Miguel Marano

The linear discrete Polya algorithm

Abstract We give a detailed analysis of the rate of convergence of the pth power minimum of an affine subspace of R n as p → ∞.

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.

UNIQUENESS OF BEST φ–APPROXIMATION BY 3-CONVEX FUNCTIONS

We prove the uniqueness of best 3-convex φ-approximation to a continuous function f ∈ L φ (J 0), where J 0 is a bounded, open interval and φ : [0, + ∞) → [0, + ∞) is a convex function that generalizes the p th–power functions, p ≥ 1.

Best φ-approximation by n-convex functions

A n-convex function defined on a bounded open interval J 0 n ≥2 is the (n−l)-st indefinite integral of a nondecreasing function. This fact and the simple structure of the latter enable to obtain concrete results about a n-convex best φ approximation g to a function f ∊ L φ on J 0, where φ: [0, ∞) → [0, ∞) is a convex function that generaJizes the pth -power functions, 1 ≤ p < ∞. It is shown that g may also be a best generalized spline φ approximation to the restriction of f on the maximal subintervals of J0 where g is a generalized spline. This is the situation in some cases, among which the Lp -approximation is includedp ≥ 1. For n = 2 it is proven that g is a polynomial of best φ-approximation to f ∊ L φ on any maximal interval where g is a polynomial. If f is in addition continuous, then this fact implies the uniqueness of g Under the same assumption, it is shown that the best 3-convex L 1-approximation is also unique whenever its derivative is bounded.

Visual interpolation of data

Following a precise definition of shape-preserving interpolating functions to data, we construct in a new and elementary manner such a cubic spline S2 ϵ C2, letting two additional knots per interval. We give an explicit description of S2, which has satisfactory properties in all the aspects. All the results in the paper—obtained with elementary calculus—are based on the behavior of the derivative of any smooth interpolating function to these data.

Monotone Iϕ-approximation

We give a construction of the maximum and the minimum of the set of nondecreasing lϕn-approximants in the discrete case, where ϕ is a positive convex function. A characterization of that set is also obtained.

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.

Rate of convergence of the Pólya algorithm from polyhedral sets

In this paper we consider a problem of best approximation in lp, 1 < p ≤ ∞. Lethp denote the best p-approximation of h ∈ Rn from a closed, convex set K of Rn 1 < p < ∞, h ∉ K, and let h∞* be the strict uniform approximation of h from K. We prove that if K satisfies locally a geometrical property, fulfilled by any polyhedral set of Rn, then lim sup p→∞ p||hp - h∞*|| < ∞.

Uniqueness of best φ-approximation from the set of splines with infinitely many simple knots

Let J be an open interval and denote by JΠ the set of all the splines of degree at most n - 1 with simple knots in Π, a countably infinite set of points in J, n≥2. In this paper, we prove that there exists a unique best φ-approximation to a continuous function in Lφ(J) from JΠ, where φ : [0, ∞) → [0, ∞) is a convex function that generalizes the pth-power functions, p≥1.

A minimizing property that characterizes Lp-SPACES

It is shown that an approximative property with respect to Orlicz or Luxemburg norms in Orlicz spaces, useful for computing best approximants from some class of functions, is generally satisfied only for the L p -norms, 1 < p < ∞, whenever the measure space contains at least four pairwise disjoint sets of finite and positive measure.