Best Uniform Approximation of Complex-Valued Functions by Generalized Polynomials Having Restricted Ranges

Abstract

We study the uniform best restricted ranges approximations of complex-valued functions by generalized polynomials. The theory, generalizing the real-valued case, embraces the theorems of existence, characterization, uniqueness, and strong uniqueness. Academic Press

Constraint Qualifications for Extended Farkas's Lemmas and Lagrangian Dualities in Convex Infinite Programming

“…In particular, Rockafellar [46] gave many interesting examples showing that a wide spectrum of problems can be cast in terms of convex-composite functions (which are, in general, non-convex and non-continuous). Another "nonclosed" situation naturally arises when one considers the best restricted range approximation in complex valued continuous function space C(Q), which has been studied extensively (see for example [34,35,37,48,49] and reference therein), consisting of finding a best approximation to f ∈ C(Q) from P Ω = {p ∈ P : p(t) ∈ Ω t for all t ∈ Q}, where Q is a compact Hausdorff space, P is a finite-dimensional subspace of C(Q) and Ω = {Ω t : t ∈ Q} is a system of nonempty convex set in the complex plane C. As done in [34,35,37,48,49], each Ω t usually is expressed as a level set of some convex function, which is not lsc if Ω t is not closed. Thus, our approach can cover the case where Ω t is not necessarily closed.…”

Limit theory of restricted range approximations of complex-valued continuous functions

“…Thus P Ω has Property U 2 and the proof of Theorem 4.2 is complete. The following corollary, which was obtained in [2][3][4][5] respectively under some stronger conditions, is now an immediate consequence of Theorem 4.1.…”

“…One basic assumption in the study of the restricted range approximation problem of complexvalued continuous functions is that P Ω has an interior point or a strong interior point which are defined as follows, see [1][2][3][4][5][6] for details.…”

“…The following two classes of admissible functions were introduced by Smirnov and Smirnov in [2,4] respectively for the study of the uniqueness problem of the best restricted range approximation of complex-valued continuous functions.…”

“…Then, for a given f ∈ C(Q), an element p * of P Ω is called a best (restricted range) approximation to f from P Ω if and only if f − p * = d(f, P Ω ), where d(f, P Ω ) is defined by d(f, P Ω ) = inf p∈PΩ f − p . This problem was first introduced and formulated by Smirnov and Smirnov in [1,2], where the characterization theorems and the uniqueness theorems of the best restricted range approximation were obtained under the special case when {Ω t : t ∈ Q} is a system of closed disks with centers and radii continuously depending on t, and P is a Haar subspace. These results were extended to some more general cases in [3][4][5].…”