On the Convergence of Kergin and Hakopian Interpolants at Leja Sequences for the Disk
Provides theoretical convergence guarantees for specific interpolation methods on the disk, addressing a gap in approximation theory for Leja sequences.
The paper proves that Kergin and Hakopian interpolation polynomials at Leja sequence points converge uniformly to sufficiently smooth functions on the unit disk, with derivatives also converging for C^∞ functions.
We prove that Kergin interpolation polynomials and Hakopian interpolation polynomials at the points of a Leja sequence for the unit disk $D$ of a sufficiently smooth function $f$ in a neighbourhood of $D$ converge uniformly to $f$ on $D$. Moreover, when $f$ is $C^\infty$ on $D$, all the derivatives of the interpolation polynomials converge uniformly to the corresponding derivatives of $f$.