Quadraturrest, Approximation und Chebyshev-Polynome
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The remainder of a quadrature formula is often treated by introducing derivatives of high order. But in many cases it is preferable to avoid derivatives and to use more robust methods. One can consider series expansions (Hilbert space, holomorphy). But there are simpler methods, employing polynomials, approximation, grids. In connection with quadrature such methods have been worked out by several authors; we mention Stroud, Locher-Zeller, Riess-Johnson and especially the recent book by Brass. In the present note we continue these investigations. We utilize approximation degrees of different order (in connection with Chebyshev expansions) and L-approximation. The general results are applied to the quadrature schemes of Clenshaw-Curtis and of Filippi. An outlook mentions extensions, cubature and fast approximation procedures.
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- BRASS, H. Quadraturverfahren. Göttingen-Zürich Vandenhoeck und Ruprecht 1977.Google Scholar
- CHENEY, E. W. Introduction to Approximation Theory. New York McGraw Hill 1966.Google Scholar
- DAVIS, P. J., RABINOWITZ, P. Methods of Numerical Integration. New York Academic Press 1975.Google Scholar
- ENGELS, H. Numerical Quadrature and Cubature. London Academic Press 1980.Google Scholar
- LOCHER, F. Optimale definite Polynome und Quadraturformeln. Internat. Ser. Numer. Math. 17, 111–121. Basel-Stuttgart Birkhäuser 1973.Google Scholar
- LORENTZ, G. G. Approximation of Functions. New York Holt, Rinehart and Winston 1966.Google Scholar
- MÜLLER, M. W. Approximationstheorie. Wiesbaden Akademische Verlagsgesellschaft 1978.Google Scholar
- SCHERER, R., ZELLER, K. Floppy vs. Fussy Approximation. Tagungsbericht Oberwolfach 1981 (in Internat. Ser. Numer. Math.).Google Scholar