Approximation Theory and Functional Analysis, Proceedings of by João B. (Ed.) Prolla PDF

By João B. (Ed.) Prolla

ISBN-10: 0444852646

ISBN-13: 9780444852649

ISBN-10: 0720419646

ISBN-13: 9780720419641

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Extra info for Approximation Theory and Functional Analysis, Proceedings of the International Symposium on Approximation Theory

Example text

6pace. opace -<-6 and only THEOREM 2: '<'6 0xx = x. It is this result which allows in many concrete examples a quick proof of a Korovkin-type theorem. In particular, Korovkin'sclassical result follows almost immediately. It states that, for a compact interval X = [a,b] on the real line functions 1, id, id 2 (id = identity lR, map the linear hull of the three x ~ x) is a Korovkin space. II. DETERMINATION AND GEOMETRICAL INTERPRETATION OF THE KOROVKIN CLOSURE In the existing literature few attention has been given to the determination of space, hence where Kor (:IC) 0XX for the case where * X.

Rl J f(x)dx £={fEC(X) f (0) }, o ~ {f E £ (f( ~) + fell) = f(O)} . It is easy to check that 10,1] • We shall see that 3) Let Kor(X,£) = X . e. the topological QXX = a£x boundary of which are harmonic in the open disk. X. We shall see that Kor (JC , £) = £ . Saskin [ 9 I announced a result that - at least for the case of a finite dimensional space and sufficient for X - the condition Kor(X,£) = £. QXX a£ X is necessary The first two examples show that 30 BAUER this is not true. £ x space with respect to is closed.

HF(K,E) must be verified for the first as- p be a continuous seminorm g E FE ( U) suc h t h at definition, FE (U) = E on compact subsets of E F (U) on ~~~ E, E U~ > K and 0 and a E B ut, aga1n . by p ( f (x) - g (x) ) <;[. (with the topology of uniform convergence U). Without loss of generality, we may assume and hence the a. p. of Schwartz's each E, f E HF(K,E). By definition, there exists an open set Em, OM e' E E' ne~ghbouILhood 0 ~o, and only ~ pace gte' ,E) E F (U) I ~o C(K) is a topological linear subspace of C(K,E) and ° HF(K) sertion.

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Approximation Theory and Functional Analysis, Proceedings of the International Symposium on Approximation Theory by João B. (Ed.) Prolla


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