Approximation Theory: Proceedings of an International by Hans-Peter Blatt (auth.), Robert Schaback, Karl Scherer

By Hans-Peter Blatt (auth.), Robert Schaback, Karl Scherer (eds.)

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Extra resources for Approximation Theory: Proceedings of an International Colloquium Held at Bonn, Germany, June 8–11, 1976

Example text

D J , the latter a certain constant in- dependent of t . )l- r : 2mJ

Last 5 points and use the derivatives of these polynomials as approximations of the Tqo~ h (see Swartz- Varga for the derivatives [13]. To check if N is appropriately the defects computed for N and 2 N. For the ~-th step Of iteration the quotient 22(~+1) again. chosen we compare should behave approximatively like If that is not the case we go from N to 2 N and start If the corrections are smaller than a certain tolerance we stop. We made some numerical experiments on a UNIVAC 1108 with normal precision and compared the results gained by just one correction.

Explicitlyj assume that r for some m, aij = 0 whene_Eve_~rli-Jl > m, and that, for some positive K_ and K, and some p m [ l ~ ] , K_ flXllp <_ IfAxIlp < Z flXIlp, an x. 4O Then the entries of the inverse IbijZ _< A "l =: (bij) const ~li-jl for some const and some , satisf~ all l,J, X~[O,1) which depend only on m, p, K and K. I_~n particular ~ these constants do not depend on the order of the matrix A. The interested reader will have no difficulty in proving this theorem after a study of the following proof of Lemma 2, a proof which makes essential use of Demko's ideas, even though the inequality of Douglas, Dupont and Wahlbin fails to make an explicit appearance.

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