Hector H. Cuenya

Isotonic approximation in L 1

Let (ΩA,P) be a measurable space and L⊆A a sub-σ-lattice of the σ-algebra A. For X∈L1(Ω,A,P) we denote by pLX the set of conditional 1-mean (or best approximants) of X given L1(L) (the set of all L measurable and integrable functions). In this paper, we obtain characterizations of the elements in PLX, similar to those obtained by Landers and Rogge for conditional s-means with 1<s<∞. Moreover, using these characterizations we can extend the operator PL to a bigger space L0(Ω,A, P). When, in certain sense, Ln goes to L∞ we will be able to prove theorems about convergence and we will obtain bounds for the maximal function |supnPLnX|. A sharper characterization of conditional 1-means for certain particular σlattice was proved in previous papers. In the last section of this paper we generalize those results to all totally ordered σ-lattices.

Best constant approximants in Lorentz spaces

In this paper, we give a characterization of best constant approximants in Lorentz spaces Lw,q, 1 ≤ q < ∞ and we establish a way to obtain the best constant approximants maximum and minimum. We also study monotony of the best constant approximation operator.

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.

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

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.