A New Proof of the Snake Theorem

Abstract

AbstractThe Snake Theorem (terminology of Krein), due to Karlin in its original form, has been periodically improved. The theorem shows under appropriate conditions the existence of a function p* from a Tchebycheff space T, with a graph that alternately "touches" the graphs of functions f and g where f < g and f ≤ p* ≤ g on a compact interval [a, b]. The number of "touchings" depends upon the dimension of T. In this paper the conditions assumed are not the weakest known (see Gopinath and Kurshan, J. of Approximation Theory 21 (1977), 151–173), but the apparently new proof offered is elementary and fairly short. f and g are not assumed continuous.