Optimising the Choice of Interpolation Nodes with a Forbidden Region

We consider the problem of optimizing the choice of interpolation nodes such that the interpolation error is minimized, given the constraint that none of the nodes may be placed inside a forbidden region. Restricting the problem to using one-dimensional polynomial interpolants, we explore different ways of quantifying the interpolation error; such as the integral of the absolute/squared difference between the interpolated function and the interpolant, or the Lebesgue constant, which compares the interpolant with the best possible approximating polynomial of a given degree. The interpolation error then serves as a cost function that we intend to minimize using gradient-based optimization algorithms. The results are compared with existing theory about the optimal choice of interpolation nodes in the absence of a forbidden region (mainly due to Chebyshev) and indicate that the Chebyshev points of the second kind are near-optimal as interpolation nodes for optimizing the Lebesgue constant, whereas placing the points as close as possible to the forbidden region seems optimal for minimizing the integral of the difference between the interpolated function and the interpolant. We conclude that the Chebyshev points of the second kind serve as a great choice of interpolation nodes, even with the constraint on the placement of the nodes explored in this paper, and that the interpolation nodes should be placed as close as possible to the forbidden region in order to minimize the interpolation error