We introduce and study the King type operators associated to a couple $$ \left( \mathcal {A},\tau \right) $$ A , τ where $$\mathcal {A}=\left( A_{n}\right) _{n\in \mathbb {N}}$$ A = A n n ∈ N is a sequence of linear positive operators from $$C\left[ 0,1\right] $$ C 0 , 1 into $$C\left[ 0,1\right] $$ C 0 , 1 and $$\tau :\left[ 0,1\right] \rightarrow \left[ 0,\infty \right) $$ τ : 0 , 1 → 0 , ∞ a continuous strictly increasing function. Given a sequence $$\Lambda =\left( \lambda _{n}\right) _{n\in \mathbb {N}}$$ Λ = λ n n ∈ N with $$\lim \nolimits _{n\rightarrow \infty }\lambda _{n}=\infty $$ lim n → ∞ λ n = ∞ we introduce the concept of the $$\Lambda $$ Λ -Voronovskaja property of a function $$f\in C\left[ 0,1\right] $$ f ∈ C 0 , 1 with respect to the sequence $$\mathcal {A} $$ A . We show that there is a natural connection between the $$\Lambda $$ Λ -Voronovskaja property with respect to the sequence $$\mathcal {A}$$ A and the $$ \Lambda $$ Λ -Voronovskaja property with respect to the sequence of King type operators. We apply these general results to the case of Bernstein, Kantorovich type operators and thus obtain entirely new Voronovskaja type theorems for such a kind of positive linear operators.

中文翻译：

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王型算子的 Voronovskaja 型定理

我们介绍并研究与一对 $$ \left( \mathcal {A},\tau \right) $$ A , τ 相关的 King 类型运算符，其中 $$\mathcal {A}=\left( A_{n}\ right) _{n\in \mathbb {N}}$$ A = A nn ∈ N 是从 $$C\left[ 0,1\right] $$ C 0 , 1 到 $$ 的线性正算子序列C\left[ 0,1\right] $$ C 0 , 1 和 $$\tau :\left[ 0,1\right] \rightarrow \left[ 0,\infty \right) $$ τ : 0 , 1 → 0 , ∞ 一个连续严格递增的函数。给定一个序列 $$\Lambda =\left( \lambda _{n}\right) _{n\in \mathbb {N}}$$ Λ = λ nn ∈ N with $$\lim \nolimits _{n\ rightarrow \infty }\lambda _{n}=\infty $$ lim n → ∞ λ n = ∞ 我们引入了函数 $$f\in C\left 的 $$\Lambda $$ Λ -Voronovskaja 性质的概念[ 0,1\right] $$ f ∈ C 0 , 1 关于序列 $$\mathcal {A} $$ A 。我们表明，关于序列 $$\mathcal {A}$$ A 的 $$\Lambda $$ Λ -Voronovskaja 性质与关于序列 $$\Lambda $$ Λ -Voronovskaja 的性质之间存在自然联系King 类型运算符的序列。我们将这些一般结果应用于 Bernstein、Kantorovich 类型算子的情况，从而为这种正线性算子获得全新的 Voronovskaja 类型定理。