By João B. (Ed.) Prolla
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Extra info for Approximation Theory and Functional Analysis, Proceedings of the International Symposium on Approximation Theory
If, in concrete examples, one examines the methods to a proof of AF(K) methods also prove spaces E. p. c. spaces E. p. AF (K,E) = HF (K,E) holds for all complete Lc. spaces by Corollary 5, too. In fact, Corollary 5 demonstrates equ~valent. the two approaches which we have just outlined are REMARK: Similarly, if then the a. p. c. space and i f AF(K) =HF(K), E or of AF (K) = HF (K) also implies AF(K,E) =HF(K,E) gene~al. 2. APPROXIMATION ON PRODUCT SETS Let us now turn to a description of the £-product plete £-tensor product of two (or more) spaces of type resp.
T-ion -in uniquelydet~ned is ex S, hence in particular to onto a certain linear subspace of quently, g E A(Y,S) C(ex:s). tU- 6-ied: (al qEA(exS,S); (b) f qdv o and aU Condition (b) is still redundant. two Fy o qdv V E U B zEF z Y We return now to the discussion of the E x n amp l e: We choose for u a concave polygon proper vertices. •. ,a n + 1 on u is of the form are affine functions on [a,b J such that a j ~ 'i< KOROVKIN APPROXIMATION IN FUNCTION SPACES holds only in the trivial case point y in the interior of Furthermore ex S j =k 27 (n =1,2, ...
The following result generalizes only one part of Theorem 1. PROPOSITION 3: Ev~~y (JC,£)-a66~n~ 6unet~on ~~ ~eiative a Ko~ovkin 6unetion: ,,£ JC We sketch the proof: "fix) = fix) a on the set " given number hi,···,h~ and € C Kor (JC, £) . Let S = a£ be a function in f x. lex) < £ for all This implies for an arbitrary (JC,£)-admissible sequence (Tn) xES. that BAUER 32 (Tnf) converges uniformly on principle follows that Indeed, a function Ig(x) I < g S to f. From this and (Tnf) converges uniformly to II gil < satisfies £ E E the maximum f even on if and only X.
Approximation Theory and Functional Analysis, Proceedings of the International Symposium on Approximation Theory by João B. (Ed.) Prolla