Uniform convergence for sequences of best L^p approximation

11/30/2021

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by   Donatella Bongiorno, et al.

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Let f be a continuous monotone real function defined on a compact interval [a,b] of the real line. Given a sequence of partitions of [a,b], Δ_n, ‖Δ_n‖→ 0, and given l≥ 0,m≥ 1, let 𝐒_m^l(Δ _n) be the space of all functions with the same monotonicity of f that are polynomial of ordermand that belong to the smoothness classC^l[a,b]. In this paper we show that, for anym≥2l+1,∙sequences of bestL^p-approximation in𝐒_m^l(Δ_n)converge uniformly tofon any compact subinterval of(a,b);∙sequences of bestL^p-approximation in𝐒_m^0(Δ_n)converge uniformly tofon the whole interval[a,b] .