jk = u jk u j,k, 0 u jk = u jk u j,k, ad u jk = u jk u j,k u j,k u j,k. Now cosider the differece u m σm. 0 From the equatio m p j u j = P j P j u j = P j u j, P j u j, j=0 = j=0 j= P j u j, u j, P m u m, j= where P = Q = 0, we have Similarly we have u m σ 0 m = P m u m σ 0 m = Q j= k= j=0 P j 0 u j =: V 00 m 0 u. Q k 0 u mk =: V 00 m 0 u. The sequeces V m 00 0 u ad V m 00 0 u are called weighted geerator sequece of u m i the sese,0 ad 0,, respectively. By applyig the Abel geeralized trasformatio for double sequeces to p j q k u jk see [], we obtai p j q k u jk = P m Q k 0 u mk Q k= j= k= m j= P j 0 u j P j Q k u jk P m Q u m.
Some Tauberia theorems 3 The multiplyig each side of the last equality by /P m Q, we have the double Kroecker idetity give by u m σm =: Vm 0 u where Vm 0 u = Vm 00 0 u Vm 00 0 u P m Q j= k= P j Q k u jk. The sequece Vm 0 u is called weighted geerator sequece of u m i the sese,. For each v 0, we defie σm v ad Vm v by σ v m := σ v m u = p j q k σjk v u, v P m Q u m, v = 0 ad V v m := V v m u = p j q k Vjk v u, v P m Q Vm 0 u, v = 0 respectively. Throughout the paper, we will use the otatio σm istead of σm. The followig lemma gives us two differet represetatios of the differece u m σm ad it ca be easily proved by Lemma 2 of [0] with suitable modificatios. Lemma.2 i For λ >, > m ad λ > u m σm = σ λ P Q m, σ λ m σ m, Q λ Q σ Q λ P m Q λ Q P m Q λ Q ii For 0 < λ <, < m ad λ <, m, σ,λ σ, σ m,λ σ m λ j=m k= p j q k u jk u m. u m σm = σ m σ Q λ λ P m P m, σ m σ λm Q m Q λ Q λ P m Q m Q λ P m Q m Q λ m,λ σ m σλ m, σ m,λ σλ m,λ j= k=λ p j q k u m u jk.
4 Cemal Bele 2 Mai results I this sectio we state ad prove Tauberia theorems for N, p, q;, method. Theorem 2. Let p, q SV A ad p m P m = O m ad q = O. 2. Q If a bouded double sequece of real umbers u m is N, p, q;, summable to L ad P m 0 Vm 0 u H ad Q 0 Vm 0 u H, 2.2 p m q for some H 0, the u m is coverget to L. Proof. Assume that u m is N, p, q;, summable to L ad coditios 2.-2.2 hold. Sice σm is coverget to L ad the weighted mea method is bouded regular, we obtai that σm 2 is coverget to L. It follows from the double weighted Kroecker idetity that Vm u is coverget to zero. If we replace u m by Vm 0 u i Lemma.2 i, we have Vm 0 Vm = P V λ P m, Vm Q λ Vm,λ m Q λ Q Q λ Vλ P m Q λ Q m,λ Vλ V m, m,λ Vm 2.3 P m Q λ Q Sice p SV A we have by Lemma. that Hece m P m = λ j=m k= lim if m p j q k Vjk 0 Vm 0. <. Pm P λm Vλ m, P V m, m = 0. m Similar coclusios ca be obtaied for the secod ad third terms i the right-had side of 2.3. Sice V 0 jk V 0 m = j µ=m H 0 V 0 µk j µ=m p µ P µ k l= k 0 V 0 ml q l Q l l=,
we have P m Q λ Q H log m log λ λ j=m k=, Some Tauberia theorems 5 p j q k Vjk 0 0 for some H > 0. Hece takig m, ad the limit as λ of both sides of 2.3, we obtai that Vm 0 0. 2.4 m, From Lemma.2 ii, we have Vm 0 V m = P Vm V λ P m P m, Q λ λm Q Q λ Q λ P m Q Q λ P m Q Q λ V m V, j= k=λ V m Vm,λ V m,λ Vλ m,λ p j q k V 0 m V 0 jk. 2.5 Agai by usig Lemma., it is obvious that the first, secod ad third terms i the right-had side of 2.5 vaishes uder the operator lim if. I additio, sice m, we have V 0 m V 0 jk = µ=j H P m Q Q λ 0 V 0 µk j µ=m H log m log λ p µ P µ j= k=λ, l=k 0 V 0 ml q l Q l l=k p j q k Vm 0 Vjk 0 for some H > 0. Hece takig lim if m, ad the limit as λ of both sides of 2.5, we obtai that lim if V 0 m, m 0. 2.6 From 2.4 ad 2.6 we have lim m, Vm 0 = lim m, Vm = 0. Hece u m is coverget to L by the double weighted Kroecker idetity.,
6 Cemal Bele Corollary 2.2 Let p, q SV A ad pm P m = O m ad q Q = O. If a bouded double sequece of complex umbers u m is N, p, q;, summable to L ad P m p m 0 Vm 0 u = O ad Q q 0 Vm 0 u = O the u m is coverget to L. Followig Schmidt [2] ad also Móricz [9], we say that a double sequece u jk is slowly decreasig with respect to the first idex if if mi u j u m 0 or 2.7 λ> m<j if m, mi u m u j 0. 2.8 <j m 0<λ< m, Note that coditios 2.7 ad 2.8 are equivalet see [5, 0]. Aalogously we say that x jk is slowly decreasig with respect to the secod idex if if λ> mi u mk u m 0. We also say that u jk is slowly decreasig i m, the strog sese with respect to the first idex if if λ> mi u jk u mk 0 m, m<j ad u jk is slowly decreasig i the strog sese with respect to the secod idex if if λ> m, mi m<j λm u jk u j 0. I these defiitios we ca replace if λ> ad if 0<λ< by lim λ ad lim λ, respectively see [5,0]. For λ > we have P m Q λ Q mi m<j λ j=m k= Vjk 0 Vmk 0 p j q k Vjk 0 0 mi Vmk 0 0 Hece, if Vm 0 u is slowly decreasig with respect to the secod idex ad slowly decreasig i the strog sese with respect to the first idex, the 2.4 holds. Aalogously 2.6 holds. Hece we have the followig result which ca be proved by the same idea i proof of Theorem 2.. Theorem 2.3 Let p, q SV A. If a bouded sequece of real umbers u m is N, p, q;, summable to L ad Vjk 0 u is slowly decreasig with respect to both idices ad, i additio, slowly decreasig i the strog sese with respect to oe of idices, the u m is coverget to L. Followig Hardy [8] ad also Móricz [9], we say that a double sequece u jk of complex umbers is slowly oscillatig with respect to the first idex if lim u j u m = 0 ad u jk is slowly oscillatig i the strog sese λ m, m<j with respect to the first idex if lim λ u jk u mk = 0. m, m<j.
Some Tauberia theorems 7 Slowly oscillatig property with respect to the secod idex ca be defied aalogous to that of slowly decreasig property. Sice P m Q λ Q m<j V 0 λ j=m k= p j q k Vjk 0 0 V jk Vmk 0 0 mk 0, accordig to the proof of Theorem 2. we have lim λ V m, m<j 0 jk Vmk 0 lim λ m, V 0 m, V 0 m V m mk 0 if p, q SV A ad if the bouded sequece of complex umbers u m is N, p, q;, summable to some L. Also, if we assume that Vjk 0 u is slowly oscillatig with respect to both idices ad, i additio, slowly oscillatig i the strog sese with respect to oe of idices, we obtai from the last iequality that Vm 0 u = o. Hece we proved the followig result: Theorem 2.4 Let p, q SV A. If a bouded sequece of complex umbers u m is N, p, q;, summable to L ad Vjk 0 u is slowly oscillatig with respect to both idices ad, i additio, slowly oscillatig i the strog sese with respect to oe of idices, the u m is coverget to L. Cosider the followig two-sided Ladau s coditios for the complex case: j Vj 0 Vj, 0 H j, > N 2.9 k Vmk 0 Vm,k 0 H j, > N 2.0 where N > 0 ad H are suitable costats. For λ > ad m, > N, we have m<j V 0 jk Vmk 0 j µ=m m<j H µ H log m. { j µ=m µ sup m<µ j µ V 0 µk Vµ,k 0 } This idicates that if 2.9 holds the V 0 jk u is slowly oscillatig i the strog sese with respect to the first idex. Similarly 2.0 implies slow oscillatio property with respect to the secod idex. Hece 2.9 ad 2.0 are Tauberia coditios for covergece followed by N, p, q;, summability.
8 Cemal Bele Remark 2. Aalogously to N, p, q;, summability oe ca obtai Tauberia theorems for N, p, q;, 0 ad N, p, q; 0, summability of double sequeces. I these cases, our Tauberia coditios are replaced either by a Schmidt type slow decrease/ oscillatio coditio or a Ladau type oe/two sided boudedess coditio for the weighted geerator sequeces i the seses, 0 ad 0,, respectively. Remark 2.2 I case of p j = ad q k =, for all j, k, our results are also valid for Cesàro summability of double sequeces. Ackowledgemets The author would like to thak the referee for the detailed list of correctios, suggestios to the paper, ad all his/her efforts i order to improve the paper. Refereces. Altay, B.; Başar, F. Some ew spaces of double sequeces, J. Math. Aal. Appl., 309 2005, 70 90. 2. Baro, S.; Stadtmüller, U. Tauberia theorems for power series methods applied to double sequeces, J. Math. Aal. Appl., 2 997, 574 589. 3. Che, C.-P.; Hsu, J.-M. Tauberia theorems for weighted meas of double sequeces, Aal. Math., 26 2000, 243 262. 4. Che, C.-P.; Chag, C.-T. Tauberia theorems i the statistical sese for the weighted meas of double sequeces, Taiwaese J. Math., 2007, 327 342. 5. Che, C.-P.; Chag, C.-T. Tauberia coditios uder which the origial covergece of double sequeces follows from the statistical covergece of their weighted meas, J. Math. Aal. Appl., 332 2007, 242 248. 6. Çaak, İ.; Totur, Ü. Some Tauberia theorems for the weighted mea methods of summability, Comput. Math. Appl., 62 20, 2609 265. 7. Hamilto, H.J. Trasformatios of multiple sequeces, Duke Math. J., 2 936, 29 60. 8. Hardy, G.H. Theorems relatig to the summability ad slowly osciilatig series, Proc. Lodo Math. Soc., 8 90, 30 320. 9. Móricz, F. Tauberia theorems for Cesàro summable double sequeces, Studia Math., 0 994, 83 96. 0. Móricz, F. Tauberia theorems for double sequeces that are statistically summable C,,, J. Math. Aal. Appl., 286 2003, 340 350.. Móricz, F.; Stadtmüller, U. Summability of double sequeces by weighted mea methods ad Tauberia coditios for covergece i Prigsheim s sese, It. J. Math. Math. Sci., 2004, 65 68, 3499 35. 2. Schmidt, R. Über divergete Folge ud lieare Mittelbilduge, Math. Z., 22 925, 89 52. 3. Stadtmüller, U. Tauberia theorems for weighted meas of double sequeces, Aal. Math., 25 999, 57 68. 4. Totur, Ü.; Çaak, İ. Some geeral Tauberia coditios for the weighted mea summability method, Comput. Math. Appl., 63 202, 999 006.