DOUBLE SEQUENCE CORE THEOREMS

In 1900, Pringsheim gave a definition of the convergence of double sequences. In this paper, that notion is extended by presenting definitions for the limit inferior and limit superior of double sequences. Also the core of a double sequence is defined. By using these definitions and the notion of regularity for 4-dimensional matrices, extensions, and variations of the Knopp Core theorem are proved.


Introduction.
The notion of convergence for double sequences was presented by Pringsheim.Also, in [2,3,4,5,10] the 4-dimensional matrix transformation (Ax) m,n = ∞,∞ k,l=0,0 a m,n,k,l x k,l was studied extensively by Robison and Hamilton.In their work and throughout this paper, the 4-dimensional matrices and double sequences have complex-valued entries unless specified otherwise.In this paper, we extend the notion of convergence by defining new double sequence spaces and consider the behavior of 4-dimensional matrix transformations on our new spaces.We also present definitions for limit inferior/limit superior of a double sequence, regularity of a 4-dimensional matrix, and the core of a double sequence.Using these definitions and the notion of regularity for a 4-dimensional matrix, we present multidimensional analogues to the Knopp Core theorem.We also present extensions and variations of this theorem.

Definitions and preliminary results
Definition 2.1 [Pringsheim, 1900].A double sequence [x] has Pringsheim limit L (denoted by P-lim[x] = L) provided that given > 0 there exists N ∈ N such that |x k,l − L| < whenever k,l > N. We shall describe such an [x] more briefly as "P-convergent."A double sequence [x] is bounded if and only if there exists a positive number M such that |x k,l | < M for all k and l (which shall be denoted by [|x|] < M).Note that a convergent double sequence need not be bounded.In 1900, Pringsheim gave the following definition: a double sequence [x] is called definite divergent if for every (arbitrarily large) G > 0 there exist two natural numbers n 1 and Definition 2.2.The sequence [y] is a subsequence of the double sequence [x] provided that there exist two increasing double index sequences {n i j } and A two dimensional matrix transformation is said to be regular if it maps every convergent sequence into a convergent sequence with the same limit.In 1926, Robison presented a 4-dimensional analogue of regularity for double sequences in which he added an additional assumption of boundedness: a 4-dimensional matrix A is said to be RH-regular if it maps every bounded P-convergent sequence into a P-convergent sequence with the same P-limit.
The following is a 4-dimensional analogue of the well-known Silverman-Toeplitz theorem [6].
Theorem 2.1 (Hamilton [2], Robison [10]).The 4-dimensional matrix A is RH-regular if and only if (RH 1 ) P-lim m,n a m,n,k,l = 0 for each k and l; Remark 2.1.The definition of a Pringsheim limit point is equivalent to the following statement: β is a Pringsheim limit point of [x] if and only if there exist two increasing index sequences {n i } and {k i } such that lim i x n i ,k i = β.A double sequence [x] is divergent in the Pringsheim sense (P-divergent) provided that [x] is not P-convergent.This is equivalent to the following: a double sequence [x] is P-divergent if and only if either [x] contains two subsequences with distinct finite limit points or [x] contains an unbounded subsequence.Also note that, if [x] contains an unbounded subsequence then [x] also contains a definite divergent subsequence.
In [7] Knopp introduced the concept of the core a complex number sequence.We follow that idea in defining the core of a double sequence.Definition 2.4.Let P-C n {x} be the least closed convex set that includes all points x k,l for k,l > n; then the Pringsheim core of the double sequence Theorem 2.2 [Knopp, 1930].If A is a nonnegative regular matrix then the core of

Main results.
In a manner similar to the classical definitions of the limit superior and the limit inferior of a sequence, we present definitions for the limit superior and the limit inferior of a double sequence.Using these definitions one can characterize the Pringsheim core of a real-valued double sequence as the closed interval [P-lim inf x, P-lim sup x].Definition 3.1.Let [x] = {x k,l } be a double sequence of real numbers and for each n, let α n = sup n {x k,l : k, l ≥ n}.The Pringsheim limit superior of [x] is defined as follows: (1) if α = +∞ for each n, then P-lim sup[x] := +∞; (2) if α < ∞ for some n, then P-lim sup[x] := inf n {α n }.Similarly, let β n = inf n {x k,l : k, l ≥ n} then the Pringsheim limit inferior of [x] is defined as follows: (1) if Example 3.1.The following is an example of an [x] which is neither bounded above nor bounded below; however, the Pringsheim limit superior and inferior are both finite numbers The proof of the following proposition is the same as the proof for single dimensional sequences and is therefore left to the reader.Proof.Note that if P-C{x} is the complex plane then the result is trivial.We shall establish our theorem by considering separately the cases where [x] is bounded or unbounded.In both cases the result will be established by proving the following: if there exists a q such that for ω ∈ P-C q {x}, then there exists a p such that ω ∈ P-C p {Ax}.When [x] is bounded, P-C{x} is not the complex plane, thus there exists an ω ∈ P-C{x}.This implies that there exists a q for which ω ∈ P-C n {x}.Since ω is finite, we may assume that ω = 0 by the linearity of A. Since we are also given that P-C q {x} is a convex set, we can rotate P-C q {x} so that the distance from zero to P-C q {x} is the minimum of {|y| : y ∈ P-C q {x}} and is on the positive real axis; say that this minimum is 3d.Since P-C q {x} is convex, all points of P-C q {x} have a real part which is at least 3d.Let M = max{|x k,l |}.By the regularity conditions (RH 1 )-(RH 4 ) and the assumption a m,n,k,l ≥ 0, there exists an N such that for m,n > N the following holds: where Therefore, {Ax} > d which implies that there exists a p for which ω = 0 is also outside of P-C p {Ax}.Now suppose that [x] is unbounded; the ω may be the point at infinity or not.If ω is not the point at infinity then choose N such that for m,n > N the following holds: In a manner similar to the first part we obtain {Ax} > d.In the case when ω is the point at infinity, P-C q {x} is bounded for all q, which implies that x k,l is bounded for k,l > q.We may assume that [|x|] < B for some positive number B without loss of generality.Thus for m and n large we obtain the following: Hence, there exists a p such that the point at infinity is outside of P-C p {Ax}.This completes the proof of our theorem.
The following lemma is a multidimensional analogue of a lemma of Agnew in [1].We use this lemma to prove Theorem 3.2, below.
,0 is a real or complex-valued 4-dimensional matrix such that (RH 1 ), (RH 3 ), (RH 4 ), and P-lim sup m,n ∞,∞ k,l=0,0 |a m,n,k,l | = M hold, then for any bounded double sequence [x] we obtain the following: where (3.9) In addition, there exists a real-valued double sequence [x] such that if a m,n,k,l is real with 0 < P-lim sup[|x|] < ∞ then Proof.Let [x] be bounded and define Given > 0 we can choose an N such that |x k,l | < (B + )/3 for each k, and/or l > N. Thus, we may assume that M > 0 without loss of generality.Using the RH-regularity conditions we choose m 0 ,n 0 ,l 0 , and k 0 , so large that , and [l q−1 ] be four chosen strictly increasing index sequences with p, q = 1 •••i−1,j −1 with k 0 = l 0 > 0. Using the RH-regularity conditions we now choose m i > m i−1 and n j > n j−1 such that (3.17) In addition, we also choose k i > k i−1 and l j > l j−1 such that (3.18) Let us define [x] as follows: and a m i ,n j ,k,l ≠ 0; 0, otherwise . (3.19) Consider the following: we take Pringsheim limits and get the desired result.

Example 2 . 1 .
The double sequences whose n, k-terms are y n,k = 1 and z n,k = −1 for each n and k are both subsequences of the double sequence whose n, kth term is x n,k = (−1) n+k .Indeed, every double sequence of 1's and −1's is a subsequence of this [x].
and (RH 6 ) there exist positive numbers A and B such that k,l>B |a m,n,k,l | < A. Definition 2.3.A number β is called a Pringsheim limit point of the double sequence [x] provided that there exists a subsequence [y] of [x] that has Pringsheim limit β: P-lim[y] = β.