Linear interpolation and Sobolev orthogonality

There is a strong connection between Sobolev orthogonality and Simultaneous Best Approximation and Interpolation. In particular, we consider very general interpolatory constraints , defined by

where f belongs to a certain Sobolev space, a_ij() are piecewise continuous functions over a,b], b_ijk are real numbers, and the points t_k belong to a,b] (the nonnegative integer m depends on each concrete interpolation scheme). For each f in this Sobolev space and for each integer l greater than or equal to the number of constraints considered, we compute the unique best approximation of f in , denoted by p_f, which fulfills the interpolatory data , and also the condition that best approximates f⁽ⁿ⁾ in (with respect to the norm induced by the continuous part of the original discrete–continuous bilinear form considered).