By Christopher T. H. Baker, M. S. Derakhshan (auth.), Prof. Dr. H. Braß, Prof. Dr. G. Hämmerlin (eds.)
Read Online or Download Numerical Integration III: Proceedings of the Conference held at the Mathematisches Forschungsinstitut, Oberwolfach, Nov. 8 – 14, 1987 PDF
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Extra resources for Numerical Integration III: Proceedings of the Conference held at the Mathematisches Forschungsinstitut, Oberwolfach, Nov. 8 – 14, 1987
Inequality (14) we obtain the following result. Corollary 5. Let the assumptions be as in Theorem 2, and suppose that w is Riemann integrable in [aj,b j ], j=1,2, ••• ,k. Then it follows that lim i+oo inf n i Var(Q ni ) (29) ~ (l-A) 2 W 2 1-B 0 + k 27T L c j=l Finally, we consider the weight function w = 1. Applying Corollary 5 an explicit calculation of the right hand side of (29) shows that (30) lim i+oo ~ inf 2 4 cos ex 1-exc/7T + 2 (ex7T / c sin ex : = c/7T , - cos ex) , ex E(0,7T/2). 86861 ••.
Lemma 3. f. 52J) Let w> 0 almost everywhere in [-l,lJ, and let x G (v=l, ... ,n) be the nodes of the corresponG v,n . ding Gaussian formula Qn . Then, (48) lim n+oo 1 n n I v=l 1 f(x G ) v,n 11 f -1 for every function f, which is Riemann integrable in [-l,lJ. Lemma 4. f. BRASS [1977, p. 93J) ve quadrature formula with deg(Qn)~2m-1 , m E :IN, and let Let Q be a positinG Qn" ~ . Then each interval (49) contains at least one node of Qn Lemma 5. •• ) be positive and let lim l/deg(Q )=0. Then ni i+oo (50) (c f .
4 =. 1I'p' 18 On the other hand we have, using s-fold partial integration 2". Ua(QO) = sup 1I/(')11~1 II[f]1 = ~ sup 11/(')119 7rp jf(a)(x)sinpxdx 0 j Ismpxldx . = -7rpa4 . 2". = 7rpa -1 o So (5) is proved. From what has been said it should be clear that, leaving aside exceptional cases, QSt is the best quadrature rule for the estimation problem (3), (4) . 3. ,sinvx is called a trigonometric polynomial of order 1. Let 1/ denote the set of all trigonometric polynomials of order 1. Definition The quadrature rule Q has degree 1 if Q[t] = I[t] for all t E 1/.