ANALOGUES OF SOME TAUBERIAN THEOREMS FOR STRETCHINGS

We investigate the effect of four-dimensional matrix transformation on new classes of double sequences. Stretchings of a double sequence is defined, and this definition is used to present a four-dimensional analogue of D. Dawson’s copy theorem for stretching of a double sequence. In addition, the multidimensional analogue of D. Dawson’s copy theorem is used to characterize convergent double sequences using stretchings. 2000 Mathematics Subject Classification. 40B05, 40C05.

The 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. A two-dimensional matrix transformation is said to be regular if it maps every convergent sequence into a convergent sequence with the same limit. The Silverman-Toeplitz theorem [5,6] characterizes the regularity of two-dimensional matrix transformations. In [4], Robison presented a fourdimensional analog of regularity for double sequences in which he added an additional assumption of boundedness. This assumption was made because a double sequence which is P -convergent is not necessarily bounded. The definition of regularity for four-dimensional matrices will be stated below along with the Robison-Hamilton characterization of the regularity of four-dimensional matrices.
Definition 2.4. The four-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.
Theorem 2.5 (see [2,4]). The four-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; (RH 2 ) P -lim m,n ∞,∞ k,l=1,1 a m,n,k,l = 1; (RH 3 ) P -lim m,n ∞ k=1 |a m,n,k,l | = 0 for each l; 6 ) there exist finite positive integers A and B such that k,l>B |a m,n,k,l | < A.
Example 2.6. The sequences [y n,k ] = 1 and [y 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 . In addition to the two subsequences given, every double sequence of 1's and −1's is a subsequence of this x. Example 2.7. As another example of a subsequence of a double sequence, we define x as follows: Then the double sequence is clearly a subsequence of x.
Remark 2.8. Note that if the double sequence x contains at most a finite number of unbounded rows and/or columns, then every subsequence of x is bounded. In addition, the finite number of unbounded rows and/or columns does not affect the P -convergence or P -divergence of x and its subsequences. This double sequence has five Pringsheim limit points, namely −2, −1, 0, 1, and 2.
Remark 2.11. The definition of a Pringsheim limit point can also be stated as follows: β is a Pringsheim limit point of x provided that there exist two increasing index sequences {n i } and {k i } such that lim i x n i ,k i = β. Definition 2.12. A double sequence x is divergent in the Pringsheim sense (Pdivergent) provided that x does not converge in the Pringsheim sense (P -convergent).
Remark 2.13. Definition 2.12 can also be stated as follows: a double sequence x is P -divergent provided that either x contains at least two subsequences with distinct finite Pringsheim 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.
Example 2.14. This is an example of a convergent double sequence whose terms form an unbounded set Example 2.15. This is an example of an unbounded divergent double sequence with three finite Pringsheim limit points, namely −1, 0, and 1: (2.6) Example 2.16. This is an example of a double sequence which contains an unbounded subsequence (2.7) Example 2.17. For an example of a definite divergent sequence take x n,k = n for each n and k; then it is also clear that x contains an unbounded subsequence.
The following propositions are easily verified.
Remark 2.20. For an ordinary single-dimensional sequence, any sequence is a subsequence of itself. This, however, is not the case in the two-dimensional plane, as illustrated by the following example.
contains only two subsequences, namely, [y n,k ] = 0 for each n and k, and neither subsequences is x.
The following propositions are easily verified.
Proposition 2.23. The double sequence x is P -convergent to L if and only if every subsequence of x is P -convergent to L.
Definition 2.24. The double sequence y contains an -Pringsheim-copy of x provided that y contains a subsequence y n i ,k j such that |y Observe that, not only does y contain an -Pringsheim-copy of x, but y itself is an -Pringsheim-copy of x.
(2.12) Remark 2.27. This definition demonstrates the procedure which is used to construct a stretching of a double sequence x. This procedure uses a sequence of stages to construct the stretching of x. These stages are constructed using a sequence of abutting rows and columns of x. These rows and columns are constructed as follows.  . . .

Stage i. Begin by repeating the 1 +
Note that in each stage we repeat the number of rows and then repeat the number of columns. However the resulting stretching y of x is the same, if we first repeat the number of columns and then repeat the numbers of rows. Also note that every sequence itself is a stretching of itself and the sequences that induce this kind of stretching are R i = i and S j = j.
Example 2.28. The sequence is a stretching of x induced by R i = 3i and S j = 3j.

Main results.
The following theorem is given its name because of its similarity to the copy theorem of Dawson in [1]. x k,l , δ i,j := min Then by (RH 2 ) there exist m α 1 and n β 1 such that for m > m α 1 >B and n > n β 1 >B, whereB is defined by the sixth RH-condition, Also by (RH 1 ) and (RH 2 ) there exist a α 1 and b β 1 such that In addition, there existm α 1 ,n β 1 ,α 2 , and β 2 such that if 1≤ ψ ≤ a α 1 and 1 ≤ ω ≤ b β 1 , then Also, there exist r α 1 > 1 and s β 1 > 1 such that if 1 ≤ m ≤m α 1 and 1 ≤ n ≤n β 1 then Now, without loss of generality, we set α p = p and β q = q. Having chosen m p ,m p ,a p ,r p n q ,n q ,b q ,s q i−1,j−1 p=0,q=0

7)
(k,l)∈c i−1,j−1 (r ,s) a m,n,k,l < δ i,j 8Q i−1,j−1 2 i+j . (3.8) Also choose a i > a i−1 and b j > b j−1 such that . (3.10) Then choose r i > r i−1 and s j > s j−1 such that if 1 ≤ m ≤m i and 1 ≤ n ≤n j then (k,l)∈c i,j (r ,s) a m,n,k,l < δ i,j 2 4+i+j Q i+1,j+1 , (3.11) where m i ,n j ,m i ,n j ,r i , and s j are chosen using (RH 1 ), (RH 2 ), (RH 3 ), and (RH 4 ) such that if 1 ≤ p ≤ j − 1 and 1 ≤ q ≤ i − 1 the following is obtained: (3.12) Therefore by (3.9) and (3.10) we have if i, j > 1, with m i ≤ m ≤m i and n j ≤ n ≤n j the following is obtained: a m,n,k,l y k,l ≤ max |x k,l | a m,n,k,l y k,l .
(3.16) By (3.8), the following holds: the following also is obtained: . Therefore, Note that the inequality (3.23) is true for m 1 ≤ m ≤m 1 and n 1 ≤ n ≤n 1 , and also this inequality is true for i, j ≥ 1 with m i ≤ m ≤m i and n j ≤ n ≤n j . Hence where |u i,j | ≤ δ i,j /2K. Note that ifm i−1 ≤ m ≤ m i andn j−1 ≤ n ≤ n j , then the following is obtained:   The next two results are immediate corollaries of the extended copy theorem.

Corollary 3.2. If T is any RH-regular matrix summability method and
A is an RH-regular matrix such that Ay is T -summable for every stretching y of x, then x is P -convergent. Corollary 3.3. If T is any RH-regular matrix summability method and A is an RH-regular matrix such that Ay is absolutely T -summable for every stretching y of x, then x is P -convergent.