The riesz convergence and riesz core of double sequences

* Correspondence: celal. cakan@inonu.edu.tr Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Full list of author information is available at the end of the article Abstract In this article, we have introduced the Riesz convergence and Riesz core of double sequences and determined the necessary and sufficient conditions on a fourdimensional matrix A to yield PR core{Ax} ⊆ P core{x} and PR core{Ax} ⊆ st2 core {x} for all x ∈ ∞. Mathematics Subject Classification 2000: 40C05; 40J05; 46A45.


Introduction
j,k=0 is said to be convergent in the Pringsheim sense or Pconvergent if for every ε >0 there exists an N N such that | x jk -ℓ | <ε whenever j, k > N, [1]. In this case, we write P-lim x = ℓ. By c 2 , we mean the space of all P-convergent sequences.
A double sequence x is bounded if x jk < ∞.
By 2 ∞ we denote the space of all bounded double sequences. Note that, in contrast to the case for single sequences, a convergent double sequence need not be bounded. So, we denote by c ∞ 2 the space of double sequences which are bounded and convergent.
Let E ⊆ N × N and E(m, n) = {(j, k):j ≤ m,k ≤ n}. Then, the double natural density of E is defined by if the limit on the right hand side exists; where the vertical bars denotes the cardinality of the set E(m,n).
A real double sequence x = [x jk ] is said to be statistical (or briefly st-) convergent [2] to the number L if for every ε >0, the set {(j,k): |x jk -L| > ε} has double natural density zero. In this case, we write st 2 -lim x = L. Let st 2 be the space of all st-convergent double sequences. Clearly, a convergent double sequence is also st-conver-gent but the converse it is not true, in general. Also, note that a st-convergent double sequence need not be bounded.  We say that a matrix A is bounded-regular if every bounded-convergent sequence x is A-summable to the same limit and the A-means are also bounded. The necessary and sufficient conditions for A to be bounded-regular or RH-regular are known (see [4,5] and is denoted by C 1 -lim x = L. We denote the space of all Cesáro convergent double sequences by C 1 . That is, The concept core of single sequences (see [6]) was extended by Patterson [7] to the double sequences by defining the Pringsheim core (or P-core) of a real bounded double sequence x = [x jk ] as the closed interval [Plim inf x,P -lim sup x]. Later this concept has been studied by many authors. For example we refer [2,[8][9][10].
The Cesáro core (or P C -core) of a real-valued bounded double sequence x = [x jk ] has been defined by the closed interval [11]. Also; where an inequality related to the P C and P-cores has been investigated.
In this article we have introduced the Riesz convergence and Riesz core of a double sequence and also we have investigated some inequalities related to the P-, statistical and Riesz cores.

Main results
Definition 2.1. Let (q i ), (p j ) be sequences of non-negative numbers which are not all zero and Q m = q 1 + q 2 + · · · + q m , q 1 >0, P n = p 1 + p 2 + · · · + p n , p 1 >0. Then, the transformation given by In what follows c 2 R will denote the set of all Riesz convergent sequences. Since a Riesz convergent double sequence need not be bounded, by c 2,∞ R we will denote the set of all bounded and Riesz convergent double sequences. c 2,∞ 0,R will denote the set of all double sequences which bounded and Riesz convergent to zero.
Note that in the case q i = 1 for all i and p j = 1 for all j, the Riesz mean reduced to the Cesáro mean and the Riesz convergence is said to be Cesáro convergence. Now, we will give some lemmas characterized some classes of matrices related to the  , n j , r, s, q, p), Then, the necessity of (2.5) follows from P − lim t qp rs (Ax). It is known by the assumption that   n, r, s, q, p) .

So, by letting m,n
∞ under the light of the assumption, we get that P − lim t qp rs (Ax) = L. This completes the proof. for every q,p is in the class ( 2 ∞ , c 2,∞ 0,R ). Therefore, the necessity of (2.8) follows from the condition (2.5) of Lemma 2.3.
For the converse take a sequence x = [x rs ] ∈ st 2 ∩ 2 ∞ with st 2 -lim x = l. Then; it is known that δ 2 = δ 2 ({(r, s): |x rs -l| ≥ ε}) = 0 and |x rs -l| < ε whenever r,s ∉ E. Now, write . Note that in the case q i = 1 for all i and p j = 1 for all j, Riesz core is reduced to the Cesáro core, [11]. Now; we are ready to give some inequalities related to the P-, P R -and st 2 -core of double sequences.