On the error of compound quadrature formulas forr-convex functions

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=">C _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 , denotes the modulus of continuity of orderr:

Keywords:	AMS (1980) subject classification" target="_blank">AMS (1980) subject classification  Primary 41A55  65D32
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