Padua points

Example

Let $n \ge 1$ and consider the curve $\gamma \colon [0, \pi] \to [-1, 1]^2$, $$\gamma(t) = \bigl(-\cos((n + 1) t), \ -\cos(n t)\bigr).$$ The Padua points of degree $n$ are the points $\gamma\bigl(k\pi / (n(n + 1))\bigr)$ for $k = 0, 1, \dots, n(n + 1)$. These are the self-intersections of $\gamma$ together with its points on the boundary of the square. For $n \ge 2$ several of these parameter values give the same point, and there are exactly $(n + 1)(n + 2)/2$ distinct Padua points.

This number is the dimension of the space of polynomials in two variables of total degree at most $n$. The Padua points are a unisolvent point set for this space. So each choice of values at the Padua points is taken by exactly one polynomial of total degree at most $n$.

Check for $n = 1$: the parameters $0$, $\pi/2$, $\pi$ give the points $(-1, -1)$, $(1, 0)$, $(-1, 1)$. They are not collinear, so they are unisolvent for the polynomials $a + bx + cy$.

The Lebesgue constant of interpolation at the Padua points is $O(\log^2 n)$ as $n \to \infty$.