A Best Nonlinear Quadrature For The Sobolev Class KWr[a, b]

As usual, denote by KW [a, b] the Sobolev class consisting of every function whose (r − 1)st derivative is absolutely continuous on the interval [a, b] and its rth derivative is bounded by K a.e. in [a, b]. For a function f ∈ KW [a, b], its values and derivatives up to r−1 order at a set of nodes x are known. These values are said to be given Hermite information. This work reports results on best quadrature based on the given Hermite information for the class KW [a, b]. Existence and concrete construction issue of the best quadrature is settled down by perfect spline interpolation. It turns out that the best quadrature depends on a system of algebraic equations satisfied by a set of free nodes of the interpolation perfect spline. From our another new result, it is shown that the system can be converted in a closed form to two single-variable polynomial equations, each being of degree approximately r/2. It is interesting to mention that the best quadrature and its worst case error bound, although nonlinear in nature, can be recursively expressed in terms of the given Hermite information via combinatorial analysis, obviating solving the nonlinear system. As a by-product, best interpolation formula for the class KW [a, b] is also obtained.