Generalized Kantorovich modifications of positive linear operators

Abstract

<p style='text-indent:20px;'>Starting with a positive linear operator we apply the Kantorovich modification and a related modification. The resulting operators are investigated. We are interested in the eigenstructure, Voronovskaya formula, the induced generalized convexity, invariant measures and iterates. Some known results from the literature are extended.</p>

Modified positive linear operators, iterates and systems of linear equations

“…. , m, and compute successively x (1) , x (2) , • • • , using (17) or alternatively (18). So, we get the sequence (x (k) ) k≥0 .…”

“…Similar studies involve modifications of classical positive linear operators. The sequence of Kantorovich operators is a prominent example of modification of classical Bernstein operators, see e.g., [1], [9], [11], [18] and the references therein. This paper is devoted to such problems for the Kantorovich modifications of linking operators and Stancu modifications of Bernstein operators.…”

Semi-exponential operators connected to $ x^3 $. A probabilistic perspective

“…Certainly, the operators Q β n , having an integral form, are also suitable for L papproximation, 1 ≤ p < ∞. Actually, some operators of discrete type have been appropriately modified in order to approximate functions in L p (see, for instance, [2], [14], and [17], among others). The corresponding results have potential applications in machine learning generalization analysis (cf.…”

Estimation Features by Transformed Bernstein kind Polynomials

“…Satisfy the reduction formula(Sharma, 2016;Acu and Agrawal, 2019;Acu and Tachev, 2021;Acu et al, 2023;Adell and Cárdenas-Morales, 2022 ) 𝑆(𝑣, 𝑛, 𝑥, 𝑦) = 𝑥𝑆(𝜈 − 1, 𝑛, 𝑥, 𝑦) + 𝑛𝛽𝑆(𝑣, 𝑛 − 1, 𝑥 + 𝛽, 𝑦) By repeated use of the reduction formula, we can show that 𝑆(1, 𝑛, 𝑥, 𝑦) = ∑ 𝑛 𝑘=0 ( 𝑛 𝑣 ) 𝑣! 𝛽 𝑘 (𝑥 + 𝑦 + 𝑛𝛽) 𝑛−𝑣 as 𝑥𝑆(0, 𝑛, 𝑥, 𝑦) = (𝑥 + 𝑦 + 𝑛𝛽) 𝑛…”