Abstract
In this paper, we define the quasi-density of subsets of the set of natural numbers and show several of the properties of this density. The quasi-density dp(A) of the set A⊆N is dependent on the sequence p=(pn). Different sequences (pn), for the same set A, will yield new and distinct densities. If the sequence (pn) does not differ from the sequence (n) in its order of magnitude, i.e., limn→∞pnn=1, then the resulting quasi-density is very close to the asymptotic density. The results for sequences that do not satisfy this condition are more interesting. In the next part, we deal with the necessary and sufficient conditions so that the quasi-statistical convergence will be equivalent to the matrix summability method for a special class of triangular matrices with real coefficients.
Highlights
IntroductionWe will present the connection between statistical convergence and the matrix method of summability of the sequence of real numbers
L ∈ R stq − limxk = L) if and only if it is summable to L ∈ R for each matrix B = ∈ T p
The quasi-density d p ( A) of subsets of natural numbers, which we use to define the quasi-statistical convergence of sequences
Summary
We will present the connection between statistical convergence and the matrix method of summability of the sequence of real numbers. In the paper [12] the authors defined the quasi-statistical convergence as: Let p = ( pn ) be a sequence of positive real numbers with the properties: lim pn = +∞. We say that the sequence x = ( xk ) converges quasi-statistically (given the sequence p = ( pn )) to the number L ∈ R (stq p − limxk = L), if ∀ε > 0 the set Eε has a quasi-density equal to zero (t.j. d p ( Eε ) = 0), where Eε = {keN, | xk − L| ≥ ε}. It converges quasi-statistically as well (see [12])
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