Conjectures of Bernstein and Erd\H os for weighted Lagrange interpolation on the halfline with exponential weights

Let I=[a,b] and consider the degree n Lagrange interpolation at the nodes x, where x\in S:={x=(x_0,x_1,...,x_n):a=x_0<x_1<...<x_n=b}. Then the norm of the Lagrange interpolation operator is the maximum of the Lebesgue function L(x,t) on I.
Bernstein conjectured that the norm of the Lagrange interpolation operator becomes minimal exactly for node systems which exhibit an equioscillation property in that the interval maxima m_k(x):=max_{[x_{k-1},x_k]} L(x,.)}, (k=1,...,n) are all equal. Erdős added to the conjecture the sandwich property that if y is an extremal (minimal norm) system, then for any other node system x there have to be indices i,j with m_i(y)<m_i(x) and m_j(y)> m_j(x).
The conjectures were proved by Kilgore and de Boor--Pinkus in 1978. Since then, analogous results were obtained only for a few cases when interpolation is made to certain very special spaces of polynomials, or when we apply weighted interpolation with rather special weights. Worse than that, it turned out that published proofs of results on infinite intervals and weighted interpolation were seriously flawed.
Here we prove the Bernstein and Erd\H os Conjectures for the case of exponentially weighted polynomials on the halfline. This is the first proof of these conjectures in a situation where, contrary to all existing successful proofs, we encounter singularity of certain derivative matrices.