caa a22 4500 465824986 CHVBK 20180323112146.0 cr unu---uuuuu 170327e19900201xx s 000 0 eng 10.1007/BF01833939 doi (NATIONALLICENCE)springer-10.1007/BF01833939 Köhler Peter Institut für Angewandte Mathematik, TU Braunschweig, Pockelsstr. 14, D-3300, Braunschweig, West Germany aut On the error of compound quadrature formulas for r -convex functions [Elektronische Daten] [Peter Köhler] Summary: LetC m be a compound quadrature formula, i.e.C m is obtained by dividing the interval of integration [a, b] intom subintervals of equal length, and applying the same quadrature formulaQ n to every subinterval. LetR m be the corresponding error functional. Iff (r) > 0 impliesR m [f] > 0 (orR m [f] < 0), then we say thatC m is positive definite (or negative definite, respectively) of orderr. This is the case for most of the well-known quadrature formulas. The assumption thatf (r) > 0 may be weakened to the requirement that all divided differences of orderr off are non-negative. Thenf is calledr-convex. Now letC m be positive definite or negative definite of orderr, and letf be continuous andr-convex. We prove the following direct and inverse theorems for the errorR m [f], where ω, denotes the modulus of continuity of orderr: $$\left| {R_m [f]} \right| \leqslant \delta \frac{{b - a}}{m}\omega _{r - 1} \left( {f,\frac{{b - a}}{m}} \right),$$ , and, forμ sufficiently large, $$\omega _r \left( {F,\frac{{b - a}}{{2^\mu }}} \right) \leqslant \gamma 2^{ - \mu r} \left( {(b - a)\left\| f \right\| + \sum\limits_{k = 0}^{\mu - 1} {2^{kr} \left| {R_{2k} [f]} \right|} } \right),$$ , whereF is a primitive off, i.e.F′ = f, andδ andγ depend onQ n only. IfQ n is a Gauss, Lobatto or Radau rule, then, for 1 < α ⩽r, $$\omega _{r - 1} (f,t) = O(t^{\alpha - 1} ) \Leftrightarrow \left| {R_m [f]} \right| = O(m^{ - \alpha } )$$ and $$\int \in P_{r - 1} \Leftrightarrow R_m [f] = o(m^{ - r} ),$$ where P r−1 denotes the polynomials of degree less thanr. This generalizes results of Brass [3], Gaier [5] and Wolfe [12]. Birkhäuser Verlag, 1990 Primary 41A55, 65D32 nationallicence aequationes mathematicae Birkhäuser-Verlag 39/1(1990-02-01), 6-18 0001-9054 39:1<6 1990 39 10 https://doi.org/10.1007/BF01833939 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF01833939 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Köhler Peter Institut für Angewandte Mathematik, TU Braunschweig, Pockelsstr. 14, D-3300, Braunschweig, West Germany aut NATIONALLICENCE 773 0- aequationes mathematicae Birkhäuser-Verlag 39/1(1990-02-01), 6-18 0001-9054 39:1<6 1990 39 10 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer