Numerical Integration IV: Proceedings of the Conference at by Prof. Dr. Günter Baszenki, Prof. Dr. Franz-Jürgen Delvos PDF

By Prof. Dr. Günter Baszenki, Prof. Dr. Franz-Jürgen Delvos (auth.), H. Brass, G. Hämmerlin (eds.)

ISBN-10: 3034863381

ISBN-13: 9783034863384

ISBN-10: 3034863403

ISBN-13: 9783034863407

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Additional resources for Numerical Integration IV: Proceedings of the Conference at the Mathematical Research Institute, Oberwolfach, November 8–14, 1992

Example text

Let " (:e = II q(:e): ;=1 z;) be a polynomial with all zeros simple, and in [-1, I]. • , II,,: ;=1 Then Discr (II", arcsin) s C log0' X log (-I 0' ). ogX Here C is an absolute constant. Proof. 3. Let E,,: {:e 1", :e 2 " , . } be the points in I" . Choose a O'(n)-th Fekete set of points for E" (note that necessarily m" $ 2n - 1, so O'(n) $ n - 1) and let us denote them by :e;", 1 s j s O'(n). Let ,,(,,) q,,(:e) : = II (:e ;=1 :e;,,). Distribution of points in convergent sequences of . 1(b) . Since q" is monic of degree q(n), so 1~jg"lq~(x;,,)1 ~ Here we have also used ra(,,) 1/{2 c7(n)r" II I ,,1I 1/ 2 } = : 1/{B,,2a ( ,, ) } .

There are analogous results for interpolatory integration rules associated with weights on R. Again, convergent interpolatory integration rules have half their points distributed like zeros of appropriate orthogonal polynomials, and half arbitrarily distributed. Here the arcsin distribution is replaced by a Nevai-Ullman distribution, and the integration points are first scaled to [-1, IJ before forming a (now contracted) zero counting measure. See Bloom et al. (1992b). 2(a) assert that a suitable proportion of the integration po ints converge to the arcsin distribution.

4, 247-258 . HAMMERLIN, G. -H . (1989), Numerische Mathematik, Springer Verlag, Berlin, Heidelberg, New York. PETRAS, K. (1988), Asymptotic behaviour of Peano kernels of fixed order. In: H. Brass/ G. ) : Numerical integration III. Birkhauser Verlag, Basel, Boston, Berlin, 186-198. PETRAS, K. (1992a), Error bounds for Gaussian and related quadrature and applications to r-convex functions . SIAM J. Numer. Anal . 29, 578-585. PETRAS, K. (1992b), One sided L1-Approximation and bounds for Peano kernels.

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Numerical Integration IV: Proceedings of the Conference at the Mathematical Research Institute, Oberwolfach, November 8–14, 1992 by Prof. Dr. Günter Baszenki, Prof. Dr. Franz-Jürgen Delvos (auth.), H. Brass, G. Hämmerlin (eds.)


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