Fabian E. Levis

Best Local Approximation in Orlicz Spaces

We get results in Orlicz spaces L φ about best local approximation on non-balanced neighborhoods when φ satisfies a certain asymptotic condition. This fact generalizes known previous results in L p spaces.

The Best Multipoint Padé Approximant

Abstract This paper is dealing with the problem of finding the “best” multipoint Padé approximant of an analytic function when data in some neighborhoods of sampling points are more important than others. More exactly, we obtain a multipoint Padé approximants as limits of best rational Lp-approximations on union of disks, when the measure of them tends to zero with different speeds. As such, this technique provides useful qualitative and analytic information concerning the approximants, which is difficult to obtain from a strictly numerical treatment.

Nonlinear Chebyshev approximation to set-valued functions

In this paper, we give a characterization of best Chebyshev approximation to set-valued functions from a family of continuous functions with the weak betweeness property. As a consequence, we obtain a characterization of Kolmogorov type for best simultaneous approximation to an infinity set of functions. We introduce the concept of a set-sun and give a characterization of it. In addition, we prove a property of Amir–Ziegler type for a family of real functions and we get a characterization of best simultaneous approximation to two functions

Best Simultaneous Monotone Approximants in Orlicz Spaces

Let f = (f 1,…, f m ), where f j belongs to the Orlicz space ℒφ[0, 1], and let w = (w 1,…, w m ) be an m-tuple of m positive weights. If 𝒟 ⊂ ℒφ[0, 1] is the class of nondecreasing functions, we denote by ℳφ, w (f, 𝒟) the set of best simultaneous monotone approximants to f, that is, all the elements g ∈ 𝒟 minimizing , where φ is a convex function, φ(t) > 0 for t > 0, and φ(0) = 0. In this work, we show an explicit formula to calculate the maximum and minimum elements in ℳφ, w (f, 𝒟). In addition, we study the continuity of the best simultaneous monotone approximants.

Pólya-type polynomial inequalities in Orlicz spaces and best local approximation

We obtain an extension of Pólya-type inequalities for univariate real polynomials in Orlicz spaces. We also give an application to a best local approximation problem.MSC 2010: 41A10; 41A17.