ISSN:
1420-8903
Keywords:
Primary 41A55, 65D32
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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].
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01833939
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