Uniform and pointwise convergence of Bernstein-Durrmeyer operators with respect to monotone and submodular set functions

作者：Gal Sorin G^*; Opris Bogdan D

来源：Journal of Mathematical Analysis and Applications, 2015, 424(2): 1374-1379.

DOI：10.1016/j.jmaa.2014.12.012

• Monotone and submodular set function
• Choquet integral
• Bernstein-Durrmeyer operator
• Uniform convergence
• Pointwise convergence

We consider the multivariate Bernstein-Durrmeyer operator M-n,M-mu in terms of the Choquet integral with respect to a monotone and submodular set function mu on the standard d-dimensional simplex. This operator is nonlinear and generalizes the Bernstein-Durrmeyer linear operator with respect to a nonnegative, bounded Borel measure (including the Lebesgue measure). We prove uniform and pointwise convergence of M-n,M-mu(f)(x) to f(x) as n -> infinity, generalizing thus the results obtained in the recent papers [1] and [2].

• 出版日期2015-4-15