Tauberian theorems for the product of Borel and Hölder summability methods

Transcription

1 A. Ştiiţ. Uiv. Al. I. Cuza Iaşi. Mat. (N.S.) Tomul LXIII, 2017, f. 1 Tauberia theorems for the product of Borel ad Hölder summability methods İbrahim Çaak Received: Received: 17.X.2012 / Revised: 5.XI.2012 / Accepted: 9.XI.2012 Abstract I this paper we prove some Tauberia theorems for the product of Borel ad Hölder summability methods which improve the classical Tauberia theorems for the Borel summability method. Keywords Borel method Hölder method Tauberia theorems Mathematics Subect Classificatio (2010) 40E05 40G05 40G10 1 Itroductio A umber of authors icludig Lord [13], Hardy ad Littlewood [9, 8], Jakimovski [1], Raagopal [15], Zeller ad Beekma [18], Parameswara [14], Kwee [12], ad Borwei ad Markovich [2] obtaied some results related to the Borel summability method. I [3] Çaak ad Totur have ivestigated coditios uder which Borel summability implies covergece (see Sectio 2 below for the defiitio ad otatios). I this paper our obect is to prove some ew Tauberia theorems for the product of Borel ad Hölder summability methods which improve classical Tauberia theorems of Hardy ad Littlewood [9]. Our results are based o the followig theorems. Theorem 1.1 ([7]) If s s (B) ad s = o( ) ( ), the (s ) coverges to s. Theorem 1.2 ([7]) If s s (B) ad s = O( ) ( ), the (s ) coverges to s. The geeralizatio of Theorem 1.1 above is give by Hardy ad Littlewood [10]. For aother proof, we refer the reader to Hardy ad Littlewood [8]. İbrahim Çaak Ege Uiversity, Departmet of Mathematics, İzmir, Turkey ibrahimcaak@yahoo.com

2 2 İbrahim Çaak 2 Defiitio ad otatios A sequece (s ) is said to be summable by Borel s method to s ad we write s s (B) if the correspodig series =0 s! x coverges for all x ad e x =0 s! x s, x. Borel summability method is regular; that is, if s s as, the s s (B). The Hölder summability method was itroduced by Hölder [11] as a geeralizatio of the limitatio method of the arithmetical averages. For a sequece (s ), we defie σ (1) = 1 +1 s for all 0. Repeatig this averagig process, we defie σ (k) σ(k). A sequece (s ) is said to be s for all k 0 by σ (0) = s ad σ (k+1) = 1 +1 summable by the Hölder s method (H, k) to s ad we write s s (H, k) if σ (k) as. Note that (H, 0) summability of a sequece meas that it coverges i the ordiary sese ad (H, 1) method is the limitatio method of the arithmetical averages. The Hölder method (H, k) are regular for ay k. It is well kow that if s s (H, k), where k 0, ad k > k, the s s (H, k ) (see [7]). If (σ (k) ) is Borel summable to s, we say that (s ) is (B)(H, k) summable to s ad we write s s (B)(H, k). We use the symbols s = o(1) ( ) ad s = O(1) ( ) to mea that s 0 as ad that (s ) is bouded for large eough, respectively. The backward differece of a sequece (s ) is defied for all 0 by s 0 = s 0 ad s +1 = s +1 s. For ay k 0, the idetity where = σ (k) =, (2.1) V (k 1), k 1 s, k = 0, is frequetly used i the proofs of our results. Sice σ (k+1) ca be expressed i the form we may rewrite (2.1) as σ (k) = s 0 + = =1, (2.2) + =1 + s 0. A sequece (s ) is said to be slowly oscillatig (see [5,6]) if lim λ 1 + lim sup max +1 k [λ] s k s = 0, where [λ] deotes the iteger part of λ. A sequece (s ) is said to be moderately oscillatig (see [5,6]) if for λ > 1, lim sup max +1 k [λ] s k s <. It ca be easily see that if (s ) is moderately oscillatig, the V (0) = O(1) ( ) ad (σ (1) ) is slowly oscillatig (see Dik [5]).

3 Tauberia theorems 3 We ow recall what we mea by Tauberia coditios ad Tauberia theorems. If a sequece is coverget, the it is summable by ay regular summability method. However, the coverse is ot true i geeral. The coverse may be valid uder some coditios which are called Tauberia coditios. Ay theorem which states that covergece of a sequece follows from a summability method ad some Tauberia coditio(s) is called a Tauberia theorem. 3 Lemmas We eed the followig lemmas required i the proof of our theorems i the ext sectio. Lemma 3.1 For ay k 0, =. Proof. Take the backward differece of (2.2) ad the multiply by. Lemma 3.2 For ay k 0, Proof. We have for all k. V (k+1) V (k+1) = = ( V (k+1). (3.1) σ (k+2) ) = V (k+1) (3.2) Lemma 3.3 ([4]) If s s (H, 1) ad (s ) is slowly oscillatig, the (s ) coverges to s. Lemma 3.4 ([16]) If s s (B), the s s (B)(H, 1). Lemma 3.5 ([5,17]) If (s ) is slowly oscillatig, the (V (0) ) is bouded ad slowly oscillatig. Proof. Let (s ) be slowly oscillatig. For λ > 1, we defie k w (s, λ) = max +1 k [λ] s =+1 ad rewrite the fiite sum k=1 k s k as the series Hece (3.3) k s k. (3.4) 2 +1 k< 2 k s k k s k k= k< 2 w [ 2 +1 ] (s, λ) 2 C = 2C 1 = C,

4 4 İbrahim Çaak where C > 0. Cosequetly, we have V (0) = O(1) ( ). Therefore (σ (1) ) = (s 0 + V (1) k k=1 k ) is slowly oscillatig. Thus (V (0) ) is slowly oscillatig. As a cosequece of Lemma 3.5, we ote that if a sequece (s ) is slowly oscillatig, the (V (k) ) is bouded ad therefore (σ (k) ) is slowly oscillatig for all k. 4 Mai results We state ad prove the followig Tauberia theorems for (B)(H, k) summability method. Theorem 4.1 If s s (B)(H, k + 1) for some k 0 ad the s s (H, k + 1). Proof. By hypothesis we have By (4.1) ad Lemma 3.1 we have = O( ) ( ), (4.1) s (B). (4.2) = = O( ) ( ). (4.3) We obtai s from Theorem 1.2, which meas that s s(h, k + 1). From Theorem 4.1 we deduce the followig corollary. Corollary 4.2 If (s ) s (B)(H, k+1) for some k 0 ad (s ) is slowly oscillatig, the (s ) coverges to s. Proof. Slow oscillatio of (s ) implies both V (k) = O(1) ( ) ad slow oscillatio of (σ (k) ) for all k 1 as a cosequece of Lemma 3.5. Hece, we have V (k) = σ (k+1) = O( ) ( ). Sice (s ) is (H, k + 1) summable to s, σ (k+1) s by Theorem 1.2. It follows by the slow oscillatio of (σ (k) ) that σ (k) s. Cotiuig i this vei, we obtai σ (1) s. By Lemma 3.3, (s ) coverges to s. Remark 4.1 If we replace slow oscillatio of (s ) by moderate oscillatio of (s ) i Corollary 4.2, we recover covergece of (σ (1) ) from (B)(H, k+1) summability of (s ). Parallel with the Hardy ad Littlewood Tauberia coditio s = O( ) ( ) for Borel summability method, the coditio V (k) = O( ) ( ) suffices to recover (H, k) summability of (s ) from its (B)(H, k) summability for some k 0. Theorem 4.3 If s s (B)(H, k) for some k 0 ad the (s ) is (H, k) summable to s. = O( ) ( ), (4.4)

5 Tauberia theorems 5 Proof. By hypothesis, σ (k) It follows by (2.1) that s (B). By Lemma 3.4, we have σ (k) s (B). (4.5) = 0 (B). (4.6) If we apply Theorem 1.2 to (V (k) ), we coclude from (4.4) that By Lemma 3.1, we have = o(1) ( ). (4.7) It follows from (4.8) that If we apply Theorem 1.1 to (σ (k+1) ), we coclude that = = o(1) ( ). (4.8) = o( ) ( ). (4.9) s. (4.10) Substitutig (4.7) ad (4.10) ito (2.1), we obtai σ (k) s which meas that s s(h, k). If coditio (4.4) is replaced by the coditio V (k+1) = o( ) ( ), we recover (H, k) summability of (s ) from its (B)(H, k + 1) summability for some k 0. Theorem 4.4 If s s (B)(H, k + 1) for some k 0 ad the s s (H, k). V (k+1) = o(1) ( ), (4.11) Proof. By hypothesis, Replacig k by k + 1 i (2.1), we obtai It follows from (4.11) that s (B). By Lemma 3.4, we have σ (k+2) s (B). (4.12) σ (k+2) If we apply Theorem 1.1 to (V (k+1) ), we obtai = V (k+1) 0 (B). (4.13) V (k+1) = o( ) ( ). (4.14) V (k+1) = o(1) ( ). (4.15)

6 6 İbrahim Çaak Replacig k by k + 1 i Lemma 3.1, we have It follows from (4.16) that V (k+1) = σ (k+2) = o(1) ( ). (4.16) By (4.12) ad (4.17), we get σ (k+2) = o( ) ( ). (4.17) from Theorem 1.1. Replacig k by k + 1 i (2.1), we have σ (k+2) s (4.18) By (4.15), (4.18) ad (4.19), we obtai σ (k+2) = V (k+1). (4.19) By Lemma 3.2 we have s. (4.20) V (k+1) By (4.11), (4.15) ad from idetity (4.21), we have = V (k+1). (4.21) = o(1) ( ). (4.22) From (4.20) ad (4.22), we obtai σ (k) s by (2.1), which meas that s s(h, k). Also we have Theorem 4.4 whe coditio (4.11) is replaced by the coditio = o(1) ( ) for some k 0. Corollary 4.5 If s s (B)(H, k + 1) for some k 0 ad the s s (H, k). = o(1) ( ), (4.23) Proof. The proof easily follows from (3.1) ad the regularity of the Hölder s method of summability.

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