The S-Property and Best Approximation

The S-Property and Best Approximation

Zajel / An-Najah National University

Sawsan Azmi Sabri Al-Dwaik The problem     of best approximation is the problem of finding, for a given point x and a given set G in a normed space (X,~1.11), a point go in G which should be nearest to x a mono all points of the set G . However, in our study, we shall mainly take as X not an arbitrary normed space but Orlicz space, we shall denote by P(x,G), the set of all elements of best approximants of x in G . i.e P(x,G) = { go E G: lix - gold = inf { lix - gl~ : g E G} I The problem of best approximation began, in 1 853, with P. L. Chebyshev who considered the problem in the space of all real valued continuous function defined on [a,b], a closed real interval in R. My thesis consist of four chapters. Each chapter is divided into sections. A number like 2.1.3 indicates item (definition, theorem, corollary or lemma ) number 3 in section 1 of chapter 2 . Each chapter begins with a clear statement of the pertinent definitions and theorems together with illustrative and descriptive material. At the end of this thesis we present a collection of references . Some interesting results have been a chieved. A mong of which it is shown that if G has the S-property then 0(~t,G) has the S-property . It is also proved that if G has the S-property then Sawsan Azmi Sabri Al-Dwaik Supervisors Dr. Abdallah A. Hakawati Dr. Waleed Deeb 2000