A Unified Approach to Uniform Real Approximation by Polynomials with Linear Restrictions

Abstract

Abstract. Problems concerning approximation of real-valued continuous functions of a real variable by polynomials of degree smaller than n with various linear restrictions have been studied by several authors. This paper is an attempt to provide a unified approach to these problems. In particular, the notion of restricted derivatives approximation is seen to fit into the theory and includes as special cases the notions of monotone approximation and restricted range approximation. Also bounded coefficients approximation, c-interpolator approximation, and polynomial approximation with interpolation fit into our scheme.

On the Existence of Strong Unicity of Arbitrarily Small Order.

“…-'~~~ ~ Now, by referring to the general theory of ( 1), one can prove that corresponding to each fEC(a , b], there exists a unique best approximation , Hennlte-Birkhoff interpolation problems in which every supported block Is even, It is relatively straightforward to prove that the maximal augmented extremal set for f and its best approximation , P f~ from K must have n+2 elements which span the dual of n , . Thus, K is generalized Haar and uniqueness of best approximations hol ds ( 1).…”

The Remez exchange algorithm for approximation with linear restrictions

“…Existence and uniqueness of best approximation in all these examples was shown in [2] as a consequence of the fact that a generalized Haar condition held. We emphasize that the above stated Haar condition emplies the generalized Haar condition and therefore uniqueness of best approximation holds from Vq whenever Vn is Haar (on Í2).…”

“…This paper demonstrates a Remez exchange algorithm applicable to approximation of real-valued continuous functions of a real variable by polynomials of degree smaller than n with various linear restrictions. In [2] existence and uniqueness results were proved in this setting. As special cases are included the notion of restricted derivatives approximation (examples of which are monotone and convex approximation and restricted range approximation) and the notion of approximation with restrictions at poised Birkhoff data (examples of which are bounded coefficients approximation, e-interpolator approximation, and polynomial approximation with restrictions at Hermite and "Ferguson-Atkinson-Sharma" data and pyramid matrix data).…”

“…We are concerned then with approximating / by elements of PSWe showed in [2] that, if V% is not empty and Vn satisfies a generalized Haar condition, then there exists a unique best approximation p in Vq to /.…”

“…We emphasize that the above stated Haar condition emplies the generalized Haar condition and therefore uniqueness of best approximation holds from Vq whenever Vn is Haar (on Í2). For bibliographical references to previous work on these examples, see [2]. In the sequel P"_i will represent the space of real polynomials of degree no larger than « -1, i.e.…”