A remark on extremal sets in the theory of polynomial interpolation

Let the numbers x1, x2, ..., x~ (n 2) be prescribed on the interval [ ~, 1] with — I ~ x1 <x~ ~ ... ~ x,~ ~ 1. Denoting the fundamental Lagrange interpolating polynomials by 4(x), we introduce the Lebesgue function )~~(x) ~ 1(x). The question arises as how to choose the nodes x1, x~,..., x~ in such a way that max %~(x) will be as small as possible. (A set of nodes with this~ property lxi is called an extremal set.) This is at famous unsolved problem in interpolation theory. interesting results in connection with it can be found in BERNsTEIN [1], ERDOs [4] and LurrMANN RIvLIN [5]; the paper by CI-IENEY—PRIcE [3] deals with problems of the same kind in a more~gençfal setting. A footnote in [his concerned with the particular cas&of three nodes. An extremal 2— 2—~. .. . 5 set Qonsisting of the points x1 —~-~-j/2, x2 0, x3 -~-~2 is exhibited with (Actually, the nodesdx1 x2, x3 with x1 ‘x3 ~- 1~2, x2. 0 are also extremal sets, cf. CHENEY[2], p. 65.) . Assuming 1 x1 and x,, 1, BERNSTEIN ([1], p. 1026 1027) conjectured that an extremal set of nodes is characterized by the prope~f~’ that the values of the n +1 local maxima of 2~(x) on the intervals [‘ 1, x1], [x1,x2], ..., [x~, 1] are all equal. This conjecture can also be found in [4]. Attention should also be drawn to a paper by LuTTMANN and RIyuN [5]. Thçy show e.g. that one can always take the ~ndpoints of [—1, 1] as nodes in an extrernal set.’ . The object of this note is to determine all extrernal sets in case of three nodes. Furthermore, we propose a modification of Bernstein’s conjecture.