Of Kopovikins Theory

Abstract

Tde by now classical tlleorem of P.P. Korovkin first appeared in [6] in 1953. It arose from the study of the role of Bernstein polynomials in the proof of the Weei rs trass approximation theorem. The theorem is simple in its staterrent and surprising in its conclusion:

Suppose L : C([O,l]) ~ C([O,l]) is a sequence of positive n

linear operators such that (L f) converges uniformly to f for n

the three test functions f(x)

1 , x,

2 , xc] 0, 1] . Then (1. f)

n converges uniformly to f on [0,1] for all functions fE C([O,1] ).

After the appearance of KorovkinI s theorem, a nurrber of extensions of his original results have been obtained by different authors. Aroundthe year 1966, Saskin [5] replaced [0,1] by an arbitrary corrpact Hausdorff space~_}{ and the set {L, f(x) = x,

of the three test fun" ctions by a special subset M of .. C(X). Subsequently, different rrodifications of Korovkin's theorem

and sorre of its irrplications were studied by Wulbert [4 ] , Bauer [1,2] etc. /Bauer used the well knownenvelope technique of the theory of integral representation in corrpact convex sets.

The first chapter shall be de,voted.to mathematical prelimi- naries which we shall need in the sequel. Howeverthe reader shall be assurred to have a working knowledgeof basic mathematical definitions and results as might be covered at undergraduate level.

In chapter II we shall give the proof of the classical

Korovkin theorem. Weshall also give sorre applications of the

3

theorem, including the proof of the classical Weierstrass approximation theorem with the help of this theorem. We shall also present Saskin's generalization of the classical Korovkirr theorem for compact Hausdorff spaces. Here, the compact interval [0,1 J shall be replaced by an arbitrary compact Hausdorff space

X and the set {I, f(x) = x, arbitrary subset M of CCX).

f(x) = 2

will be replaced by an

The Korovki n 's theorem and its generalization by Saskin as given in chapter II led to a quest for a deeper understanding on the role of the :test functions involved therein. In the case when [0, I] Ls replaced by an arbitrary corrpact Hausdorff space X, some G~estions naturally arise: If (In) is a sequence of positive operators (linear or non-linear)

on CCX) into itself, what size should the set of test functions

M be so that the convergence in the norm of the sequence (~t) -".......- ... to t for every test flllction t € M implies the convergence of

the sequence (II/) to f in the •

for every function

f € CCX).

Secondly for a given test set M, \\hich functions f € CCX)

wi l I satisfy the condition that 'the sequence (In£) converges to f in the norm whenever' (Ln) is a sequence of positive (linear or non-linear) operators on CCX) into itself such that the sequence (Lnt) converges in the norm to t for every

function t € M. These questions have been raised and answered by II, railer in [I] with the help of the w211 known envelope technique of notential theory. 'Ihey will be presented

4

in chapter III. Also the rrore recent work of H. Bauer on Korovkin's theorem for an arbitrary topological space X will be discussed in chapter III. Here the interval [ 0, I] will be replaced by an arbitrary topological space X and an arbitrary subset M of CCX) vri l L replace the set n, Ux) = x ,

f(x) =

2 }.

In the fourth and final chapter Vl(~ shall discuss KorovkinI s theorem for locally compact spaces. Here a locally compact space X will replace trie compact interval [0, I] and the set 2 {I, f(x) = x , f(x) = x} will be replaced by a subset H of

CCX)~lich shall be assumed to be adapted in the sense deter; bed ; n

[ 3 ].