SOME SEQUENCE SPACES AND STATISTICAL CONVERGENCE

for sets of sequences x = (xk) which are strongly (V ,λ)-summable to L, that is, xk→ L[V,λ]. We recall that a modulus f is a function from [0,∞) to [0,∞) such that (i) f(x)= 0 if and only if x = 0; (ii) f(x+y)≤ f(x)+f(y) for all x, y ≥ 0; (iii) f is increasing; (iv) f is continuous from the right at 0. It follows that f must be continuous on [0,∞). A modulus may be bounded or unbounded. Maddox [6] and Ruckle [9] used the modulus f to construct sequence spaces. In this paper, we introduce the strongly (V ,λ)-convergent sequences and give the relation between strongly (V ,λ)-convergence and strongly (V ,λ)-convergence with respect to a modulus.

The generalized de la Vallée-Poussin mean is defined by where ) is said to be (V , λ)-summable to a number L (see [5]) if t n (x) → L as n → ∞.If λ n = n, then (V , λ)-summability is reduced to (C, 1)-summability.We write for sets of sequences x = (x k ) which are strongly (V , λ)-summable to L, that is, (iv) f is continuous from the right at 0. It follows that f must be continuous on [0, ∞).A modulus may be bounded or unbounded.Maddox [6] and Ruckle [9] used the modulus f to construct sequence spaces.In this paper, we introduce the strongly (V , λ)-convergent sequences and give the relation between strongly (V , λ)-convergence and strongly (V , λ)-convergence with respect to a modulus.

Some sequence spaces
Definition 2.1.Let f be a modulus.We define the spaces, When λ n = n then the sequence spaces defined above become w 0 (f ) and w(f ), respectively, where w 0 (f ) and w(f ) are defined by Maddox [6].
Note that if we put f (x) = x, then we have We have the following result.
Proof.We consider only [V ,λ,f ].Suppose that x i → L and y j → L [V ,λ,f ] and that α, β are in C. Then there exists integers T α and M β such that |α| ≤ T α and |β| ≤ M β .We therefore have (2.3) This completes the proof.
This can be proved by using the techniques similar to those used in Maddox [6] and hence we omit the proof.

Theorem 2.5. Let f be any modulus. If lim
Let ε > 0 and choose δ with 0 < δ < 1 such that f (t) < ε for every t with 0 ≤ t ≤ δ.We can write and this completes the proof.

λ-statistical convergence.
In [3], Fast introduced the idea of statistical convergence, which is closely related to the concept of natural density or asymptotic density of subsets of the positive integers N. In recent years, statistical convergence has been studied by several authors [1,2,4,8,10].
A sequence x = (x k ) is said to be statistically convergent to the number L if for every ε > 0, where the vertical bars indicate the number of elements in the enclosed set.In this case we write s −lim x = L or x k → L(s) and s denotes the set of all statistically convergent sequences.
In this section, we introduce and study the concept of λ-statistical convergence and find its relation with [V ,λ,f ] and s λ .
In this case, we write s λ − lim x = L or x k → L(s λ ) and s λ = {x : for some L, s λ − lim x = L}.Note that if λ n = n, then s λ is same as s.
The following definition was introduced by Connor [2] as an extension of the original definition of statistical convergence which appeared in [3].Definition 3.2.Let A be a nonnegative regular summability method and let x be a sequence.Then x is said to be A-statistically convergent to L if χ S(x−Le:ε) is contained in w 0 (A) for every ε > 0, where In the above definition, if we define the matrix by we get λ-statistical convergence as a special case of A-statistical convergence.Let ∇ denote the set of all nondecreasing sequences λ = (λ n ) of positive numbers tending to ∞ such that λ n+1 ≤ λ n + 1 and λ 1 = 1.
We have the following result.