Asymptotic Behaviour of Peanokernels of Fixed Order

Abstract

Many interpolation quadrature rules

$$ {Q_n}\left[ f \right] = \sum\limits_{v = 1}^n {{a_v}f({x_v}) \approx I\left[ f \right]} = \int_{ - 1}^1 {f(x)} w(x)dx,\quad - 1{x_1} < {x_2} < \ldots < {x_n}1 $$

, where w is a nonnegative integrable weight function, are known to be appropriate for the integration of functions having sufficiently high derivatives. On the other hand, little is known on the quadrature error R _n = I − Q _n in the spaces A ^s[−1, 1], s ≪ n, consisting of all functions with an s − 1th absolutely continuous derivative. If Q _n is exact for polynomials of degree s − 1, the error can be estimated for every f ∈ A ^s [−1, 1] by using the s th Peano kernel:

$$ \begin{array}{*{20}{c}} {{R_n}\left[ f \right] = \int_{ - 1}^1 {{f^s}(x)} {K_s}(x)dx,} \\ {{K_s}(x) = {R_n}\left[ {\frac{{(\cdot - x)_ + ^{s - 1}}}{{(s - 1)!}}} \right],\quad where\;t_ + ^{s - 1}: = \frac{1}{2}(1 + \operatorname{sgn} t)\cdot{t^{s - 1}}} \end{array} $$

(cf. Braß [1], p.39). We therefore have the unimprovable bounds

$$ \left| {{R_n}\left[ f \right]\left| {{{\left\| {{K_s}} \right\|}_p}\cdot{{\left\| {{f^{\left( s \right)}}} \right\|}_q};\;\frac{1}{p} + \frac{1}{q} = 1,1 < q\infty } \right.} \right. $$

and

$$ \left| {{R_n}\left[ f \right]\left| \leqslant \right.{{\left\| {{K_s}} \right\|}_\infty } \cdot Var\;{f^{\left( {s - 1} \right)}}.} \right. $$

(1.1)