SUMMABILITY OF DOUBLE SEQUENCES BY WEIGHTED MEAN METHODS AND TAUBERIAN CONDITIONS FOR CONVERGENCE IN PRINGSHEIM

After a brief summary of Tauberian conditions for ordinary sequences of numbers, we consider summability of double sequences of real or complex numbers by weighted mean methods which are not necessarily products of related weighted mean methods in one variable. Our goal is to obtain Tauberian conditions under which convergence of a double sequence follows from its summability, where convergence is understood in Pringsheim’s sense. In the case of double sequences of real numbers, we present necessary and sufficient Tauberian conditions, which are so-called one-sided conditions. Corollaries allow these Tauberian conditions to be replaced by Schmidt-type slow decrease conditions. For double sequences of complex numbers, we present necessary and sufficient so-called two-sided Tauberian conditions. In particular, these conditions are satisfied if the summable double sequence is slowly oscillating.


Introduction.
We begin with a brief and concise summary of the corresponding well-known results for single sequences.For basic facts on summability theory, we refer to [4,9,12] for ordinary sequences and to [1] for double sequences.
Let p = (p k : k = 0, 1, 2,...) be a fixed sequence of nonnegative numbers with p 0 > 0, and set P m := m k=0 p k , k= 0, 1,.... (1.1) Weighted means of a sequence (s k : k = 0, 1,...) of complex numbers are defined by The sequence (s k ) is said to be summable by the weighted mean method determined by the sequence p, in short, summable (N, p), if the sequence (t m ) converges to a finite limit s; in symbols, s m → s(N, p).The summability method (N, p) is said to be regular, if s m → s implies s m → s(N, p).It is well known that (N, p) is regular if and only if P m → ∞ as m → ∞ (see, e.g., [9, page 16]), which we assume in the following.
We are interested in converse conclusions.In general, s m → s(N, p) implies s m → s only under additional so-called Tauberian conditions.Set a k := s k − s k−1 , k= 0, 1,... ; s −1 := 0. (1.3) Then, each of the following conditions is Tauberian for the method (N, p): ) since they yield t m − s m → 0 as m → ∞.The o-type condition (1.6) can be weakened to an O-type condition or even to a one-sided condition where c is a positive constant and in the last case we suppose that p also satisfies the condition p m /P m → 0 as m → ∞.
A necessary and sufficient Tauberian condition was given in [8], which is implied by either of the conditions (1.4)- (1.8).To present it, we recall the following two definitions.Let λ = (λ(m)), where λ(m) > m for all m, be an increasing sequence of natural numbers such that lim inf and denote by Λ u the set of all such sequences λ.Similarly, let µ = (µ(m)), where µ(m) < m for all m, be a nondecreasing sequence of natural numbers such that lim inf and denote by Λ the set of all such sequences µ.Now, the following theorem was proved in [8]: for a given sequence (s m ) of real numbers, s m → s(N, p) implies s m → s if and only if inf λ∈Λu lim sup (1.11b) In case of smooth weights, for example, when implies both (1.11a) and (1.11b) (see [8] for details).
In the same paper, the following theorem for complex sequences was proved: s m → s(N, p) implies s m → s if and only if one of the following Tauberian conditions is satisfied: (1.15b) The following special case is called the condition of slow oscillation (see, e.g., [6]): In this case (1.15a) (and a fortiori, (1.15b)) is clearly satisfied.The symmetric counterpart involving the class Λ can be formulated analogously.
From now on, we will consider double sequences.Let p = (p k, : k, = 0, 1,...) be a fixed double sequence of nonnegative numbers with p 0,0 > 0, and set The weighted means of a double sequence (s k, : k, = 0, 1,. Furthermore, we also consider the so-called bounded convergence (in Pringsheim's sense); in symbols, b − lim s m,n = s, which means the following: where It is known (see, e.g., [3]) that In particular, it follows from (1.22) that P m,n → ∞ as m, n → ∞.
Of particular interest are the weighted means of multiplicative structure, that is, when p k, = p k q , where (p k ) and (q ) are sequences of nonnegative real numbers with p 0 ,q 0 > 0. Given a double sequence (s m,n ), we set with the agreement that s m,n = 0 if m < 0 or n < 0. It is known (see [11]) that if where c is some constant, is a Tauberian condition; that is, under conditions (1.25) and (1.26), lim s m,n = s(N, p) implies lim s m,n = s.We observe that (1.26) can be considered to be a multiplicative version of the Tauberian condition (1.7).However, ) is not a Tauberian condition, as it has been shown in [5] for the (C, 1, 1)-mean.

Main results.
In the sequel we will need the following notations.Let m, n, µ, ν be nonnegative integers and set where a µ,ν is defined in (1.24).Clearly, we have Furthermore, set Now, the multivariate version of the Tauberian condition (1.4) reads as follows.
Theorem 2.1.If s m,n → s(N, p) and one of the following Tauberian conditions is satisfied, The multivariate version of Tauberian condition (1.5) is more involved.
Theorem 2.2.Assume that (1.22) holds.If b − lim s m,n = s(N, p) and one of the following Tauberian conditions is satisfied: for every ε > 0 there exist natural numbers µ 0 , ν 0 such that Now, we turn to our main result which provides a necessary and sufficient Tauberian condition to deduce the conclusion s m,n → s from s m,n → s(N, p).A result of this type was already discussed in [7] for the (C, 1, 1)-mean and for more general weighted means in [11], however, the assumptions there can be simplified and the proof in the present paper is direct.
To formulate Theorem 2.5, we introduce the notation where k > m and > n.We consider a pair of nondecreasing sequences (λ and denote by Λ u the set of all such pairs of sequences.Furthermore, we consider a pair of nondecreasing sequences and denote by Λ the set of all such pairs of sequences.
We remind the reader that the limit infimum of a double sequence of real numbers is defined by and a typical example of a sequence from We cannot expect that (2.8) is always satisfied, since for example with p k, = e −k we have P m,n = n + m + o (1).However, in this case (1.22) is not satisfied.Lemma 2.4.Under (1.22), the sets Λ u and Λ are not empty.
The next result contains a multivariate version of the Tauberian conditions in (1.11).
Theorem 2.5.Assume that the sets Λ u and Λ are nonempty and s m,n is a double sequence of real numbers.Then, s m,n → s(N, p) implies s m,n → s if and only if both of the following conditions are satisfied: (ii) In the special (but important) case of multiplicative weights p k, = p k q with sequences p, q as before, conditions (2.7) and (2.8) are satisfied if the sequences (λ 1 (m)) and (λ 2 (n)) of natural numbers are chosen such that lim inf Then, condition (2.7) is obviously satisfied.In addition, we have respectively.Now, one can verify that the last two conditions are equivalent.Furthermore, all particular cases discussed in [11] can be deduced from (2.24a).Finally, the counterpart of Theorem 2.5 when (s m,n ) is a double sequence of complex numbers reads as follows.
Theorem 2.7.Assume that the sets Λ u and Λ are nonempty.Then, s m,n → s(N, p) implies s m,n → s if and only if one of the following conditions is satisfied: (2.26a) Remark 2.8.The following special case may be called the condition of slow oscillation.
then condition (2.25a) is obviously satisfied.The symmetric counterpart of (2.27) from which (2.26a) follows can be formulated analogously.
Originally, these conditions were considered in the case of multiplicative weights in [2,11].

Proofs
Proof of Theorem 2.1.It hinges on the following decomposition.By (2.1), we may write  Taking into account (3.1) and the boundedness of the double sequence (t m,n ), from the inequalities above we conclude that (s m,n ) is also bounded, that is, with some constant K.By (2.2), it follows that Now, we can proceed as follows.For any ε > 0, choose natural numbers µ 0 , ν 0 according to the assumptions in Theorem 2.2 and follow the estimations above to obtain provided that m, n are large enough (observe (1.22)).Similarly, we obtain For the proof of Theorem 2.5, we need the following auxiliary result which is interesting in itself.