Another extremal property of perfect splines

Abstract:

Let ${\mathbf {t}} = \{ {t_i}\} ,i = 1,2, \ldots ,n + k$, be a given sequence in [a, b], ${f_0} \in W_\infty ^k[a,b],A > {\left \| {f_0^{(k)}} \right \|_\infty }$, and $F = \{ f \in W_\infty ^k[a,b]:f|{\mathbf {t}} = {f_0}|{\mathbf {t}}$ and ${\left \| {{f^{(k)}}} \right \|_\infty } \leqslant A\}$. We show that F contains precisely two perfect splines g, h of degree k with $|{g^{(k)}}| = |{h^{(k)}}| = A$ and n interior nodes, and for all $f \in F$, $\min (g(x),h(x)) \leqslant f(x) \leqslant \max (g(x),h(x));\forall x \in [a,b]$.