On the Divergence of Polynomial Interpolation in the Complex Plane

We extend the results in [1] and [2] from the divergence of Hermite–Fejér interpolation in the complex plane to the divergence of arbitrary polynomial interpolation in the complex plane. In particular, we prove the following theorem: Let 1n = −1 ≤ t (n) 1 < · · · < t (n) n < 1. Let φ k be polynomials of arbitrary degree such that φ k (t (n) j ) = δk j . Then the Lebesgue function3n(x) = ∑n j=1 |φ (n) j (x)| tends to infinity at every complex neighborhood of some point in [−1, 1].