The double descent and Runge phenomena in overparametrized polynomial interpolation

The Runge phenomenon in polynomial interpolation is often considered a classical analogue of the double descent phenomenon in machine learning. In this note, we explore overparameterized polynomial interpolation in three popular polynomial bases: Monomial, Chebyshev and Legendre basis with coefficients that are minimal in the ${\mathrm{\ell }}^2$-norm (and, for the monomial basis, also those minimal in the ${\mathrm{\ell }}^1$-norm). We present our results primarily for equidistant and Chebyshev data points, but many results are independent of the exact form of sampling.