TAUBERIAN THEOREMS FOR THE PRODUCT OF ...

˘ ¸ II “AL.I. CUZA” DIN IAS ANALELE S ¸ TIINT ¸ IFICE ALE UNIVERSITAT ¸ I (S.N.) ˘ Tomul ..., ..., f.... MATEMATICA, DOI: 10.2478/aicu-2014-0039

TAUBERIAN THEOREMS FOR THE PRODUCT OF BOREL ¨ AND HOLDER SUMMABILITY METHODS BY

˙ IBRAHIM C ¸ ANAK

Abstract. In this paper we prove some Tauberian theorems for the product of Borel and H¨ older summability methods which improve the classical Tauberian theorems for the Borel summability method. Mathematics Subject Classification 2010: 40E05, 40G05, 40G10. Key words: Borel method, H¨ older method, Tauberian theorems.

1. Introduction A number of authors including Lord [13], Hardy and Littlewood [9, 8], Jakimovski [1], Rajagopal [15], Zeller and Beekmann [18], Parameswaran [14], Kwee [12], and Borwein and Markovich [2] obtained some results related to the Borel summability method. In [3] C ¸ anak and Totur have investigated conditions under which Borel summability implies convergence (see Section 2 below for the definition and notations). In this paper our object is to prove some new Tauberian theorems for the product of Borel and Hölder summability methods which improve classical Tauberian theorems of Hardy and Littlewood [9]. Our results are based on the following theorems. √ Theorem 1.1 ([7]). If sn → s (B) and n∆sn = o( n) (n → ∞), then (sn ) converges to s. √ Theorem 1.2 ([7]). If sn → s (B) and n∆sn = O( n) (n → ∞), then (sn ) converges to s.

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The generalization of Theorem 1.1 above is given by Hardy and Littlewood [10]. For another proof, we refer the reader to Hardy and Littlewood [8]. 2. Definition and notations A sequence (sn ) is said to be summable by P Borel’s method to s and we sn n write sn →Ps (B) if the corresponding series ∞ n=0 n! x converges for all sn n x and e−x ∞ n=0 n! x → s, x → ∞. Borel summability method is regular; that is, if sn → s as n → ∞, then sn → s (B). ¨ lder [11] as a The Hölder summability method was introduced by Ho generalization of the limitation method of the arithmetical averages. For a (1) 1 Pn sequence (sn ), we define σn = n+1 j=0 sj for all n ≥ 0. Repeating this (k)

(0)

(k+1)

averaging process, we define σn for all k ≥ 0 by σn = sn and σn = (k) 1 Pn older’s j=0 σj . A sequence (sn ) is said to be summable by the H¨ n+1 (k)

method (H, k) to s and we write sn → s (H, k) if σn → s as n → ∞. Note that (H, 0) summability of a sequence means that it converges in the ordinary sense and (H, 1) method is the limitation method of the arithmetical averages. The Hölder method (H, k) are regular for any k. It is well known that if sn → s (H, k), where k ≥ 0, and k′ > k, then sn → s (H, k′ ) (see [7]). (k) If (σn ) is Borel summable to s, we say that (sn ) is (B)(H, k) summable to s and we write sn → s (B)(H, k). We use the symbols sn = o(1) (n → ∞) and sn = O(1) (n → ∞) to mean that sn → 0 as n → ∞ and that (sn ) is bounded for large enough n, respectively. The backward difference of a sequence (sn ) is defined for all n ≥ 0 by ∆s0 = s0 and ∆sn+1 = sn+1 − sn . For any k ≥ 0, the identity σn(k) − σn(k+1) = Vn(k) ,

(1) where Vn(k)

 n 1 X (k−1)    Vj ,k ≥ 1   n+1 j=0 = n  1 X   j∆sj , k = 0,   n+1 j=0

is frequently used in the proofs of our results.


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TAUBERIAN THEOREMS (k+1)

Since σn

can be expressed in the form σn(k+1)

(2)

= s0 +

(k) n X Vj j=1

j

,

(k) V (k) (k) P we may rewrite (1) as σn = Vn + nj=1 jj +s0 . A sequence (sn ) is said to be slowly oscillating (see [5, 6]) if limλ→1+ lim supn→∞ maxn+1≤k≤[λn] |sk − sn | = 0, where [λn] denotes the integer part of λn. A sequence (sn ) is said to be moderately oscillating (see [5, 6]) if for λ > 1, lim supn→∞ maxn+1≤k≤[λn] |sk − sn | < ∞.

(0)

It can be easily seen that if (sn ) is moderately oscillating, then Vn = (1) O(1) (n → ∞) and (σn ) is slowly oscillating (see Dik [5]). We now recall what we mean by Tauberian conditions and Tauberian theorems. If a sequence is convergent, then it is summable by any regular summability method. However, the converse is not true in general. The converse may be valid under some conditions which are called Tauberian conditions. Any theorem which states that convergence of a sequence follows from a summability method and some Tauberian condition(s) is called a Tauberian theorem. 3. Lemmas We need the following lemmas required in the proof of our theorems in the next section. (k)

Lemma 3.1. For any k ≥ 0, Vn

(k+1)

= n∆σn

.

Proof. Take the backward difference of (2) and then multiply by n. Lemma 3.2. For any k ≥ 0, n∆Vn(k+1) = Vn(k) − Vn(k+1) .

(3) Proof. We have (4)

Vn(k) − Vn(k+1) = n∆(σn(k+1) − σn(k+2) ) = n∆Vn(k+1)

for all k.



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Lemma 3.3 ([4]). If sn → s (H, 1) and (sn ) is slowly oscillating, then (sn ) converges to s. Lemma 3.4 ([16]). If sn → s (B), then sn → s (B)(H, 1). (0)

Lemma 3.5 ([5, 17]). If (sn ) is slowly oscillating, then (Vn ) is bounded and slowly oscillating. Proof. Let (sn ) be slowly oscillating. For λ > 1, we define k X (5) wn (s, λ) = max ∆sj n+1≤k≤[λn] j=n+1 and rewrite the finite sum

Pn

k=1 k∆sk

∞ X

(6)

j=0

X

as the series k∆sk .

n ≤k< nj 2j+1 2

Hence n ∞ X X k∆sk ≤ k=1

j=0

≤ nC1

X

n ≤k< nj 2j+1 2

  ∞ X n n k∆sk ≤  w[ j+1 ] (s, λ) 2 2j j=0

∞ X 1 = 2nC1 = nC, 2j j=0

(0)

where C > 0. Consequently, we have Vn = O(1) (n → ∞). There(1) P V (1) (0) fore (σn ) = (s0 + nk=1 kk ) is slowly oscillating. Thus (Vn ) is slowly oscillating. As a consequence of Lemma 3.5 we note that if a sequence (sn ) is slowly (k) (k) oscillating, then (Vn ) is bounded and therefore (σn ) is slowly oscillating for all k. 4. Main results We state and prove the following Tauberian theorems for (B)(H, k) summability method.


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TAUBERIAN THEOREMS

Theorem 4.1. If sn → s (B)(H, k + 1) for some k ≥ 0 and √ (7) Vn(k) = O( n) (n → ∞), then sn → s (H, k + 1). Proof. By hypothesis we have σn(k+1) → s (B).

(8)

By (7) and Lemma 3.1 we have √ Vn(k) = n∆σn(k+1) = O( n) (n → ∞).

(9) (k+1)

We obtain σn →s from Theorem 1.2, which means that sn →s(H, k+1). From Theorem 4.1 we deduce the following corollary. Corollary 4.2. If (sn ) → s (B)(H, k + 1) for some k ≥ 0 and (sn ) is slowly oscillating, then (sn ) converges to s. (k)

Proof. Slow oscillation of (sn ) implies both Vn = O(1) (n → ∞) (k) and slow oscillation of (σn ) for all k ≥ 1 as a consequence of Lemma 3.5. √ (k) (k+1) Hence, we have Vn = n∆σn = O( n) (n → ∞). Since (sn ) is (H, k+1) (k+1) summable to s, σn → s by Theorem 1.2. It follows by the slow oscillation (k) (k) (1) of (σn ) that σn → s. Continuing in this vein, we obtain σn → s. By Lemma 3.3, (sn ) converges to s. Remark 4.3. If we replace slow oscillation of (sn ) by moderate os(1) cillation of (sn ) in Corollary 4.2, we recover convergence of (σn ) from (B)(H, k + 1) summability of (sn ). Parallel with the Hardy and Littlewood Tauberian condition n∆sn = √ (k) O( n) (n → ∞) for Borel summability method, the condition n∆Vn = √ O( n) (n → ∞) suffices to recover (H, k) summability of (sn ) from its (B)(H, k) summability for some k ≥ 0. Theorem 4.4. If sn → s (B)(H, k) for some k ≥ 0 and √ (10) n∆Vn(k) = O( n) (n → ∞), then (sn ) is (H, k) summable to s.



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(k)

Proof. By hypothesis, σn → s (B). By Lemma 3.4, we have σn(k+1) → s (B).

(11) It follows by (1) that

σn(k) − σn(k+1) = Vn(k) → 0 (B).

(12)

(k)

If we apply Theorem 1.2 to (Vn ), we conclude from (10) that Vn(k) = o(1) (n → ∞).

(13) By Lemma 3.1, we have

Vn(k) = n∆σn(k+1) = o(1) (n → ∞).

(14)

It follows from (14) that √ n∆σn(k+1) = o( n) (n → ∞).

(15)

(k+1)

If we apply Theorem 1.1 to (σn

), we conclude that

σn(k+1) → s.

(16)

(k)

Substituting (13) and (16) into (1), we obtain σn → s which means that sn → s(H, k). √ (k+1) If condition (10) is replaced by the condition n∆Vn = o( n) (n → ∞), we recover (H, k) summability of (sn ) from its (B)(H, k + 1) summability for some k ≥ 0. Theorem 4.5. If sn → s (B)(H, k + 1) for some k ≥ 0 and n∆Vn(k+1) = o(1) (n → ∞),

(17) then sn → s (H, k).

(k+1)

Proof. By hypothesis, σn (18)

→ s (B). By Lemma 3.4, we have

σn(k+2) → s (B).

Replacing k by k + 1 in (1), we obtain (19)

σn(k+1) − σn(k+2) = Vn(k+1) → 0 (B).


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TAUBERIAN THEOREMS

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It follows from (17) that (20)

√ n∆Vn(k+1) = o( n) (n → ∞). (k+1)

If we apply Theorem 1.1 to (Vn

), we obtain

Vn(k+1) = o(1) (n → ∞).

(21)

Replacing k by k + 1 in Lemma 3.1, we have (22)

Vn(k+1) = n∆σn(k+2) = o(1) (n → ∞).

It follows from (22) that (23)

√ n∆σn(k+2) = o( n) (n → ∞).

By (18) and (23), we get (24)

σn(k+2) → s

from Theorem 1.1. Replacing k by k + 1 in (1), we have (25)

σn(k+1) − σn(k+2) = Vn(k+1) .

By (21), (24) and (25), we obtain (26)

σn(k+1) → s.

By Lemma 3.2 we have (27)

Vn(k) − Vn(k+1) = n∆Vn(k+1) .

By (17), (21) and from identity (27), we have (28)

Vn(k) = o(1) (n → ∞). (k)

From (26) and (28), we obtain σn → s by (1), which means that sn → s(H, k). Also we have Theorem 4.5 when condition (17) is replaced by the con(k) dition Vn = o(1) (n → ∞) for some k ≥ 0. Corollary 4.6. If sn → s (B)(H, k + 1) for some k ≥ 0 and (29)

Vn(k) = o(1) (n → ∞),

then sn → s (H, k). Proof. The proof easily follows from (3) and the regularity of the Hölder’s method of summability.


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REFERENCES

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´ sz, O. – Introduction to the theory of divergent series, Revised ed. Department 17. Sza of Mathematics, Graduate School of Arts and Sciences, University of Cincinnati, Cincinnati, Ohio, 1952. 18. Zeller, K.; Beekmann, W. – Theorie der Limitierungsverfahren, (German) Zweite, erweiterte und verbesserte Auflage, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band, 15 Springer-Verlag, Berlin-New York, 1970. Received: 17.X.2012 Revised: 5.XI.2012 Accepted: 9.XI.2012

Ege University, Department of Mathematics, ˙ Izmir, TURKEY [email protected]
