A Differential Equation Approach to the Bernstein Problem
In 1932 Bernstein  conjectured that the optimal set of nodes for polynomial interpolation were the set of nodes which made the values of the extreme points equal for the corresponding Lebesgue function. He believed also that this set of nodes was unique. In 1947 Erdos [2,3] conjectured that for each set of nodes a de La Vallée Poussin type theorem was valid for the extremal points. In 1977 Kilgore [4,5] and de Boor, Pinkus  gave independent proofs of these conjectures. In this paper we present a constructive proof of these results and outline an algorithm based on the construction. We believe that these two conjectures are valid for a general Tchebycheff system and we hope to test out these propositions using an algorithm based on our methods. In this connection one is referred to Cheney and Kilgore  for some results in this direction.
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