By Erkus E., Duman O.

During this paper, utilizing the idea that ofA-statistical convergence that is a regular(non-matrix) summability approach, we receive a common Korovkin variety approximation theorem which issues the matter of approximating a functionality f through a series {Lnf } of optimistic linear operators.

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N Proof: The implication (i) =⇒ (ii) is trivial. To see that (ii) implies (iii), first observe that if ψ is absolutely monotone on (a, b) and h ∈ (0, b − a), then ∆h ψ is absolutely monotone on (a, b − h). Indeed, because D ◦ ∆h ψ = ∆h ◦ Dψ on (a, b − h), we have that t+h h Dm ◦ ∆h ψ (t) = Dm+1 ψ(s) ds ≥ 0, t ∈ (a, b − h), t for any m ∈ N. Returning to the function ϕ, we now know that ∆m h ϕ is absolutely monotone on (0, 1 − mh) for all m ∈ N and h > 0 with mh < 1. In particular, m [∆m h ϕ](0) = lim [∆h ϕ](t) ≥ 0 t and so ∆m h ϕ (0) ≥ 0 when h = 0 1 n if mh < 1, and 0 ≤ m < n.

Indeed, because D ◦ ∆h ψ = ∆h ◦ Dψ on (a, b − h), we have that t+h h Dm ◦ ∆h ψ (t) = Dm+1 ψ(s) ds ≥ 0, t ∈ (a, b − h), t for any m ∈ N. Returning to the function ϕ, we now know that ∆m h ϕ is absolutely monotone on (0, 1 − mh) for all m ∈ N and h > 0 with mh < 1. In particular, m [∆m h ϕ](0) = lim [∆h ϕ](t) ≥ 0 t and so ∆m h ϕ (0) ≥ 0 when h = 0 1 n if mh < 1, and 0 ≤ m < n. Moreover, since [∆n1 ϕ](0) = lim1 [∆nh ϕ](0), n h n we also know that ∆nh ϕ (0) ≥ 0 when h = n1 , and this completes the proof that (ii) implies (iii).

5) |y|2 1 , g(y) ≡ √ exp − 2 2π y ∈ R, and recall that a random variable X is standard normal if P X∈Γ = g(y) dy, Γ ∈ BR . Γ In spite of their somewhat insultingly bland moniker, standard normal random variables are the building blocks for the most honored family in all of probability theory. Indeed, given m ∈ R and σ ∈ [0, ∞), the random variable Y is said to be normal (or Gaussian) with mean value m and variance σ 2 (often this is abbreviated by saying that X is an N m, σ 2 -random variable) if and only if the distribution of Y is γm,σ2 , where γm,σ2 is the distribution of the variable σX + m when X is standard normal.

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A -Statistical extension of the Korovkin type approximation theorem by Erkus E., Duman O.


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