Some Tauberian theorems for regularly generated sequences

Computers and Mathematics with Applications 62 (2011) 4486–4491

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Some Tauberian theorems for regularly generated sequences İbrahim Çanak, Ferhat Hasekiler 1 , Duygu Kebapcı ∗ Ege University, Department of Mathematics, 35100 Izmir, Turkey

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Article history: Received 9 June 2011 Received in revised form 10 October 2011 Accepted 10 October 2011 Keywords: Abel summability method Regularly generated sequences Slow oscillation Moderate oscillation General control modulo

In this paper, we establish some Tauberian theorems for the Abel summability method in terms of regularly generated sequences which generalizes some results obtained in Çanak and Totur [İ. Çanak, Ü. Totur, A note on Tauberian theorems for regularly generated sequences, Tamkang J. Math. 39 (2) (2008) 187–191]. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Throughout this article, N and N0 will denote the set of all positive integers and all nonnegative integers, respectively. Let s = (sn ) be a sequence of real numbers and any term with a negative index be zero. For a sequence (sn ), we define (m)

σn

 n  1  (m−1) σ (s), (s) = n + 1 k=0 k  sn ,

m∈N m=0

where

(m)

Vn

(∆s) =

 n 1  (m−1)    Vk (∆s),  n + 1

m∈N

n  1     k∆sk , n + 1

m=0

k=0

k=0

and

∆ sn =

sn − sn−1 , s0 ,



n∈N n = 0.

For a sequence (sn ), we have the identity sn − σn(1) (s) = Vn(0) (∆s),

( n ∈ N0 )

∗

Corresponding author. E-mail addresses: [email protected] (İ. Çanak), [email protected] (F. Hasekiler), [email protected], [email protected] (D. Kebapcı). 1 Tel.: +90 232 311 5418; fax: +90 232 388 1036. 0898-1221/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2011.10.027

(1.1)

İ. Çanak et al. / Computers and Mathematics with Applications 62 (2011) 4486–4491

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which is well-known as the Kronecker identity. (1)

Since σn (s) = s0 + sn = Vn(0) (∆s) +

(0) Vk (∆s)

n

k=1

k

(0) n  V (∆s) k

k

k=1

, identity (1.1) can be written as

+ s0 .

(1.2) (0)

Define the classical control modulo of the oscillatory behavior of (sn ) by ωn (s) = n∆sn and the general control modulo (m) (m−1) of the oscillatory behavior of (sn ) of order m ∈ N by ωn (s) = ωn (s) − σn(1) (ω(m−1) (s)), recursively. Definition 1.1. A sequence (sn ) is said to be Cesàro summable to L, and we write sn → L(C ) if lim σn(1) (s) = L.

n→∞

Definition 1.2. A sequence (sn ) is said to be Abel summable to L, and we write sn → L (A) if (1 − x) for 0 < x < 1 and tends to L as x → 1− .

∞

n =0 s n x

n

converges

(j)

If (σn (s)) is Abel summable to L for some j ∈ N0 , we write sn → L (A, j). Definition 1.3 ([1]). A sequence (sn ) is said to be slowly oscillating if lim lim sup

λ→1+

n→∞

max n+1≤k≤[λn]

|sk − sn | = 0.

Denote by S the class of slowly oscillating sequences. (1) A sequence (sn ) is said to be (C , 1) slowly oscillating if (σn (s)) is slowly oscillating. (0) It is proved in [2] that a sequence (sn ) is slowly oscillating if and only if (Vn (∆s)) is bounded and slowly oscillating. Definition 1.4. A sequence (sn ) is said to be moderately oscillating if, for λ > 1, lim sup

max

n→∞

n+1≤k≤[λn]

|sk − sn | < ∞.

Denote by M the class of moderately oscillating sequences. Let B be a subset of any linear space of sequences L. Let m ∈ N. We define the set B (m) =



(b(nm) )|b(nm) =

n

(m−1) bk

k =1

k

 ,

(0)

where (bn ) := (bn ) ∈ B . Definition 1.5. Let (sn ) ∈ L. If sn = b(nm) +

(m) n  b k

k=1

k

(m)

(m)

for some b(m) = (bn ) ∈ B (m) , we say that (sn ) is regularly generated by (bn ) and b(m) is called a generator of (sn ). We denote the set of all sequences regularly generated by the sequences in B (m) by U (B (m) ). Let B> denote the set of all sequences b = (bn ) such that for every (bn ) ∈ B> there exists Cb ≥ 0 such that bn ≥ −Cb . (m) The set U (B> ) can be defined in the same manner as in definition above. (0)

(1)

Let B = S . If (sn ) ∈ U (S ), then (Vn (∆s)) ∈ U (S ) and (σn (s)) ∈ U (S (1) ). If B is the set of all bounded and slowly sequences, then U (B ) is the set of all slowly oscillating sequences. Çanak [3] obtained a short proof of the generalized Littlewood Tauberian theorem [4] which asserts that if sn → L(A) and (sn ) ∈ S , then sn → L. Later, Dik et al. [5] have generalized Çanak’s theorem by means of the concept of a regularly generated sequence. Recently, Çanak and Totur [6,7] have proved some Tauberian theorems for which Tauberian conditions are given in terms of the control modulo of oscillatory behavior of a sequence. Our aim in this work is to establish several Tauberian theorems for the Abel summability method in terms of regularly generated sequences which generalize the results given in [8]. Our new results are based on the following theorems. (m)

Theorem 1.6 ([6]). If sn → L(A) and ωn (s) ≥ −C for some C > 0 and for some m ∈ N, then sn → L. (m)

Theorem 1.7 ([7]). If sn → L(A) and (ωn (s)) is (C , 1) slowly oscillating for some m ∈ N, then sn → L. The proof of Theorem 1.7 for m = 0 was given by Tauber [9].

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2. Identities and lemmas For a sequence (sn ), we define inductively

(n∆)0 sn = sn ,

(n∆)m sn = n∆((n∆)m−1 sn ),

where m ∈ N. We use the following identities in the proofs extensively. Identity 2.1. For a sequence (sn ),

σn(i) ((n∆)j s) = (n∆)j (σ (i) (s)) (i, j ∈ N0 ). Identity 2.2. For a sequence (sn ),

σn(i) ((n∆)j s) = (n∆)j (σ (i) (s)) (i, j ∈ N0 ). We need a number of lemmas for the proof of the theorems stated below. Lemma 2.3 ([6]). For a sequence (sn ),

ωn(m) (s) = (n∆)m Vn(m−1) (∆s), where m ∈ N. (k)

(k+1)

Lemma 2.4 ([8]). Let s = (sn ) ∈ L. If (Vn (∆s)) ∈ U (B (m) ) for some k, m ∈ N0 , then (n∆)m+1 Vn

(∆s) = bn .

3. Main results (m)

Theorem 3.1. If sn → L(A, j) for some j ∈ N0 and (ωn (s)) is (C , 1) slowly oscillating for some m ∈ N, then sn → L. (m)

(1)

Proof. Since (ωn (s)) is (C , 1) slowly oscillating, (σn (ω(m) (s))) is slowly oscillating. By the fact that the sequence of (j) arithmetic means of a slowly oscillating sequence is slowly oscillating, we have that (σn (σ (1) (ω(m) (s)))) is slowly oscillating. It follows from Identity 2.1 that

σn(j) (σ (1) (ω(m) (s))) = σn(j+1) (ω(m) (s)) = σn(1) (ω(m) (σ (j) (s))). (j)

(m)

Hence we conclude that (ωn (σ (j) (s))) is (C , 1) slowly oscillating. Since sn → L(A, j), we have, by Theorem 1.7, σn (s) → L, (j−1) which means that σn (s) → L(C ). By the fact that Cesàro summability implies Abel summability, we have σn(j−1) (s) → (j−1) L(A). Continuing in this way, we obtain σn (s) → L. If we repeat this j times, we have sn → L.  If we take j = 0 and j = 1 in Theorem 3.1, we obtain Corollary 5.1 in [7] and Theorem 3.7 in [8], respectively. (0)

Theorem 3.2. If sn → L(A, j) for some j ∈ N0 and (ωn (s)) is (C , 1) slowly oscillating, then sn → L. (0)

(1)

(0)

Proof. Since (ωn (s)) is (C , 1) slowly oscillating, (σn (n∆s)) = (Vn (∆s)) is slowly oscillating. By the fact that the (j) sequence of arithmetic means of a slowly oscillating sequence is slowly oscillating, (σn (V (0) (∆s))) is slowly oscillating. (j) (0) (0) (0) (j) (j) Since σn (V (∆s)) = Vn (∆σ (s)), (Vn (∆σ (s))) is slowly oscillating. Since sn → L(A, j), we have, by Tauber’s second (j) (j−1) (s) → L(C ). The rest of the proof is as in Theorem 3.1.  theorem [9], σn (s) → L, which means that σn (m)

Theorem 3.3. If sn → L(A, j) for some j ∈ N0 and ωn (s) ≥ −C for some C > 0 and for some m ∈ N, then sn → L. (m)

Proof. Since the sequence of arithmetic means of a bounded sequence is bounded, it follows that ωn (s) ≥ −C for some (j) C > 0 implies σn (ω(m) (s)) ≥ −C . By Identity 2.1, we have

σn(j) (ω(m) (s)) = ωn(m) (σ (j) (s)) ≥ −C . (j)

(j−1)

Since sn → L(A, j), we have, by Theorem 1.7, σn (s) → L, which means that σn (s) → L(C ). By the fact that Cesàro (j−1) summability implies Abel summability, we have σn (s) → L(A). Continuing in this way, we obtain σn(j−1) (s) → L. If we repeat this j times, we have sn → L. If we take j = 0 in Theorem 3.3, we obtain Corollary 2.9 in [6].

İ. Çanak et al. / Computers and Mathematics with Applications 62 (2011) 4486–4491

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(k)

(m) Theorem 3.4. If sn → L(A, j) for some j ∈ N0 and (Vn (∆s)) ∈ U (B> ) for some k, m ∈ N0 where k < m, then sn → L.

Proof. If we take B = B> in Lemma 2.4, we have

(n∆)m+1 Vn(k+1) (∆s) = bn ∈ B> (k)

(k+1)

(m) since (Vn (∆s)) ∈ U (B> ). By the definition of B> , there exists some C ≥ 0 such that (n∆)m+1 Vn Identity 2.1 and 2.2, and Lemma 2.3, we have

(∆s) ≥ −C . By

σn(m−k+j−1) ((n∆)m+1 V (k+1) (∆s)) = σn(j) ((n∆)m+1 V (m) (∆s)) = σn(j) (ω(m+1) (s)) = ωn(m+1) (σ (j) (s)). From

σn(m−k+j−1) ((n∆)m+1 V (k+1) (∆s)) = σn(m−k+j−1) (b), we obtain

ωn(m+1) (σ (j) (s)) = σn(m−k+j−1) (b). Since the sequence of arithmetic means of a bounded sequence is bounded,

ωn(m+1) (σ (j) (s)) ≥ −C (j)

(j−1)

for some C > 0. Since sn → L(A, j), we have, by Theorem 1.6, σn (s) → L, which means that σn (s) → L(C ). By (j−1) the fact that Cesàro summability implies Abel summability, we have σn (s) → L(A). Continuing in this way, we obtain σn(j−1) (s) → L. If we repeat this j times, we have sn → L.  (k)

Theorem 3.5. If sn → L(A, j) for some j ∈ N0 and (Vn (∆s)) ∈ U (S (m) ) for some k, m ∈ N0 , where k ≤ m, then sn → L. Proof. Taking B = S in Lemma 2.4, we have

(n∆)m+1 Vn(k+1) (∆s) = bn ∈ S (k)

since (Vn (∆s)) ∈ U (S (m) ). By Identity 2.1 and 2.2 and Lemma 2.3, we obtain

σn(m−k) ((n∆)m+1 V (k+1) (∆s)) = σn(1) ((n∆)m+1 V (m) (∆s)) = σn(1) (ω(m+1) (s)). Since

σn(m−k) ((n∆)m+1 V (k+1) (∆s)) = σn(m−k) (b), we have

σn(1) (ω(m+1) (s)) = σn(m−k) (b). (1)

By the fact that the sequence of arithmetic means of a slowly oscillating sequence is slowly oscillating, (σn (ω(m+1) (s))) is (m+1) slowly oscillating. This means that (ωn (s)) is (C , 1) slowly oscillating. By sn → L(A, j), it follows from Theorem 3.1 that s n → L.  In Theorem 3.5, if we take j = 0 and k = m we have Theorem 3.2, j = 0 and k = m − 1, we have Theorem 3.3, j = 1, k = 0 and m = 1, we have Corollary 3.6 in [8]. (k)

Theorem 3.6. If sn → L(A, j) for some j ∈ N and (Vn (∆s)) ∈ U (M (m) ) for some k, m ∈ N0 , where k ≤ m, then sn → L. Proof. If we take B = M in Lemma 2.4, we have

(n∆)m+1 Vn(k+1) (∆s) = bn ∈ M (k)

since (Vn (∆s)) ∈ U (M (m) ). From Identity 2.2 and Lemma 2.3,

σn(m−k+j) ((n∆)m+1 V (k+1) (∆s)) = σn(j+1) ((n∆)m+1 V (m) (∆s)) = σn(j+1) (ω(m+1) (s)) = σn(1) (ω(m+1) (σ (j) (s))).

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From

σn(m−k+j) ((n∆)m+1 V (k+1) (∆s)) = σn(m−k+j) (b) we have

σn(1) (ω(m+1) (σ (j) (s))) = σn(m−k+j) (b). (1)

Since the sequence of arithmetic means of a moderately oscillating sequence is slowly oscillating, (σn (ω(m+1) (σ (j) (s)))) (m+1) is slowly oscillating. Hence, (ωn (σ (j) (s))) is (C , 1) slowly oscillating. Since sn → L(A, j), we have, by Theorem 1.7, (j) (j−1) σn (s) → L, which means that σn (s) → L(C ). By the fact that Cesàro summability implies Abel summability, we have σn(j−1) (s) → L(A). Continuing in this way, we obtain σn(j−1) (s) → L. If we repeat this j times, we have sn → L.  Note that for the proof of the case j = 0 in Theorem 3.6, m should be greater than k. If we take j = 1, k = 0 and m = 1 in Theorem 3.6, we have Corollary 3.8 in [8]. (m) Theorem 3.7. If sn → L(A, j) for some j ∈ N0 and (sn ) ∈ U (B> ) for some m ∈ N, then sn → L. (m) Proof. If (sn ) ∈ U (B> ), then

sn = b(nm) +

(m−1) n  b k

(m)

for some bn

(3.1)

k

k=1

∈ B>(m) . From identity (3.1), we have

(n∆)sn = n∆bn(m) + b(nm) .

(3.2)

Taking the arithmetic means of both sides of (3.2), we have

σn(1) (n∆s) = Vn(0) (∆b(m) ) + σn(1) (b(m) ) = b(nm) . (0)

(1)

Since σn (n∆s) = Vn (∆s), Vn(0) (∆s) = b(nm) +

(m−1) n  b k

k=1

k

.

(3.3)

Applying (n∆)m to both sides of (3.3), we have

(n∆)m Vn(0) (∆s) = bn . Hence we obtain

σn(m−1) ((n∆)m V (0) (∆s)) = σn(m−1) (b). By Identity 2.2 and Lemma 2.3, we have

(n∆)m Vn(m−1) (∆s) = ωn(m) (s). Since the sequence of arithmetic means of a bounded sequence is bounded,

ωn(m) (s) ∈ B> . By sn → L(A, j) it follows from Theorem 3.3 that sn → L.



If we take j = 1 and m = 1 in Theorem 3.7, we have Corollary 3.9 in [8]. Theorem 3.8. If sn → L(A, j) for some j ∈ N0 and (sn ) ∈ U (S (m) ) for some m ∈ N, then sn → L. Proof. If (sn ) ∈ U (S (m) ), then sn = b(nm) +

(m−1) n  b k

k=1

(m)

for some bn

k

(3.4)

∈ S (m) . From identity (3.4), we have

(n∆)sn = n∆bn(m) + b(nm) . Taking the arithmetic means of both sides of (3.5), we have

σn(1) (n∆s) = Vn(0) (∆b(m) ) + σn(1) (b(m) ) = b(nm) .

(3.5)

İ. Çanak et al. / Computers and Mathematics with Applications 62 (2011) 4486–4491

(1)

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(0)

Since σn (n∆s) = Vn (∆s), Vn(0) (∆s) = b(nm) +

(m−1) n  b k

k=1

k

.

(3.6)

Applying (n∆)m to both sides of (3.6), we have

(n∆)m Vn(0) (∆s) = bn . Hence we obtain

σn(m−1) ((n∆)m V (0) (∆s)) = σn(m−1) (b). By Identity 2.2 and Lemma 2.3, we have

(n∆)m Vn(m−1) (∆s) = ωn(m) (s). (m)

Since the sequence of arithmetic means of a slowly oscillating sequence is slowly oscillating, (ωn (s)) ∈ S . By sn → L(A, j), (m) it follows from Theorem 3.3 that sn → L. Hence (ωn (s)) is (C , 1) slowly oscillating. Since sn → L (A, j), we have, by Theorem 3.1, sn → L.  Theorem 3.9. If sn → L(A, j) for some j ∈ N and (sn ) ∈ U (M (m) ) for some m ∈ N, then sn → L. Proof. The proof is similar to the proof of Theorem 3.8. References [1] Č.V. Stanojević, in: İ. Çanak. (Ed.), Analysis of Divergence: Control and Management of Divergent Process, in: Graduate Research Seminar Lecture Notes, University of Missouri, Rolla, Fall, 1998. [2] M. Dik, Tauberian theorems for sequences with moderately oscillatory control moduli, Doctoral Dissertation, University of Missouri-Rolla, Missouri, 2002. [3] İ. Çanak, A proof of the generalized Littlewood Tauberian theorem, Appl. Math. Lett. 23 (7) (2010) 818–820. [4] R. Schmidt, Über divergente folgen und lineare mittelbildungen, Math. Z. 22 (1925) 89–152. [5] M. Dik, F. Dik, İ. Çanak, Classical and neoclassical Tauberian theorems for regularly generated sequences, Far East J. Math. Sci. 13 (2) (2004) 233–240. [6] İ. Çanak, Ü. Totur, A Tauberian theorem with a generalized one-sided condition, Abstr. Appl. Anal. 2007 (2007) p. 12. Article ID 60360. [7] İ. Çanak, Ü. Totur, Tauberian theorems for Abel limitability method, Cent. Eur. J. Math. 6 (2) (2008) 301–306. [8] İ. Çanak, Ü. Totur, A note on Tauberian theorems for regularly generated sequences, Tamkang J. Math. 39 (2) (2008) 187–191. [9] A. Tauber, Ein Satz aus der Theorie der unendlichen Reihen, Monatsh. Math. 8 (1897) 273–277.