Jacobi polynomials on the Bernstein ellipse
Provides theoretical extensions of polynomial extremal properties for spectral interpolation, but the results are incremental and domain-specific.
The paper derives explicit representations and extremal properties of Jacobi polynomials on the Bernstein ellipse, showing that the maximum modulus is attained at the major axis endpoints when α+β≥-1, and the minimum for Gegenbauer polynomials occurs at the minor axis endpoints. These results extend known special cases and provide refined asymptotic estimates.
In this paper, we are concerned with Jacobi polynomials $P_n^{(α,β)}(x)$ on the Bernstein ellipse with motivation mainly coming from recent studies of convergence rate of spectral interpolation. An explicit representation of $P_n^{(α,β)}(x)$ is derived in the variable of parametrization. This formula further allows us to show that the maximum value of $\left|P_n^{(α,β)}(z)\right|$ over the Bernstein ellipse is attained at one of the endpoints of the major axis if $α+β\geq -1$. For the minimum value, we are able to show that for a large class of Gegenbauer polynomials (i.e., $α=β$), it is attained at two endpoints of the minor axis. These results particularly extend those previously known only for some special cases. Moreover, we obtain a more refined asymptotic estimate for Jacobi polynomials on the Bernstein ellipse.