NECESSARY AND SUFFICIENT TAUBERIANCONDITIONS UNDER WHICH CONVERGENCEFOLLOWS FROM SUMMABILITY АR,P Текст научной статьи по специальности «Математика»

DOI: 10.15393/j3.art.2021.10110

UDC 517.521

Ç. Kambak, i. Çanak

NECESSARY AND SUFFICIENT TAUBERIAN CONDITIONS UNDER WHICH CONVERGENCE FOLLOWS FROM SUMMABILITY p

Abstract. In this paper, we introduce the summability method Ar'p and obtain necessary and sufficient Tauberian conditions under which the ordinary convergence of a sequence follows from its summability Ar'p. The main results are new Tauberian theorems for the summability method Ar'p, which are generalizations of the corresponding Tauberian theorems for the summability method Ar introduced by Basar.

Key words: summability by Ar'p method, slow oscillation, slow decrease, Tauberian condition

2010 Mathematical Subject Classification: 40E05, 40G05

1. Introduction. Let p = (pn) be a sequence of non-negative numbers with p0 > 0 and

Pn = ^^ pk ^ <x>, n ^ <x>. (

k=0

Let 0 < r < 1. The class Ar'p = (anD Toeplitz matrices is given by

' Pk (1 + rk )

nr'p = I P

ank \ Pn

if 0 < k < n

0 if k > n.

Given a sequence x = (xn) of real or complex numbers, we define the Ar'p transform of x by

1n

(Ar,px)n = = pk(1 + rk)xk, n = 0,1, 2,...

Pn

n

k=0

© Petrozavodsk State University, 2021

If

lim = l, (2)

n^tt ''

we say that (xn) is summable to l by summability Ar'p.

It is clear that (1) is a necessary and sufficient condition that every convergent sequence (xn) is Ar'p-summable to the same limit.

It is easy to check that if the limit

lim xn = l (3)

n^-tt

exists, we also have (2). However, the opposite is not true in general. Let us define the sequence (xn) by xn = (—1)n ((1 + rn)pn)-1 and particularly choose pn = (n+1)-1 for all non-negative integers n; then we have a1np — 0 as n —y to. This shows that (xn) is Ar'p-summable to zero, though it does not converge. Note that (2) implies (3) under the certain condition on the sequence (xn), called Tauberian condition. Any theorem that states that convergence of sequences follows from its Ar'p-summability and some Tauberian condition(s) is said to be a Tauberian theorem for summability method Ar'p.

If pn = 1 for all non-negative integers n, we have the Ar method; it has been introduced by Basar [6] (see also [1], [2], [3], [4], and [5] for some results related to sequence spaces defined by the domain of the Ar matrices). The recent monograph [10] is devoted to the sequence spaces, summability theory and on the domain of certain triangle matrices in the normed/paranormed sequence spaces. In [12], Talo and Basar have given necessary and sufficient Tauberian conditions for the Ar method. In this paper, we extend the results of [12] to Ar'p and obtain necessary and sufficient conditions for the summability method Ar'p under which the existence of the limit (3) follows from that of (2).

2. Auxiliary Results. We need the following lemmas to prove our theorems.

Denote the integer part of the product A and n by An := [An].

Lemma 1. [8], [9] If (pn) is a sequence of non-negative numbers, the conditions

Pn

lim sup —— < 1 for every A > 1 (4)

n^tt P An

and

P\

lim sup ——— < 1 for every 0 < A < 1

P n

n^œ -i n

are equivalent.

Lemma 2. Let (4) be satisfied. If a sequence (xn) is Ar'p summable to

a finite number l, then

л

lim —-— V pk(1 + rk)xk = l for every A > 1 (6)

ra^œ pA — PL

An n k=n+1

and

lim —--— V pk(1 + r )xk = l for every 0 < A < 1. (7)

n Лп k = An + 1

Proof. Case A > 1. By definition,

1 Лп p

Pk(1 + rk)xfc = <p + p Л" P K,p - <p) . (8)

P p / y ' k n,p ' P P

PA — Pn PA — Pn

An n fc=n+1 An n

By (4), we have

0 < lim sup ——An = M — lim sup —n J < to.

n^tt pAn Pn \ n^tt pAn /

Now, (6) follows from (2) and (8). Case 0 < A < 1. By definition,

1 ™ P

^ /->n r i n ( r r

Pk(1 +rk)xk = <p + p prap --

Pn - pa ^ ' ' '~k "n'p pn - Pa V An'p '

n An k=An+1

pn Л pA^ 1

From (5) we have

0 < lim sup —--- =11 - lim su^--^ < то.

ra^œ pra pAn \ ra^œ pn /

Now, (7) follows from (2) and (9). □

3. Main Results. First, we consider sequences of real numbers and prove the following one-sided Tauberian theorem.

Theorem 1. Let (4) be satisfied, (xn) be a sequence of real numbers, Ar'p-summable to a finite limit l; then (3) holds if and only if the following two conditions are satisfied:

1 An

sup lim inf —-— V" (pfc(1 + rk)xfc - Pfcxj ^ 0 (10)

A>1 ^ pAn - pn k^l

and

1 n

sup liminf —-— Y^ (pfcXn - pfc(1 + rk)xfc) ^ 0. (11)

0<A<1 n—~ Pn - PA„ ^^V

A sequence (xn) of real numbers is said to be slowly decreasing if

lim liminf min (xk — xn) ^ 0. (12)

A—1+ n—^^o n<fc^An

Note that condition (12) can be equivalently reformulated as:

lim liminf min (xn — ) ^ 0. (13)

A—1- n—^ An<fc^n

The right-hand limit in (12) exists and can be equivalently replaced by supA>1. The concept of slow decreasing was introduced by Schmidt [11].

For sequences (xn) and (yn) of real or complex numbers, we write xn = O(yn) if there exists some positive number M, such that |xn| ^ M|yn| for all sufficiently large n.

We have the following corollary for Theorem 1.

Corollary 1. Let (4) and Pn = O(npn) be satisfied. If a sequence (xn) of real numbers is slowly decreasing, (2) implies (3).

Remark. If conditions (2) and (3) or, equivalently, the conditions (2), (10), and (11) are satisfied, then we necessarily have

1 An

lim --— V (pfc(1 + rk)xfc — pfcXn) = 0 (14)

An n fc=n+1

for every A > 1 and

1

lim n-(pfcXn - Pfc(1 + rk)xfc) = 0 (15)

n^œ PL — PA v '

n An fc=An + 1

for every 0 < A < 1.

Remark. Theorem 1 remains true if conditions (10) and (11) are replaced by their symmetric counterparts:

1 An

inf lim sup —-— V] (pfc(1 + rk)xfc - pfcXn) ^ 0 (16)

A>1 ^ pAn - Pnfc=n+i

and

1 n

inf lim sup —-— V] (pfc- pfc(1 + rk)xfc) ^ 0, (17)

0<A<1 n—^-OO Pn — P\ -

respectively.

Second, we consider sequences of complex numbers and prove the following two-sided Tauberian theorem.

Theorem 2. Let (4) be satisfied, (xn) be a Ar'p-summable sequence of complex numbers; then (xn) converges to the same limit if and only if one of the following two conditions is satisfied:

inf lim sup

A> n—s-oo

P\ — P

PA™ Pn fc=n+1

y^ (pfc(1 + rk)xfc - PfcIn)

:is)

or

inf lim sup '

0<A<1 n—s-oo

P — P

Pn fc=A„ + 1

y^ (pfcIn - Pfc(1 + rk)xfc)

:i9)

A sequence (xn) of complex numbers is said to be slowly oscillating if lim lim sup max |xk — xn| = 0. (20)

A—1+ n—^^o n<fc^An

The concept of slow oscillation was introduced by Hardy [7]. An equivalent reformulation of (20) can be given as follows:

lim lim sup max |xk — xn| = 0. (21)

A—1- An<fc^n

The right-hand limit in (20) can be equivalently replaced by infA>1. We have the following corollary for Theorem 2:

Corollary 1. Let (4) and Pn = O(npn) be satisfied. If a sequence (xn) of complex numbers is slowly oscillating, (2) implies (3).

4. Proofs. In this section we present the proofs. Proof of Theorem 1.

Necessity. Assume that (2), (3), and (4) are satisfied. Then Lemma 4 yields (10) in case A > 1 and (11) in case 0 < A < 1. Sufficiency. Assume that (2), (4), (10) and (11) are satisfied.

1

0

0

First, consider the case A > 1. Let e > 0 be given. By (10), there exists some A > 1, such that

1 An

liminf —-— V (pfc(1 + rk)xfc — pfcx—) ^ —e. (22)

n^tt PA — ^-'

An n fc=n+1

It follows from (8) that

r _ PA— ( r r .

Xn — ^n,p ~n p" V^An,p — °"n,p ' —

P\„ _ Pn

1 A—

k=n+1

By (4), we have

P P S (Pk(1 + - Pkxj . (23)

PA— — pn —

lim (aA—,p - <p) = 0. (24)

n^œ P\ — Pn

Combining (23) and (24) gives

PA—

limsup(xn - ^n,p) ^ limsup " (aAn,p - ^n,p) +

n^œ n^œ PA— Pn

A—

+ lim sup [- ———— ^ (pk (1 + rk)xk - PkXn^ ^

n^œ V —A— Pnk=n+1 7

1 A—

^ - liminf (—-— Y^ (pk (1 + rk )xk - Pk Xn)) ^ e.

n^tt \ Pa — P—

An n fc=n+1

Consequently, we have

limsupxn ^ l + e. (25)

n^tt

Second, consider the case 0 < A < 1. It follows from (9) that

p

r __1 n /r

Xn - °n,p = p _ p l°n,p - ^A—,pJ +

n - PA

1

Pn - PA

+ p ^ p X] (PkXn - Pk(1 + rk)Xk) . (26)

k=A—+1

Using a similar argument as above, we conclude by (5) and (11) for any given e > 0 that

P

liminf(xn - <P) ^ liminf An (^n,p- ^An, J +

n—ro n—ro P — P\ ^

n An

+ liminf [ 1 ^ (pfcin - Pk(1 + rk)xfc)) ^ -e. n—ro \ Pn — PA ' v 7 /

\ n An fc=An+1 /

Consequently, we have

liminf xn ^ l - e. (27)

n—ro

Combining (15) and (27) yields

l - e ^ lim inf xn ^ lim sup xn ^ l + e.

n—ro n—ro

Choose e arbitrary small; hence (3) follows. □

Proof of Corollary 1. For A > 1, we have the following inequality:

1A

V (pfc(1 + rk)xfc - pfc^ min ((1 + rk)xfc - ^

P. 7 \ / , V k V 1 / k k n /

A — P„ ' v y n<k^A

An n k=n+1

^ min (xk - xn) + min (r xk).

n<k^An n<k^An

We have

rr

_ P n^n,p P n-1^n-1,p

xn

Pn(1 + rn)

and

r r r

xn _ Pn(°n,p - ^n-1,p) + ^n-1,p

n

npn(1 + rn) n(1 + rn)

xn

Since (xn) is summable Ar,p and Pn = O(npn), we have — ^ 0 as

n

n ^ w. Therefore, rnxn ^ 0 as n ^ to. Hence, condition (12) clearly implies (10). Similarly, (13) implies (11). By Theorem 1, we have (3). □

Proof of Theorem 2.

Necessity. The proof is similar to the proof of the necessity part of Theorem 1.

Sufficiency. Assume that (2) and one of the conditions (18) and (19) are satisfied. Let any e > 0 be given. By (18), there exists some A > 1, such that

lim sup

n^œ

1

An

Pa — Pn J

An n k=n+1

(pfc (1 + rk )xfc - Pk Xn

< e.

By (23), we have

(28)

P\

limsup |xn — ^n p| ^ limsup

PA

P^- Pn

■|aAn ,p— an,p|+

+ lim sup

n

PA Pn

Y. (pfc (1 + rfc)xfc — Pk Xn)

k=n+1

(29)

By (19), there exists some 0 < A < 1, such that

lim sup

n

1

_ Y (Pk Xn — Pk (1 + rk )Xk)

n — P An k=An + 1

Pn PA

< e.

(30)

By (26), we have

limsup |xn — ^n p| ^ limsup

P

n

n^x ^ra PA

"|an,p — aAn,p| +

+ lim sup

ra^œ

Pn - PAn

Y (pk Xn — Pk (1 + rk )Xk)

k=An+1

(31)

By (29) or (31), in either case we obtain

limsup |Xn — ^n p| = 0

(32)

whence, it follows that

lim |Xn — ^n p| = 0.

(33)

Now, we conclude (3) from (2) and (33). □

Proof of Corollary 1. For A > 1, we have the following inequality:

PA Pn

Y (Pk (1+ rk )Xk

PkXn

k=n+1

^ max |(1 + r )Xk — Xn| ^

n<k^An

1

1

1

^ max |xk - xn| + max |rkxk|.

n<k^An n<k^An

xn

As in the proof of Corollary 1, we have — ^ 0 as n ^ to. Therefore,

n

rnxn ^ 0 as n ^ to. Hence, condition (20) clearly implies (18). Similarly, (21) implies (19). By Theorem 2, we have (3). □

Acknowledgment. The authors would like to thank the anonymous referee for his/her careful reading of the manuscript, correcting many errors, and useful comments that improved the manuscript.

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Received March 25, 2021. In revised form, April 23, 2021. Accepted April 25, 2021. Published online May 21, 2021.

Ege University

Faculty of Science, Department of Mathematics Erzene District, Bornova/Izmir 35040, Turkey E-mail: [email protected]

Ege University

Faculty of Science, Department of Mathematics Erzene District, Bornova/Izmir 35040, Turkey E-mail: [email protected]