A uniform bound for the Lagrange polynomials of Leja points for the unit disk and applications in multivariate Lagrange interpolation

We study uniform estimates for the family of fundamental Lagrange polynomials associated with any Leja sequence for the complex unit disk. The main result claims that all these polynomials are uniformly bounded on the disk, i.e. independently on the range $N$ of the associated $N$-Leja section. We also give an improved estimate for special values of $N$. As a first application, we get an analogous estimate for any compact subset whose boundary is an Alper-smooth Jordan curve. We then deal with some applications in multivariate Lagrange interpolation. We first give some estimates of the Lebesgue constant for the intertwining sequence of Leja sequences, a result that also requires an explicit formula for the associated fundamental Lagrange polynomials with uniform estimates. We finish by dealing with a method to get explicit bidimensional Leja sequences and then give a positive example for the bidisk.