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 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 ff belongs to a certain Sobolev space, aij(⋅)aij(⋅) are piecewise continuous functions over [a,b][a,b], bijkbijk are real numbers, and the points tktk belong to [a,b][a,b] (the nonnegative integer mm depends on each concrete interpolation scheme). For each ff in this Sobolev space and for each integer ll greater than or equal to the number of constraints considered, we compute the unique best approximation of ff in PlPl, denoted by pfpf, which fulfills the interpolatory data xi∗(pf)=xi∗(f), and also the condition that pf(n) best approximates f(n)f(n) in Pl−nPl−n (with respect to the norm induced by the continuous part of the original discrete–continuous bilinear form considered).