AbstractLet the points (1)(xi,yi) (i=l,…, k; k⩾2), a⩽x1≤x2≤⋯ ≤xk⩽b, I= [a,b] (−∞≤a≤b≤∞) be prescribed. Furthermore, let m and n be integers such that l⩽n⩽k⩽m, and define the polynomial class Πm={P(x):P(x)επm,P(xi)=yi(i=l,…, k)}. Within Πm we determine Pm(x) as the solution of the extremum problem ∫I[P(n)(x)]2dx=minimum for P(x)εMm. Finally, let S(x) = S2n− 1(x) be the natural spline interpolant of degree 2n − 1 of the k points (1). Our main result is: (1) there is a unique polynomial Pm(x) which is the solution of the minimum problem (2); (2) we have limm→∞ Pm(x)=S(x) uniformly in xεI.