) method of summability was introduced in 2013 by Natarajan. In the paper some new proper es of this method are studied.

New proper es of the Natarajan method of summability

P. N. Natarajan

Summary The (M ^, λ

) method of summability was introduced in 2013 by Natarajan. In the paper some new proper es of this method are studied.

Keywords regular matrix;

the (M, λ

) method of summability;

l-stronger;

l-equivalent;

ordered abelian semigroup;

infinite chain

MSC 2010

40C05; 40D05; 40G99 Received: 2014/05/08; Accepted: 2014/10/26

1. Introduc on and preliminaries

Throughout, the entries of matrices, sequences and series are real or complex numbers. To make the paper self-contained, we recall the following. Given an infinite matrix A ≡ (a

), n , k ∈ N

∶= {0 ^, 1 , 2 , . . . }, and a sequence x = {x

}, k ∈ N

, by the A-transform of x = {x

} we mean the sequence A (x) = {(Ax)

},

(Ax)

= ∑

k=0

a

x

, n ∈ N

,

where we assume that the series on the right converges. If lim

(Ax)

= s, we say that x = {x

} is A-summable or summable A to s. If lim

(Ax)

= s whenever lim

x

= s, we

P. N. Natarajan Old No. 2/3, New No. 3/3, Second Main Road, R.A. Puram, Chennai 600 028, India (e-mail: pinnangudinatarajan@gmail.com)

DOI 10.14708/cm.v55i1.810 © 2015 Polish Mathema cal Society

say that A is regular. The following result, which characterizes a regular matrix in terms of its entries, is well known (see, for instance, [2]).

1.1. Theorem. The infinite matrix A ≡ (a

) is regular if and only if

(i) sup

n⩾0

∞

∑

k=0

∣a

∣ < ∞ ^,

(ii) lim

n→∞

a

= 0 ^, k ∈ N

, (iii) lim

n→∞

∞

∑

k=0

a

= 1.

Let ℓ denote the Banach space of all sequences x = {x

} such that ∑

∣x

∣ < ∞. We write A ≡ (a

) ∈ ( ^{ℓ, ℓ} ) if {(Ax)

} ∈ ^ℓ whenever x = {x

} ∈ ^ℓ . The following result is due to Knopp and Lorentz (see [3]):

1.2. Theorem. A ∈ ( ^{ℓ, ℓ} ) if and only if

sup

k⩾0

∞

∑

n=0

∣a

∣ < ∞ ^.

If A , B are infinite matrices such that lim

(Ax)

= s implies that lim

(Bx)

= s, then we say that A is included in B (or B includes A) and write A ⊆ B.

2. Main results

The (M , λ

) method of summability was introduced by Natarajan in [ 4] and some of its proper es were studied in [4–6].

2.1. Defini on. Let {λ

} be a sequence such that ∑

∣λ

∣ < ∞. The (M ^, λ

) method is defined by the infinite matrix (a

), where

a

= ⎧⎪⎪ ⎪⎨ ⎪⎪⎪ ⎩

λ

, k ⩽ n ^, 0 , k > n ^.

2.2. Remark. Note that the (M ^, λ

) method reduces to the Y-method when λ

= λ

=

and λ

= 0, n ⩾ 2.

The following result is known (see [4], Theorem 2.3).

2.3. Theorem. The (M , λ

) method is regular if and only if

∞

∑

n=0

λ

= 1 ^.

Following Dafranza [1], we will study some proper es of the (M ^, λ

) method. For convenience, we write (M ^, λ ) for (M ^, λ

). Let M denote the set of all (M ^, λ ) methods. First note the following.

2.4. Theorem. For any method (M ^, λ ) ∈ M, (M ^, λ ) ∈ ( ^{ℓ, ℓ} ).

Let ℓ ((M ^, λ )) denote the set of all sequences x = {x

} such that (M ^, λ )(x) ∈ ^ℓ . 2.5. Defini on. Given two methods (M ^, λ ), (M ^, μ ) in M, we say that

– (M ^, μ ) is ^ℓ -stronger than (M ^, λ ) if ^ℓ ((M ^, λ )) ⊆ ^ℓ ((M ^, μ ));

– (M ^, μ ) is strictly ^ℓ -stronger than (M ^, λ ) if ^ℓ ((M ^, λ )) ⫋ ^ℓ ((M ^, μ ));

– (M ^, λ ) and (M ^, μ ) are ^ℓ -equivalent if ℓ ((M ^, λ )) = ^ℓ ((M ^, μ )).

Given the methods (M , λ), (M , μ) in M, we formally define

λ (x) = ∑

n=0

λ

x

, μ (x) = ∑

n=0

μ

x

and

a (x) = λ (x) μ (x) = ∑

n=0

a

x

,

b (x) = μ (x) λ (x) =

∞

∑

n=0

b

x

.

Following an argument similar to the one used in [2, Theorem 18], we can prove the following results:

2.6. Theorem. If (M ^, λ ) ∈ M, then the series ∑

λ

x

converges for ∣x∣ < 1.

2.7. Theorem. If (M , λ ) , (M , μ ) ∈ M, then ∑

a

x

, ∑

b

x

have posi ve radii of conver- gence and

(i) λ

= a

μ

+ a

μ

+ ⋅ ⋅ ⋅ + a

μ

, (ii) μ

= b

λ

+ b

λ

+ ⋅ ⋅ ⋅ + b

λ

.

Further, if s = {s

} ∈ ^ℓ ((M ^, λ )), then the series s(x) = ∑

s

x

has posi ve radius of conver-

gence.

Given sequences λ = {λ

}, μ = {μ

}, let λ ∗ μ = {g

} denote their Cauchy product, i.e., g

= ∑

λ

μ

, n ∈ N

.

2.8. Defini on. Given (M ^, λ ) ^, (M ^, μ ) ∈ M, we say that (M ^, λ ∗ μ) is the symmetric product of (M ^, λ ) and (M ^, μ ).

In the context of Defini on 2.8, ∑

∣λ

∣ < ∞, ∑

∣μ

∣ < ∞, and so ∑

∣g

∣ < ∞ in view of Abel’s theorem on Cauchy mul plica on of absolutely convergent series. Conse- quently, (M , λ ∗ μ) ∈ M.

We will need the following result.

2.9. Lemma. For sequences λ = {λ

}, μ = {μ

}, let γ = λ ∗ μ and let (M ^, λ ), (M ^, γ ) ∈ M. Then

ℓ ((M ^, λ )) ⊆ ^ℓ ((M ^, γ )) if and only if μ ∈ ^ℓ .

Proof. For any sequence x = {x

},

((M ^, γ )(x))

= γ

x

+ γ

x

+ ⋅ ⋅ ⋅ + γ

x

= (λ

μ

+ λ

μ

+ ⋅ ⋅ ⋅ + λ

μ

)x

+ (λ

μ

+ λ

μ

+ ⋅ ⋅ ⋅ + λ

μ

)x

+ ⋅ ⋅ ⋅ + (λ

μ

)x

= (λ

x

+ λ

x

+ ⋅ ⋅ ⋅ + λ

x

)μ

+ (λ

x

+ λ

x

+ ⋅ ⋅ ⋅ + λ

x

)μ

+ ⋅ ⋅ ⋅ + (λ

x

)μ

= ((M ^, λ )(x))

μ

+ ((M ^, λ )(x))

μ

+ ⋅ ⋅ ⋅ + ((M ^, λ )(x))

μ

= ∑

k=0

t

((M ^, λ )(x))

,

where

t

= ⎧⎪⎪ ⎪⎨ ⎪⎪⎪ ⎩

μ

, k ⩽ n ^, 0 , k > n ^.

In view of Theorem 1.2, ℓ ((M , λ )) ⊆ ℓ ((M , γ )) if and only if ∑

∣μ

∣ < ∞, i.e., μ ∈ ℓ , which completes the proof.

We deduce an inclusion result:

2.10. Theorem. Given the methods (M ^, λ ) ^, (M ^, μ ) ∈ M,

ℓ ((M ^, λ )) ⊆ ^ℓ ((M ^, μ )) if and only if b = {b

} ∈ ^ℓ.

Proof. In Lemma 2.9 we replace the sequence μ by the sequence b, so γ = λ ∗ b = μ, and hence

ℓ ((M ^, λ ) ⊆ ^ℓ ((M ^, μ )) if and only if b = {b

} ∈ ^ℓ .

2.11. Corollary. Let (M ^, λ ) ^, (M ^, μ ) ∈ M. Then

(i) ℓ ((M ^, λ )) = ^ℓ ((M ^, μ )) if and only if {a

} ∈ ^ℓ and {b

} ∈ ^ℓ , (ii) ℓ ((M ^, λ )) ⫋ ^ℓ ((M ^, μ )) if and only if {a

} /∈ ^ℓ and {b

} ∈ ^ℓ .

2.12. Corollary. For (M ^, λ ) ∈ M, let h(x) =

. Then ℓ ((M ^, λ )) = ^ℓ if and only if {h

} ∈ ^ℓ . Proof. Let I be the iden ty mapping, so ℓ (I) = ^ℓ . Now, I (x) = ∑

i

x

= 1, i

= 1, i

= 0, n ⩾ 1. Then a(x) =

= λ(x) and b(x) =

=

= h(x). From Theorem 2.10 it follows that {h

} ∈ ^ℓ .

2.13. Corollary. Let (M ^, λ ) ^, (M ^, μ ) ∈ M and γ = λ ∗ μ. Then ^ℓ ((M ^, λ )) ⊆ ^ℓ ((M ^, γ )).

2.14. Theorem. Let (M ^, λ )) ∈ M and let μ = {λ

, λ

, λ

, . . . }, λ

≠ λ

. Then (M ^, μ ) ∈ M and ℓ ((M , λ) ∩ ℓ ((M , μ)) = ℓ.

Proof. Since {λ

} ∈ ^ℓ , it follows that {μ

} ∈ ^ℓ , hence the method (M ^, μ ) ∈ M. For any sequence x = {x

},

((M ^, μ )(x))

= μ

x

+ μ

x

+ ⋅ ⋅ ⋅ + μ

x

= λ

x

+ λ

x

+ ⋅ ⋅ ⋅ + λ

x

= (λ

x

+ λ

x

+ ⋅ ⋅ ⋅ + λ

x

) + (λ

− λ

)x

= ((M ^, λ )(x))

+ (λ

− λ

)x

. (1) Let {x

} ∈ ^ℓ . Since (M ^, λ ) ^, (M ^, μ ) ∈ ( ^{ℓ, ℓ} ), hence by Theorem 2.4, {((M ^, λ )(x))

} ∈ ^ℓ and {((M ^, μ )(x))

} ∈ ^ℓ , so {x

} ∈ ^ℓ ((M ^, λ )) ∩ ^ℓ ((M ^, μ )). Consequently, ^ℓ ⊆ ^ℓ ((M ^, λ )) ∩ ^ℓ ((M ^, μ )). Let {x

} ∈ ^ℓ ((M ^, λ )) ∩ ^ℓ ((M ^, μ )). Then {((M ^, λ )(x))

} ∈ ^ℓ and {((M ^, μ )(x))

} ∈ ^ℓ . Now, from (1) it follows that {x

} ∈ ^ℓ . Thus

ℓ ((M ^, λ ) ∩ ^ℓ ((M ^, μ )) ⊆ ^ℓ,

and so ℓ ((M ^, λ )) ∩ ^ℓ ((M ^, μ )) = ^ℓ , which completes the proof of the theorem.

3. M as an ordered abelian semigroup

We recall that M denotes the set of all (M ^, λ ) methods. In this sec on, we prove that M is an ordered abelian semigroup, the order rela on being the set inclusion between summability fields of type ℓ ((M ^, λ )), and the binary opera on being the symmetric product ∗.

3.1. Lemma. If (M ^, λ ) ^, (M ^, μ ) ∈ M and γ = λ ∗ μ, then (M ^, γ ) ∈ M and ℓ ((M ^, λ )) ∪ ^ℓ ((M ^, μ )) ⊆ ^ℓ ((M ^, γ )) ^.

Proof. Since ∑

∣λ

∣ < ∞ and ∑

∣μ

∣ < ∞, from Abel’s theorem we have ∑

∣γ

∣ < ∞, so (M ^, γ ) ∈ M. By Corollary 2.13,

ℓ ((M , λ )) ∪ ℓ ((M , μ )) ⊆ ℓ ((M , γ )) .

3.2. Lemma. Let (M ^, λ ) ^, (M ^, μ ) ^, (M ^, γ ) ∈ M. Let θ = λ ∗ γ and ϕ = μ ∗ γ.

(i) If ℓ ((M ^, λ )) ⊆ ^ℓ ((M ^, μ )), then ^ℓ ((M ^, θ )) ⊆ ^ℓ ((M ^, ϕ )).

(ii) If ℓ ((M ^, λ )) ⫋ ^ℓ ((M ^, μ )), then ^ℓ ((M ^, θ )) ⫋ ^ℓ ((M ^, ϕ )).

Proof. Let b (x) =

and c (x) =

. By Theorem 2.10, {b

} ∈ ^ℓ . We claim that {c

} ∈ ^ℓ . First we note that

θ (x) = λ(x)γ(x) and ϕ(x) = μ(x)γ(x) ^, so

∞

∑

n=0

c

x

= c(x) = ϕ (x)

θ(x) = μ (x)γ(x) λ(x)γ(x) = μ (x)

λ(x) = ∑

n=0

b

x

.

Thus c

= b

, n ∈ N

, and hence {c

} ∈ ^ℓ . Using Theorem 2.10 again, we have

ℓ ((M ^, θ )) ⊆ ^ℓ ((M ^, ϕ )) ^. The second part of the theorem follows by Corollary 2.11(ii).

Therefore, we can state the main result of this sec on:

3.3. Theorem. With “strictly ℓ -weaker than” as the order rela on and the symmetric product ∗ as the binary opera on, M is an ordered abelian semigroup.

3.4. Lemma. Let (M ^, λ ) ∈ M. Define λ

= 0, μ

=

, n ⩾ 0. Then (M ^, μ ) ∈ M and

ℓ ((M ^, λ )) ⫋ ^ℓ ((M ^, μ )).

Proof. Since {λ

} ∈ ^ℓ , {μ

} ∈ ^ℓ , hence (M ^, μ ) ∈ M. Now,

μ (x) = ∑

n=0

μ

x

= ∑

n=0

λ

+ λ

2 x

= 1 + x 2 λ (x) ^. It is clear that

ℓ ((M ^, λ )) ⫋ ^ℓ ((M ^, μ )) in view of Corollary 2.11(ii).

We conclude the paper with the following interes ng result:

3.5. Theorem. There are infinite chains of (M ^, λ ) methods in M.

Proof. Let {λ

} be a sequence such that ∑

∣λ

∣ < ∞. Let λ

(x) = ∑

λ

x

. Then (M ^, λ

) ∈ M. For n ⩾ 2, define

λ

(x) = ( 1 + x 2 )

n−1 ∞

∑

k=0

λ

x

.

Then (M ^, λ

) ∈ M, n ⩾ 2. Applying Lemma 3.4 repeatedly, we have ℓ ((M ^, λ

)) ⫋ ^ℓ ((M ^, λ

)) ⫋ ⋅ ⋅ ⋅ ⫋ ^ℓ ((M ^, λ

)) ⫋ ^{. . . ,} which completes the proof of the theorem.

References

[1] J. Defranza, An ordered set of Nörlund means, Internat. J. Math. Math. Sci. 4 (1981), 353–364, DOI 10.1155/S0161171281000215.

[2] G.H. Hardy, Divergent Series, Oxford University Press, Oxford 1949.

[3] K. Knopp and G.G. Lorentz, Beiträge zur absoluten Limi erung, Arch. Math. 2 (1949), 10–16.

[4] P. N. Natarajan, On the (M ^, λ

) method of summability, Analysis (München) 33 (2013), no. 2, 51–56.

[5] P. N. Natarajan, A product theorem for the Euler and the Natarajan methods of summability, Analysis (München) 33 (2013), no. 2, 189–195.

[6] P. N. Natarajan, Cauchy mul plica on of (M ^, λ

) summable series, Advancement and Development in Mathema cal Sciences 3 (2012), no. 1–2, 39–46.

© 2015 Polish Mathema cal Society