On the rates of pointwise convergence for Bernstein polynomials

2024/11/15 by Adell, José A., Cárdenas-Morales, Daniel, López-Moreno, Antonio J. · 2 citations
#41A10 #41A25 #41A36 #60E05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2411.10135

Abstract

Let f be a real function defined on the interval [0,1] which is constant on (a,b)⊂ [0,1], and let B_nf be its associated nth Bernstein polynomial. We prove that, for any x∈ (a,b), |B_nf(x)-f(x)| converges to 0 as n→ ∞ at an exponential rate of decay. Moreover, we show that this property is no longer true at the boundary of (a,b). Finally, an extension to Bernstein-Kantorovich type operators is also provided

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• Tight bounds on Poisson tails with applications to Szász-Mirakyan operators
• Random positive linear operators and their applications to nonparametric statistics

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