By William Allan Light, Elliott Ward Cheney (auth.)
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Extra resources for Approximation Theory in Tensor Product Spaces
By convexity, there exists h" a O such t h a t fir(a) - h"[[ < A. Let ~ / b e a n e i g h b o r h o o d of a such t h a t ttr(8) - f(o-)ll < ~ - 11t(o9 - h"ll for all s E A/. ~) - J'(o)lJ + llf(o) Hence h " 6 ¢ ( s ) N 0 , s e 0", )/c - h"ll < ,x. 0 " , and 0 is open. 14) there exists g 6 C(S, H) such t h a t g(s) 6 ¢(8) for all s. Hence [If(s) - g(s)[[ < A and I[f - g[[ <- A. It follows t h a t dist (f, C(S, H)) < A and dist ( f , C ( S , H ) ) <_s u p d i s t ( f ( s ) , g } . s • Let H be a closed subspace of Y.
T) (~ = 3, 4 , . . 3 loglogn i f 0 < s < 1/n. One can prove the following facts. (i) The sequence ]lwn]lD is bounded. (ii) Every representation of R w , has the form (xn - c,) + (y, + c,) for some constant c,~. ) (iii) For any c,~ E JR, I1~. II + Ily. II -+ oo. 2, R(W) is not closed. 47 CHAPTER THE ALTERNATING 3 ALGORITHM Let X be a Banach space and U a subspace of X. Recall that a mapping A : X---oU is a p r o x i m i t y map if dist (x, U) = IIx - A ]J (x • x ) . Thus, A x is a best approximation to x in U.
We refer to the definition of the alternating algorithm as given at the beginning of Chapter 3. 1 L E M M A . +1) If n is odd, then (because A is a central proximity map) PROOF. ll = lix-+~ + ( ~ - + ~ - ~-)Ii = 11~- - A ~ . II- - A * . l l = 11~- - proof for even n is the same, except t h a t B replaces A. Now if This yields (ii) immediately. 2 L E M M A . A:r,~ + A x . l l = fix-li- II¢ll < 1, then | L e t A : X ..... ~ U be a central proxJmity m a p . C o r r e s p o n d i n g to each e x and ¢ ~ X* there is a~ element ¢ e X ~" such that II¢lI = II¢lI, ¢ + ¢ ~ v ± , ¢(x- Ax) = ¢(x- PROOF.
Approximation Theory in Tensor Product Spaces by William Allan Light, Elliott Ward Cheney (auth.)