GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 6, 1999, 591-600

ABSOLUTE SUMMABILITY FACTORS AND ABSOLUTE TAUBERIAN THEOREMS FOR DOUBLE SERIES AND

SEQUENCES

SH. YANETZ (CHACHANASHVILI)

Abstract. Let A and B be the linear methods of the summability of double series with fields of bounded summability A′_{b} and B′_{b}, respectively. Let T be certain set of double series. The condition x ∈ T is called B_{b}-Tauberian for A if A′_{b} ∩ T ⊂ B′_{b}.

Some theorems about summability factors enable one to find new B_{b}-Tauberian conditions for A from the already known B_{b}-Tauberian conditions for A.

The first paper on connections between summability factors and Tauberian theorems for simple series and sequences was by Kangro [1] who proved three theorems where conditions were formulated in terms of summability factors. The theorems of Kangro [1] and Baron [2, §27] are generalizations of the theorems of Meyer-König and Tietz [3]–[5] and Leviatan [6].

To prove absolute Tauberian theorems with the aid of absolute summability factors for double series and sequences we need to generalize some important theorems about absolute summability factors for simple series and sequences.

1. Basic Definitions and the Main Lemma

The following notation and definitions will be used in this section.

Let

∑_{m,n=0}^{∞} u_{mn}  (1.1)

591 be an infinite double series of real or complex numbers with partial sums

1991 Mathematics Subject Classification. 40B05, 40D15, 40E05.

Key words and phrases. Double sequences and series, Tauberian conditions, summability factors, summability methods.

1072-947X/99/1100-0591$16.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

U_{mn} = ∑_{k,ℓ=0}^{m,n} u_{kℓ}.

Let A = (a_{mnkℓ}) be an infinite normal double matrix of real or complex numbers.

Let

U′_{mn} = ∑_{k,ℓ=0}^{m,n} a_{mnkℓ}U_{kℓ}.

We say that (1.1) is summable by the method A, or, shortly, (1.1) is A-summable to the sum U′ if there exists the limit

lim_{m,n} U′_{mn} = U′.  (1.2)

In that case we call the matrix A a method of summability.

We call a double series (1.1) boundedly A-summable, or, shortly, A_{b}- summable to the sum U′ if U′_{mn} satisfies conditions (1.2) and U′_{mn} = O(1). In that case we also denote the method of summability A by A_{b}. Analogously, the B_{b}-summability will be defined with the aid of the double matrix B = (b_{mnkℓ}).

Let A′_{b} be the set of all A_{b}-summable double series. If (1.1) is A_{b}- summable to the sum U′, we write

A_{b}{ ∑_{m,n} u_{mn}} = U′.

The number U′ is called the A-sum of the double series (1.1).

If for every absolutely convergent double series (1.1) is |A|-summable (see [7], p. 141), then we say that the method A conserves absolute convergence. If the method A conserves absolute convergence and

A_{b}{ ∑_{m,n} u_{mn}} = ∑_{m,n} u_{mn}

for each absolute convergent double series (1.1), then the method A is called absolute regular.

The set of all |A|-summable double series is called the field of absolute summability of A and we denote it by A′_{ℓ} or |A|′. If

B_{b}{ ∑_{m,n} u_{mn}} = A_{b}{ ∑_{m,n} u_{mn}}

for each double series from |B|′ ∩ |A|′, then we say that A_{ℓ} is consistent with B_{ℓ}.

In the sequel we use the notation:

\overline{Δ}_{m}x_{mn} = x_{mn} − x_{m−1,n,}
\overline{Δ}_{n}x_{mn} = x_{mn} − x_{m,n−1,}
\overline{Δ}_{mn}x_{mn} = \overline{Δ}_{m}(\overline{Δ}_{n}x_{mn}) = \overline{Δ}_{n}(\overline{Δ}_{m}x_{mn}) =
= x_{mn} − x_{m−1,n} − x_{m,n−1} + x_{m−1,n−1}.

Let ρ be an arbitrary set of double series (1.1). We denote by r the set of double sequences x = (U_{kℓ}) associated with ρ, i.e.,

r = \{x : ∑_{k,ℓ} \overline{Δ}_{kℓ}U_{kℓ} ∈ ρ\}.

For example, if ρ is the set of bounded convergent double series, then r = bc, the set of bounded convergent double sequences; if ρ is the set bγ_{0} of bounded double series converging to zero, then r = bcn, the set of bounded double sequences converging to zero; if ρ is the set μ of double series with bounded partial sums, then r is the set m of bounded double sequences; if ρ is the set ℓ of absolutely convergent double series, ℓ_{0} absolutely convergent double series converging to zero, then r is the set a of absolutely converging double sequences, i.e., a = {x : U_{kℓ} = Ω(1)}. Here x_{mn} = Ω(y_{mn}) denotes ∑_{m,n} |\overline{Δ}_{mn}(x_{mn}/y_{mn})| < ∞. If ρ is the set ℓ_{0} of double series absolutely

converging to zero, then r is the set a_{0} of double sequences x, for which U_{kℓ} = ω(1). Here x_{mn} = ω(y_{mn}) denotes x_{mn} = Ω(y_{mn}) and x_{mn} = o(y_{mn}).

We will use the notation:

x_{mn} = Ω_{m}(y_{mn}) means ∑_{m} |\overline{Δ}_{m}(x_{mn}/y_{mn})| = O(1),

x_{mn} = Ω_{n}(y_{mn}) means ∑_{n} |\overline{Δ}_{n}(x_{mn}/y_{mn})| = O(1),

x_{mn} = ω_{m}(y_{mn}) means x_{mn} = Ω_{m}(y_{mn}), x_{mn} = o(1), x_{mn} = ω_{n}(y_{mn}) means x_{mn} = Ω_{n}(y_{mn}), x_{mn} = o(1).

The numbers (ε_{mn}) are called summability factors of type (ρ, B_{ℓ}) (or (ρ, B_{b})) if for every double series ∑_{m,n} u_{mn} ∈ ρ it always follows that

∑_{m,n} ε_{mn}u_{mn} ∈ B′_{ℓ} (or ∑_{m,n} ε_{mn}u_{mn} ∈ B′_{b}).

Let A and B be the methods of the summability of double series (1.1). Let T and T_{0} be certain sets of double series (1.1). We denote three double series (1.1) by x, y and z.

The condition x ∈ T will be called B_{ℓ}-Tauberian for A_{ℓ} if

A′_{ℓ} ∩ T ⊂ B′_{ℓ}.

In particular, when the convergence method E = C^{0,0} (see [7], p. 141), i.e., if B′_{ℓ} = E′_{ℓ} = ℓ, instead of an E_{ℓ}-Tauberian condition, it will simply be called an absolute Tauberian condition.

Now we formulate the main lemma.

Lemma 1.1. Let A′_{ℓ} ⊃ B′_{ℓ} and A_{ℓ} be consistent with B_{ℓ}. If the condition x ∈ T_{0} is B_{ℓ}-Tauberian for A_{ℓ}, then the condition x ∈ T is also B_{ℓ}- Tauberian for A_{ℓ} if every element x ∈ T can be represented by x = y + z, where y ∈ T_{0} and z ∈ B′_{ℓ}.

Proof. Let x ∈ A′_{ℓ} ∩ T and x = y + z with y ∈ T_{0} and z ∈ B′_{ℓ}. Then clearly y = x − z ∈ A′_{ℓ} ∩ T_{0} ⊂ B′_{ℓ}. Hence x ∈ B′_{ℓ}.

Let the numbers λ_{mn} ≠ 0, μ_{mn} ≠ 0, where \overline{Δ}_{mn}μ_{mn} ≠ 0 and μ_{−1,n} = μ_{m,−1} = 0.

Let us assume that μ_{mn} and λ_{mn} are factorizable, i.e.,

μ_{mn} = μ′_{m} · μ′′_{n}, λ_{mn} = λ′_{m} · λ′′_{n},

and for m, n > 0

h′ = \overline{Δ}_{m}μ′_{m}/λ′_{m}, h′′ = \overline{Δ}_{n}μ′′_{n}/λ′′_{n}.

Then

\frac{\overline{Δ}_{n}μ_{m−1,n}}{λ_{mn}} = \frac{μ′_{m−1}}{λ′_{m}} · \frac{\overline{Δ}_{n}μ′′_{n}}{λ′′_{n}} , \frac{\overline{Δ}_{m}μ_{m,n−1}}{λ_{mn}} = \frac{\overline{Δ}_{m}μ_{m}}{λ′_{m}} · \frac{μ′′_{n−1}}{λ′′_{n}} .

2. Connections Between Absolute Summability Factors for Double Series and Absolute Tauberian Theorems

For every double series (1.1) a double sequence (V_{mn}) will be constructed by the formula

V_{mn} = \frac{1}{μ_{mn}} ∑_{k,ℓ=0}^{m,n} λ_{kℓ}u_{kℓ}.  (2.1)

Recall

α_{mn} = μ_{m−1,n−1}/λ_{mn}, γ_{mn} = (\overline{Δ}_{mn}μ_{mn})/λ_{mn},
β′_{mn} = (\overline{Δ}_{n}μ_{m−1,n})/λ_{mn}, β′′_{mn} = (\overline{Δ}_{m}μ_{m,n−1})/λ_{mn}.

One can now state

Theorem 2.1. Let A′_{ℓ} ⊃ B′_{ℓ}, where A_{ℓ} is consistent with B_{ℓ}. If the following conditions are fulfilled:

(1) the numbers α_{mn} are summability factors of type (ρ, B_{ℓ}),

(2) the double series ω′ = ∑_{m,n} β′_{mn}\overline{Δ}_{m}V_{mn} and ω′′ = ∑_{m,n} β′′_{mn}\overline{Δ}_{n}V_{mn} are

boundedly |B|-summable,

(3) condition (u_{mn}/γ_{mn}) ∈ r is B_{ℓ}-Tauberian for A_{ℓ}, then the condition

(V_{mn}) ∈ r  (2.2)

is also B_{ℓ}-Tauberian for A.

Proof. From (2.1) we obtain

u_{mn} = \frac{1}{λ_{mn}} \overline{Δ}_{mn}(μ_{mn}V_{mn}).

We use the formula (see formula (15.31) in [2]) for the difference of products of double sequences and obtain

u_{mn} = α_{mn}\overline{Δ}_{mn}V_{mn} + β′_{mn}\overline{Δ}_{m}V_{mn} + β′′_{mn}\overline{Δ}_{n}V_{mn} + γ_{mn}V_{mn}. (2.3)

With the aid of the series

x = ∑_{m,n} u_{mn}, y = ∑_{m,n} γ_{mn}V_{mn}, z = ω + ω′ + ω′′,

where

ω = ∑_{m,n} α_{mn}\overline{Δ}_{mn}V_{mn}, ω′ = ∑_{m,n} β′_{mn}\overline{Δ}_{m}V_{mn}, ω′′ = ∑_{m,n} β′′_{mn}\overline{Δ}_{n}V_{mn},

we get x = y + z from (2.3).

If condition (2.2) is satisfied, then ∑_{m,n} \overline{Δ}_{mn}V_{mn} ∈ ρ, and from conditions

(1) and (2) it follows that z ∈ B′_{ℓ}. We denote

T_{0} = { ∑_{m,n} u_{mn} : (u_{mn}/γ_{mn}) ∈ r}.

From (2.3) we conclude that y ∈ T_{0}. According to (3), the condition x ∈ T_{0} is B_{ℓ}-Tauberian for A_{ℓ}. Thus the statement of Theorem 2.1 follows from Lemma 1.1 if one sets

T = { ∑_{m,n} u_{mn} : (V_{mn}) ∈ r}.

If in Theorem 2.1 we take B = E and r = a, then (ρ, B_{ℓ}) = (ℓ, ℓ). It is known (see [8], p. 131, Theorem 3) that ε_{mn} ∈ (C_{ℓ}^{α,β}, C_{ℓ}^{α,β}) for α, β ≥ 0 iff the following conditions

Δ_{mn}^{α,β}ε_{mn} = O[(m + 1)^{−α}(n + 1)^{−β}], Δ_{m}^{α}ε_{mn} = O[(m + 1)^{−α}],
Δ_{n}^{β}ε_{mn} = O[(n + 1)^{−β}], ε_{mn} = O(1)

are fulfilled. From here, with α = β = 0, we obtain following

Lemma 2.2. For ε_{mn} ∈ (|ℓ|, |ℓ|) it is necessary and sufficient that

ε_{mn} = O(1).

The proof of the condition ω′ = O(1) in the proof of Corollary 4.1 from [9], p. 160, and Lemma 2.2 imply

Corollary 2.3. Let A be an absolutely regular method for the summability of double series. If the following conditions are fulfilled:

(1) μ_{m−1,n−1} = O(λ_{mn}),

(2) ∑_{m,n} |h_{mn}| < ∞,

(3) ∑_{m,n} |\frac{μ′_{m−1}}{λ′_{m}} \overline{Δ}_{m}V_{mn}| = O(1), ∑_{mn} |\frac{μ′′_{n−1}}{λ′′_{n}} \overline{Δ}_{n}V_{mn}| = O(1),

(4) condition u_{mn} = Ω(γ_{mn}) is absolute Tauberian for A_{ℓ}, then V_{mn} = Ω(γ_{mn}) is also absolute Tauberian for A_{ℓ}.

As we see by Lemma 2.2, ε_{mn} ∈ (ℓ, ℓ) iff ε_{mn} = O(1). Since ℓ_{0} ⊂ ℓ, the conditions for ε_{mn} ∈ (ℓ, ℓ) are necessary and sufficient for ε_{mn} ∈ (ℓ_{0}, ℓ) as well. Therefore ε_{mn} ∈ (ℓ_{0}, ℓ) iff ε_{mn} = O(1). Thus analogously to Corollary 2.3 we obtain

Corollary 2.4. Let A be an absolutely regular method of the summability for double series. If the conditions (1), (2), and (3) of Corollary 2.3 are fulfilled and

(4) the condition u_{mn} = ω(γ_{mn}) is absolute Tauberian for A_{ℓ}, then V′_{mn} = ω(γ_{mn}) is also absolute Tauberian for A_{ℓ}.

Analogous theorems for simple series were proved by S. Baron (see [2], p. 234, Corollary 27.3).

3. Connections Between Absolute Summability Factors for Double Sequences and Absolute Tauberian Theorems

In this section we shall find connections between summability factors for a double sequence and absolute Tauberian theorems.

In (2.1) let

λ_{mn} = a_{mn}b_{mn}.

With the notation used in Theorem 2.1 we get

Theorem 3.1. Let A′_{ℓ} ⊃ B′_{ℓ}, where A_{ℓ} is consistent with B_{ℓ}. If the following conditions are fulfilled:

(1) the numbers α_{mn} are summability factors of type (ρ, B_{ℓ}),

(2) the numbers \overline{Δ}_{mn}μ_{mn}/b_{mn} are summability factors for a double sequence of type (r, r),

(3) the double series ω′ and ω′′ are B_{ℓ}-summable,

(4) condition (a_{mn}u_{mn}) ∈ r is B_{ℓ}-Tauberian for A_{ℓ}, then the condition

(V_{mn}) ∈ r  (3.1)

is also B_{ℓ}-Tauberian for A_{ℓ}.

Proof. With the aid of the formal series x, y, z as in the proof of Theorem 2.1, expansion (2.3) gives x = y+z. Then, as stated in the proof of Theorem 2.1, from condition (3.1) and conditions (1) and (3) it follows that z ∈ B′_{ℓ}. In addition, from conditions (3.1) and (2) one gets

(\frac{\overline{Δ}_{mn}μ_{mn}}{b_{mn}} V_{mn}) ∈ r.

We set

T_{0} = { ∑_{m,n} u_{mn} : (a_{mn}u_{mn}) ∈ r}

and conclude that y ∈ T_{0} if in T_{0} we take u_{mn} = γ_{mn}V_{mn}. According to (4), the condition x ∈ T_{0} is B_{ℓ}-Tauberian for A_{ℓ}. Thus the statement of Theorem 3.1 follows from Lemma 1.1 if one sets

T = { ∑_{m,n} u_{mn} : (V_{mn}) ∈ r}.

In Theorem 3.1, if we put B′_{ℓ} = ℓ and r = a, then (ρ, B_{ℓ}) = (ℓ, ℓ) and (r, r) = (a, a). Then the convergence factors α_{mn} ∈ (ℓ, ℓ) and are characterized by Lemma 2.2. In condition (2) the convergence factors Δ_{mn}μ_{mn}/b_{mn} ∈ (a, a) for the above-mentioned convergence factors, which we characterize by Lemma 3.2 below, are obtained from Theorem 1 of [7], p. 144.

If we examine B = E in Theorem 1 of [7] and assume that \overline{b}_{mnkℓ} = \overline{Δ}_{mnkℓ}δ_{mnkℓ}, then

∑_{μ,ν=k,ℓ}^{m,n}\overline{b}_{mnμν} \overline{ξ}_{μνkℓ}ε_{μν} = ∑_{μ,ν=k,ℓ}^{m,n} \overline{Δ}_{mn}δ_{mnμν}ε_{μν} =

= \overline{Δ}_{mn} ∑_{μ,ν=k,ℓ}^{m,n} δ_{mnkℓ}ε_{μν} = \overline{Δ}_{mn}ε_{mn}.  (3.2)

By substituting (3.2) into (4) of [7] we obtain

∑_{m,n} |\overline{Δ}_{mn}ε_{mn}| < ∞.  (3.3)

If in (5) of [7], p. 144, we take only the term with k = m and put ℓ = 0, we obtain

∑_{n} | ∑_{ν=0}^{n} b_{mnmν}ε_{mn}| = ∑_{n} |\overline{Δ}_{n}δ_{nν}ε_{mν}| = ∑_{n} |\overline{Δ}_{n}ε_{mν}| = O(1).

Consequently, condition (5) of Theorem 1 in [7] implies the condition

ε_{mn} = Ω_{n}(1)  (3.4)

and, analogously, the condition

ε_{mn} = Ω_{m}(1).  (3.5)

Since by Proposition 1 of [7], p. 144, condition (5) is equivalent to conditions (6), (7), (8) of Proposition 1 in [7], the latter follow from (3.3), (3.4), and (3.5).

Indeed, from (3.4) we obtain

∑_{m,n} | ∑_{μ,ν=0}^{m,ℓ−1} \overline{Δ}_{mn}δ_{mnμν}ε_{μν} | = ∑_{m,n} | ∑_{ν=0}^{ℓ−1} \overline{Δ}_{n}δ_{nν} · \overline{Δ}_{m}ε_{m}ν| =

= O(1) ∑_{n} | ∑_{ν=0}^{ℓ−1} \overline{Δ}_{n}δ_{nν} | = O(1).

Consequently, the above mentioned conditions (7) and (8) follow from (3.4) and (3.5), but in our case condition (6) is condition (3.3).

Thus we have proved

Lemma 3.2. For ε_{mn} ∈ (|𝔈|, |𝔈|) it is necessary and sufficient that conditions (3.4) and (3.5) and ε_{mn} = Ω(1) be fulfilled.

If as above we combine Theorem 3.1 with Lemmas 2.2 and 3.2, and choose ε_{mn} = μ_{m−1,n−1}/γ_{mn} and ε_{mn} = \overline{Δ}_{mn}μ_{mn}/b_{mn} (respectively), we obtain

Corollary 3.3. Let A be an absolute regular method of the summability for double series. If the following conditions are fulfilled:

(1) μ_{m−1,n−1} = O(λ_{mn}),

(2) \overline{Δ}_{m}μ_{mn} = Ω(b_{mn}), \overline{Δ}_{mn}μ_{mn} = Ω_{m}(b_{mn}), \overline{Δ}_{mn}μ_{mn} = Ω_{n}(b_{mn}),

(3) ∑_{m,n} |h_{mn}| < ∞,

(4) ∑_{m} |\frac{μ′_{m−1}}{λ′_{m}} \overline{Δ}_{m}V_{mn}| = O(1) ∑_{n} |\frac{μ′′_{n−1}}{λ′′_{n}} \overline{Δ}_{n}V_{mn}| = O(1)

(5) condition a_{mn}u_{mn} = Ω(1) is absolute Tauberian for A_{ℓ}, then the condition V_{mn} = Ω(1) is also absolute Tauberian for A_{ℓ}.

We use Theorem 3.1 with r = a_{0} and show that the conditions of Lemma 3.2 are necessary and sufficient for ε_{mn} ∈ (a_{0}, a). Indeed, from conditions (3.3)–(3.5) it follows that ε_{mn} = O(1) by the formula (see [10], p. 10)

ε_{mn} = ε_{00} + ∑_{μ,ν=0}^{m−1,n−1} Δ_{μν}ε_{μν} − ∑_{μ=0}^{m−1} Δ_{μ}ε_{μ0} − ∑_{v=0}^{n−1} Δ_{ν}ε_{0ν}, m, n = 1, 2, ….

Therefore the conditions of Lemma 3.2 are necessary and sufficient for ε_{mn} ∈ (a_{0}, a_{0}), since ε_{mn}U_{mn} = O(1)U_{mn} = O(1) · o(1) = o(1), and from (3.3)–(3.5) it follows that (ε_{mn}U_{mn}) ∈ a.

Theorem 3.1 and Lemma 3.2 imply

Corollary 3.4. Let A be an absolute regular method for the summability of double series. If conditions (1), (3), and (4) of Corollary 3.3 and conditions

(2′) \overline{Δ}_{mn}μ_{mn} = ω_{m}(b_{mn}), \overline{Δ}_{mn}μ_{mn} = ω_{n}(b_{mn}),

(5′) condition a_{mn}u_{mn} = ω(1) is absolute Tauberian for A_{ℓ}, are fulfilled, then the condition V_{mn} = ω(1) is also absolute Tauberian for A_{ℓ}.

Analogous theorems for simple series were proved by H. Tietz ([11], p. 139, Theorems 3.1 and 3.3) and S. Baron ([2], p. 236, Corollary 27.4).

References

1. G. Kangro, The weakening of Tauber’s condition. (Russian) Izv. Akad. Nauk. Eston. SSR, Mat. 19(1970), 24–33.

2. S. Baron, Introduction to the Theory of Summability of Series. (Russian) Second edition corrected and completed, Tallinn, Valgus, 1977.

3. W. Meyer-König and H. Tietz, On Tauberian conditions of type σ. Bull. Amer. Math. Soc. 73(1967), 926–927.

4. W. Meyer-König and H. Tietz, Über die Limitierungsumkehrsätze vom Typus σ. Studia Math. 31(1968), 205–216.

5. W. Meyer-König and H. Tietz, Über Umkehrbedingungen in der Limitierungstheorie. Arch. Mat. (Brno) 5(1969), 177–186.

6. D. Leviatan, Remarks on some Tauberian theorems of Meyer-König, Tietz and Stieglitz. Proc. Amer. Math. Soc. 29(1971), 126–132.

7. S. Baron and S. Ianetz, Absolute convergence and summability factors in a double sequence. Indian J. Math. 28(1986), No. 2, 141–153.

8. S. Baron, Summability factors and absoulte summability for double series which are absolutely Cesàro summable. (Russian) Tartu Ül. Toimetised No. 102(1961) 118–134.

9. S. Yanetz, Summability factors and Tauberian theorems for double series. Tartu Ül. Toimetised No. 960(1993), 151–165.

10. G. Kangro and S. Baron, Summability factors for Cesàro-summable and Cesàro-bounded double series. (Russian) Tartu Ül. Toimetised No. 73(1959), 3–49.

11. H. Tietz, Über absolute Tauber-Bedingungen. Math. Z. 113(1970), 136–144.

(Received 26.06.1998)

Author’s address: Department of Mathematics and Computer Science Bar-Ilan University 52900 Ramat-Gan Israel