A Taiiberian theorem îor the product

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ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X IX (1976) ROCZNIKI POLSKIEGO TOWAKZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE X IX (1976)

M. E. P

arames

WAPu

an

* ( Winnipeg)

A Taiiberian theorem îor the product

oî the logarithmic and Cesàro methods of summability

I . Let {sM} (n = 0 , 1 , 2 , . . . ) be a sequence of (real or complex) numbers. It is said to be summable by the logarithmic method (L ) if

1 Д snxn+l

— —--- > ---tends to a limit Я as x -> 1 — 0 and we say then th at l o g ( l - ) ^ n*

+ 1

{бп} is (L)-summable to Я. The method (L) has been mentioned by Hardy* ([2], in 81) and has been studied in detail by Borwein, Ishiguro and Ban- gachari and Sitaraman in several papers, of which [1], [3] and [4] are the most significant in the context of the present note. In this note we prove a Tauberian theorem for the product summability method (LCa) defined as ( follows. Given a sequence {jun} of real numbers, the Hausdorff matrix

H = (hnk) generated by is defined by / \ n~k

(n , h — 0,1, 2, ...). I t is well known th a t Hausdorff matrices are commu

tative and th a t the Cesàro method Ca (a > —1) is a Hausdorff matrix +d \ ^

I . (For these and other details about Hausdorff methods, see [2].) As usual, we shall use the same symbol for a m atrix and the sequence-to-sequence transformation defined by it. For a < —1 let us define Ca as (C_a)“ 1. I t is to be remarked, however, th a t the definition

€ a = ((7_J_1 for all a < 0 (which differs from the previous one if —1

< a < 0) is also consistent with the statements in Section 2 below. Given a real number a and a (real or complex) sequence {sn}, let {$"} denote the (7a-transform of {sn} ; we shall say th a t {sn} is summable by the product

* This paper is part of the work done by the author during his stay at the Uni

versity of Stuttgart. The author wishes to thank the University and Professor W. Meyer- Konig in particular for their hospitality and kindness, and the Deutscher Akademischer Austauschdienst for an invitation to visit Germany.

8 — Roczniki PTM — Prace Matematyczne XIX

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114 M. R. P a r a m e s w a r a n

method. (LGa) if the sequence {«“} is (L)-summable. The method (LGa) is linear, and is regular if a > 0 (see H ardy [2], p. 81 and Lemma 2 given below).

2. I t is the object of this note to prove the

T

heorem

. Let {$n} be a sequence of real or complex numbers and a, real numbers such that

(1) {sn} is (LGa)-summable to X (i.e. {«“} is (L)-summable to X), and

(2) logfl- = 0(1).

Then s ^ - ^ X as n~^oo, i.e. {sw} is G^~1-summable to X.

To prove the theorem we shall make use of the following lemmas, hfote th a t all sequences we consider may be real or complex.

L

emma

1. I f is (L)-summable to X and n(sn — sn_l)\ogn = 0(1), then sn -+X as n oo.

P ro o f. This is easily derived from the one-sided Tauberian theorem given by Eangachari and Sitaraman [4].

(Ishiguro [3], the first to prove a Tauberian theorem for the method (L), had о instead of О in the result stated above.)

L

emma

2 (Borwein [1]). Let H be any regular Hausdorff matrix and a sequence (L)-summable to X. Then the H-transform of {sn} is also (L)-summable to X.

L

emma

3. I f p < q, then {s i s the transform of {s%} by a regular Haus- dorff matrix; that is, Gq = HGP, where H is regular Hausdorff.

P ro o f. By definition of the Cesàro methods, H — Са(Ср)~г exists and is Hausdorff, with Gq = HGp . The regularity of H is verified as follows.

Case (i): q > p > —1. This is well known (see e.g. [2], Theorem 43).

Case (ii): q > —1 Then H = Gq(G

^{p) ~ 1}

= G qG_p. Since —p-\- q> 0 and q > —1, it follows th at II is equivalent to G_p+q, a regular Cesàro method (see e.g. [5], p. 109). Case (iii): —1 > q > p. Then — p > — q > 1 and hence H — С

⁹

(Ср)~г = (0_g)~1C'_:p is regular.

L

emma

4. For each real number p, the methods Gp and G

1

G

P__1

are equivalent. I n other words, there exists a Hausdorff method H equivalent to the identity (i.e. H and its inverse (H

^{) ~1}

are both regular Hausdorff) such that Gp = HG

1

G

P _ 1

= G

1

HGP_1.

P ro o f. Case (i): p > 0. This is well known. Case (ii): — 1 < p ^ 0.

Then by Case (i) there exist regular Hausdorff matrices А, В, А ~ г and B -1 such th at

0 , - 1 0 ! = ( С - р + х Г 1^ = ( A G ^ C J - ' G , = A ~ l ( G _ p ) ~ l = A ~ l B G p .

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Tauberian theorem

115

Then the lemma is satisiied if we take H — A B h Case (iii): p < —1.

Sow the proof in Case (ii) applies with B = I, the identity itself.

L

emma

5. For any sequence {sn}

sn- s l = п{

8

гп- 4 -

ï

) for all n ^ O (where, as usual, sLj is defined as 0).

P ro o f. From the definition of sxn we get

= (w + l ) s i - w s i _ i - 4 =

P r o o f of th e th e o re m . First we observe th at it is enough to prove th at {s is convergent. For, if {tn} is the sequence whose Gt-transform is the sequence {$£}, then Lemma 5 and condition (2) of the theorem

show th at x

K ~ 8n 1)->0 as n ^ o o .

Hence {tn} converges to Я if and only if {sp n} converges to Я. But writing {tn} = H-transform of as in Lemma 4, we see th a t {tn} converges to Я if and only if converges to Я. Thus

(3) sn~>^ (as n-+oo) implies th a t «£- 1 -»Я as n-^o o.

We now proceed to prove th a t the sequence {s^} converges to Я.

C ase (i). a = /?. The theorem reduces to Lemma 1.

C ase (ii). a < /?. Then {s^} is (L)-summable to Я, by Lemmas 2 and 3;

the desired result now follows at once from Lemma 1.

C ase (iii). a > /?. Let Tc be the smallest positive integer such th a t a + i/? |. By Lemmas 2 and 3 and condition (1) of the theorem we have :

(4) {sfj is (XC^.)-summable to Я.

Since condition (2) of the theorem implies th at n(s^ — ^ _ x)-> 0 as n->oo, we see by Lemma 5 th a t

(5) tn~ sn~>® ш п - > о о .

Since the method (LCk) is regular for h > 0 (by the regularity of (L), and by Lemma 2 when Tc > 0), the sequence {tn — s^} is certainly (LCk)- summable. From (4) and the linearity of the method (LCk) we see then th a t the sequence {tn} is (LCk)-summable to Я. Hence, by Lemma 4, the (^-transform of {tn} is (L(7A;_1)-summable to Я; th a t is,

(6) { s £ } is ( L C f c .^ - s u m m a b l e t o Я.

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116 М. К. P a r a m e s w a r a n

The argument for the passage from the statement (4) to the statement (6) can be repeated a finite number of times to obtain th at

(7) {s^} is (L)-summable to X.

[We remark th a t the condition

& > a + l / ? l

was used initially only to obtain (4) ; and only the condition Тс > 0 was needed to derive (6) from (4).]

From (7), (2) and Lemma 1 we see th a t {s converges to A. In view of (3), this completes the proof of the theorem.

References

[1] D. B orw ein, A logarithmic method of summability, J. London Math. Soc. 33 (1958), p. 212-220.

[2] G. H. H ardy, Divergent series, Oxford 1949.

[3] K. Ish igu ro, Tauberian theorems concerning the summability methods of logarithmic type, Proc. Japan Acad. 39 (1963), p. 156-159.

[4] M. S. R a n gach ari and Y. Sitaram an , Tauberian theorems for logarithmic summability ( L ), Tohôku Math. J. 16 (1964), p. 257-269.

[5] K. Z eller and W. B eek m an n , Théorie der Limitierungsverfahren Springer- Verlag, 1970.

DEPARTMENT OF MATHEMATICS UNIVERSITY OF MANITOBA WINNIPEG, CANADA and

MATHEMATISCHES INSTITUT A UNIVERSITÂT STUTTGART STUTTGART

FEDERAL REPUBLIC OF GERMANY

A Taiiberian theorem îor the product

M. E. P

WAPu

* ( Winnipeg)

A Taiiberian theorem îor the product

oî the logarithmic and Cesàro methods of summability

I . Let {sM} (n = 0 , 1 , 2 , . . . ) be a sequence of (real or complex) numbers. It is said to be summable by the logarithmic method (L ) if

1 Д snxn+l

— —--- > ---tends to a limit Я as x -> 1 — 0 and we say then th at l o g ( l - * ) ^ n

H = (hnk) generated by is defined by / \ n~k

(n , h — 0,1, 2, ...). I t is well known th a t Hausdorff matrices are commu­

tative and th a t the Cesàro method Ca (a > —1) is a Hausdorff matrix +d \ ^

I . (For these and other details about Hausdorff methods, see [2].) As usual, we shall use the same symbol for a m atrix and the sequence-to-sequence transformation defined by it. For a < —1 let us define Ca as (C_a)“ 1. I t is to be remarked, however, th a t the definition

€ a = ((7_J_1 for all a < 0 (which differs from the previous one if —1

< a < 0) is also consistent with the statements in Section 2 below. Given a real number a and a (real or complex) sequence {sn}, let {$"} denote the (7a-transform of {sn} ; we shall say th a t {sn} is summable by the product

* This paper is part of the work done by the author during his stay at the Uni­

versity of Stuttgart. The author wishes to thank the University and Professor W. Meyer- Konig in particular for their hospitality and kindness, and the Deutscher Akademischer Austauschdienst for an invitation to visit Germany.

method. (LGa) if the sequence {«“} is (L)-summable. The method (LGa) is linear, and is regular if a > 0 (see H ardy [2], p. 81 and Lemma 2 given below).

2. I t is the object of this note to prove the

T

. Let {$n} be a sequence of real or complex numbers and a, real numbers such that

(1) {sn} is (LGa)-summable to X (i.e. {«“} is (L)-summable to X), and

(2) logfl- = 0(1).

Then s ^ - ^ X as n~^oo, i.e. {sw} is G^~1-summable to X.

To prove the theorem we shall make use of the following lemmas, hfote th a t all sequences we consider may be real or complex.

L

1. I f is (L)-summable to X and n(sn — sn_l)\ogn = 0(1), then sn -+X as n oo.

P ro o f. This is easily derived from the one-sided Tauberian theorem given by Eangachari and Sitaraman [4].

(Ishiguro [3], the first to prove a Tauberian theorem for the method (L), had о instead of О in the result stated above.)

L

2 (Borwein [1]). Let H be any regular Hausdorff matrix and a sequence (L)-summable to X. Then the H-transform of {sn} is also (L)-summable to X.

L

3. I f p < q, then {s i s the transform of {s%} by a regular Haus- dorff matrix; that is, Gq = HGP, where H is regular Hausdorff.

P ro o f. By definition of the Cesàro methods, H — Са(Ср)~г exists and is Hausdorff, with Gq = HGp . The regularity of H is verified as follows.

Case (i): q > p > —1. This is well known (see e.g. [2], Theorem 43).

Case (ii): q > —1 Then H = Gq(G

= G qG_p. Since —p-\- q> 0 and q > —1, it follows th at II is equivalent to G_p+q, a regular Cesàro method (see e.g. [5], p. 109). Case (iii): —1 > q > p. Then — p > — q > 1 and hence H — С

(Ср)~г = (0_g)~1C'_:p is regular.

L

4. For each real number p, the methods Gp and G

G

are equivalent. I n other words, there exists a Hausdorff method H equivalent to the identity (i.e. H and its inverse (H

are both regular Hausdorff) such that Gp = HG

G

= G

HGP_1.

P ro o f. Case (i): p > 0. This is well known. Case (ii): — 1 < p ^ 0.

Then by Case (i) there exist regular Hausdorff matrices А, В, А ~ г and B -1 such th at

Tauberian theorem

Then the lemma is satisiied if we take H — A B h Case (iii): p < —1.

Sow the proof in Case (ii) applies with B = I, the identity itself.

L

5. For any sequence {sn}

sn- s l = п{

гп- 4 -

) for all n ^ O (where, as usual, sLj is defined as 0).

P ro o f. From the definition of sxn we get

= (w + l ) s i - w s i _ i - 4 =

P r o o f of th e th e o re m . First we observe th at it is enough to prove th at {s is convergent. For, if {tn} is the sequence whose Gt-transform is the sequence {$£}, then Lemma 5 and condition (2) of the theorem

show th at x

K ~ 8n 1)->0 as n ^ o o .

Hence {tn} converges to Я if and only if {sp n} converges to Я. But writing {tn} = H-transform of as in Lemma 4, we see th a t {tn} converges to Я if and only if converges to Я. Thus

(3) sn~>^ (as n-+oo) implies th a t «£- 1 -»Я as n-^o o.

We now proceed to prove th a t the sequence {s^} converges to Я.

C ase (i). a = /?. The theorem reduces to Lemma 1.

C ase (ii). a < /?. Then {s^} is (L)-summable to Я, by Lemmas 2 and 3;

the desired result now follows at once from Lemma 1.

C ase (iii). a > /?. Let Tc be the smallest positive integer such th a t a + i/? |. By Lemmas 2 and 3 and condition (1) of the theorem we have :

(4) {sfj is (XC^.)-summable to Я.

Since condition (2) of the theorem implies th at n(s^ — ^ _ x)-> 0 as n->oo, we see by Lemma 5 th a t

(5) tn~ sn~>® ш п - > о о .

The argument for the passage from the statement (4) to the statement (6) can be repeated a finite number of times to obtain th at

(7) {s^} is (L)-summable to X.

[We remark th a t the condition

was used initially only to obtain (4) ; and only the condition Тс > 0 was needed to derive (6) from (4).]

From (7), (2) and Lemma 1 we see th a t {s converges to A. In view of (3), this completes the proof of the theorem.

[1] D. B orw ein, A logarithmic method of summability, J. London Math. Soc. 33 (1958), p. 212-220.

[2] G. H. H ardy, Divergent series, Oxford 1949.

[3] K. Ish igu ro, Tauberian theorems concerning the summability methods of logarithmic type, Proc. Japan Acad. 39 (1963), p. 156-159.

[4] M. S. R a n gach ari and Y. Sitaram an , Tauberian theorems for logarithmic summability ( L ), Tohôku Math. J. 16 (1964), p. 257-269.

— —--- > ---tends to a limit Я as x -> 1 — 0 and we say then th at l o g ( l - ) ^ n*

(n , h — 0,1, 2, ...). I t is well known th a t Hausdorff matrices are commu

* This paper is part of the work done by the author during his stay at the Uni