BestL 1 approximation by convex functions

It is shown that a function inL ₁ has a best approximation by convex functions, and that the net of bestL _p approximations converges asp decreases to one.     

Closed convex sets and their best simultaneous approximation properties with applications

We develop a theory of best simultaneous approximation for closed convex sets in a conditionally complete lattice Banach space X with a strong unit. We study best simultaneous approximation in X by elements of closed convex sets, and give necessary and sufficient conditions for the uniqueness of best simultaneous approximation. We give a characterization of simultaneous pseudo-Chebyshev and quasi-Chebyshev closed convex sets in X. Also, we present various characterizations of best simultaneous approximation of elements by closed convex sets in terms of the extremal points of the closed unit ball B _X* of X*.     

Characterization theorem of generalized polynomial of best approximation having bounded coefficients

Let the set of generalized polynomials having bounded coefficients beK={p= _jg_j. _{j j j},j=1, 2, ...,n}, whereg ₁,g ₂, ...,g _n are linearly independent continuous functions defined on the interval [a, b], _j, _j are extended real numbers satisfying _j<+, _j>-, and _{j j}. Assume thatf is a continuous function defined on a compact setX [a, b]. This paper gives the characterization theorem forp being the best uniform approximation tof fromK, and points out that the characterization theorem can be applied in calculating the approximate solution of best approximation tof fromK.     

On the best polynomial approximation of entire transcendental functions of generalized order

We prove a Hadamard-type theorem that associates the generalized order of growth of an entire transcendental function ƒ with the coefficients of its expansion in a Faber series. This theorem is an extension of one result of Balashov to the case of a finite simply connected domain G with boundary γ belonging to the Al'per class Λ*. Using this theorem, we obtain limit equalities that associate with a sequence of the best polynomial approximations of ƒ in certain Banach spaces of functions analytic in G. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 8, pp. 1011–1026, August, 2008.     

Some best approximation results in locally convex spaces

In this note we obtain generalization of well known results of carbone and Conti, Sehgal and Singh and Tanimoto concerning the existence of best approximation and simultaneous best approximation of continuous funcitons from the set up of a normed space to the case of a Hausdorff locally convex space.     

On generalized means and generalized convex functions

Properties of generalized convex functions, defined in terms of the generalized means introduced by Hardy, Littlewood, and Polya, are easily obtained by showing that generalized means and generalized convex functions are in fact ordinary arithmetic means and ordinary convex functions, respectively, defined on linear spaces with suitably chosen operations of addition and multiplication. The results are applied to some problems in statistical decision theory.This research was supported by Project No. NR-047-021, Contract No. N00014-75-C-0569 with the Center for Cybernetic Studies, The University of Texas, Austin, Texas, and by NSF Grant No. ENG-76-10260 at Northwestern University, Evanston, Illinois.     

Note on generalized convex functions

In this note, an important class of generalized convex functions, called invex functions, is defined under a general framework, and some properties of the functions in this class are derived. It is also shown that a function is (generalized) pseudoconvex if and only if it is quasiconvex and invex.     

Alternants for generalized convex functions

In this paper we consider the problem of best uniform approximation by elements of WT-spaces. In particular, we investigate the structure of the corresponding error function when the function to be approximated is generalized convex with respect to a WT-space. The principal concept involved is that of an alternation element, an element for which the error function takes on its norm with alternating signs a specified number of times. This approach has been employed by Jones, Karlovitz [4], Sommer, Strauss [10], Nurnberger, Sommer [7] and Barrar, Loeb [2]. Much of the material in this paper was inspired by a paper of Amir and Ziegler [1] . A new characterization of WT-spaces in terms of alternation elements is given.     

On the approximation of convex functions by Bernstein-type operators

As a consequence of Jensen's inequality, centered operators of probabilistic type (also called Bernstein-type operators) approximate convex functions from above. Starting from this fact, we consider several pairs of classical operators and determine, in each case, which one is better to approximate convex functions. In almost all the discussed examples, the conclusion follows from a simple argument concerning composition of operators. However, when comparing Szász-Mirakyan operators with Bernstein operators over the positive semi-axis, the result is derived from the convex ordering of the involved probability distributions. Analogous results for non-centered operators are also considered.     

Linear best approximation using a class of polyhedral norms

A class of polyhedral norms is introduced, which contains thel ₁ andl norms as special cases. Of primary interest is the solution of linear best approximation problems using these norms. Best approximations are characterized, and an algorithm is developed. This is a methods of descent type which may be interpreted as a generalization of existing well-known methods for solving thel ₁ andl problems. Numerical results are given to illustrate the performance of two variants of the algorithm on some problems.Communicated by C. Brezinski