Linear interpolation and Sobolev orthogonality

AbstractThere is a strong connection between Sobolev orthogonality and Simultaneous Best Approximation and Interpolation. In particular, we consider very general interpolatory constraints xi∗, defined by xi∗(f)=∫ab(∑j=0n−1aij(t)f(j)(t))dt+∑j=0n−1∑k=0mbijkf(j)(tk),0≤i≤n−1, where f belongs to a certain Sobolev space, aij(⋅) are piecewise continuous functions over [a,b], bijk are real numbers, and the points tk 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 Pl, denoted by pf, which fulfills the interpolatory data xi∗(pf)=xi∗(f), and also the condition that pf(n) best approximates f(n) in Pl−n (with respect to the norm induced by the continuous part of the original discrete–continuous bilinear form considered)