Leírás
For classical polynomial interpolation on a closed interval Bernstein conjectured in 1931 that for the minimal norm of the interpolation operator a characterising property is equioscillation of the interval maxima of the Lebesgue function on the intervals between the nodes. This famous conjecture, together with an additional one by Erdős, were proved in 1978. Later, T. Kilgore published several extensions of the result to various settings of interpolation operators including exponentially weighted interpolation on the halfline.
Both in the classical solution and in the known other cases the crux of the proof was always a nonsingularity statement about Jacobi (derivative) matrices. However, recently Patrícia Szokol found counterexamples to the nonsingularity statement in Kilgore's argument for the exponential weight, turning his statement to a conjecture. We present a proof of this conjecture. This is the first proof of a Bernstein-type equioscillation result in a setup where singularity is present.