On the error bounds of the Gauss-type quadrature formulae associated with spaces of parabolic and cubic spline functions with double equidistant knots

In two papers from 1995 P.~Köhler and G. Nikolov showed that Gauss-type quadrature formulae associated with spaces of spline functions with equidistant knots are asymptotically optimal in certain Sobolev classes of functions. In particular, Gauss-type quadratures associated with the spaces of spline functions of degree $r-1$ with double equispaced knots are asymptotically optimal definite quadrature formulae of order $r$ when $r$ is even, and it is conjectured that the asymptotical optimality property persists also in the case of odd $r$. For $r=3,\,4$, these quadrature formulae have been constructed by G. Nikolov, who also proved estimates for their error constants. The aim of this note is to refine the estimates for the error constant in the case $r=3$, and to point out to some error estimates in both cases $r=3$ and $r=4$, which are easier to evaluate and could be sharper than those which involve the uniform norm of the $r$-th derivative of the integrand.