Padua points

In polynomial interpolation of two variables, the Padua points are the first known example (and up to now the only one) of a unisolvent point set (that is, the interpolating polynomial is unique) with minimal growth of their Lebesgue constant, proven to be $$O(\log^2 n)$$. Their name is due to the University of Padua, where they were originally discovered.

The points are defined in the domain $$[-1,1] \times [-1,1] \subset \mathbb{R}^2$$. It is possible to use the points with four orientations, obtained with subsequent 90-degree rotations: this way we get four different families of Padua points.

The four families


We can see the Padua point as a "sampling" of a parametric curve, called generating curve, which is slightly different for each of the four families, so that the points for interpolation degree $$n$$ and family $$s$$ can be defined as


 * $$\text{Pad}_n^s=\lbrace\mathbf{\xi}=(\xi_1,\xi_2)\rbrace=\left\lbrace\gamma_s\left(\frac{k\pi}{n(n+1)}\right),k=0,\ldots,n(n+1)\right\rbrace.$$

Actually, the Padua points lie exactly on the self-intersections of the curve, and on the intersections of the curve with the boundaries of the square $$[-1,1]^2$$. The cardinality of the set $$\operatorname{Pad}_n^s$$ is. Moreover, for each family of Padua points, two points lie on consecutive vertices of the square $$[-1,1]^2$$, $$2n-1$$ points lie on the edges of the square, and the remaining points lie on the self-intersections of the generating curve inside the square.

The four generating curves are closed parametric curves in the interval $$[0,2\pi]$$, and are a special case of Lissajous curves.

The first family
The generating curve of Padua points of the first family is


 * $$\gamma_1(t)=[-\cos((n+1)t),-\cos(nt)],\quad t\in [0,\pi].$$

If we sample it as written above, we have:


 * $$\operatorname{Pad}_n^1=\lbrace\mathbf{\xi}=(\mu_j,\eta_k), 0\le j\le n; 1\le k\le\lfloor\frac{n}{2}\rfloor+1+\delta_j\rbrace,$$

where $$\delta_j=0$$ when $$n$$ is even or odd but $$j$$ is even, $$\delta_j=1$$ if $$n$$ and $$k$$ are both odd

with


 * $$\mu_j=\cos\left(\frac{j\pi}{n}\right), \eta_k=

\begin{cases} \cos\left(\frac{(2k-2)\pi}{n+1}\right) & j\mbox{ odd} \\ \cos\left(\frac{(2k-1)\pi}{n+1}\right) & j\mbox{ even.} \end{cases} $$

From this follows that the Padua points of first family will have two vertices on the bottom if $$n$$ is even, or on the left if $$n$$ is odd.

The second family
The generating curve of Padua points of the second family is


 * $$\gamma_2(t)=[-\cos(nt),-\cos((n+1)t)],\quad t\in [0,\pi],$$

which leads to have vertices on the left if $$n$$ is even and on the bottom if $$n$$ is odd.

The third family
The generating curve of Padua points of the third family is


 * $$\gamma_3(t)=[\cos((n+1)t),\cos(nt)],\quad t\in [0,\pi],$$

which leads to have vertices on the top if $$n$$ is even and on the right if $$n$$ is odd.

The fourth family
The generating curve of Padua points of the fourth family is


 * $$\gamma_4(t)=[\cos(nt),\cos((n+1)t)],\quad t\in [0,\pi],$$

which leads to have vertices on the right if $$n$$ is even and on the top if $$n$$ is odd.

The interpolation formula
The explicit representation of their fundamental Lagrange polynomial is based on the reproducing kernel $$K_n(\mathbf{x},\mathbf{y})$$, $$\mathbf{x}=(x_1,x_2)$$ and $$\mathbf{y}=(y_1,y_2)$$, of the space $$\Pi_n^2([-1,1]^2)$$ equipped with the inner product


 * $$\langle f,g\rangle =\frac{1}{\pi^2} \int_{[-1,1]^2} f(x_1,x_2)g(x_1,x_2)\frac{dx_1}{\sqrt{1-x_1^2}}\frac{dx_2}{\sqrt{1-x_2^2}}

$$

defined by


 * $$K_n(\mathbf{x},\mathbf{y})=\sum_{k=0}^n\sum_{j=0}^k \hat T_j(x_1)\hat T_{k-j}(x_2)\hat T_j(y_1)\hat T_{k-j}(y_2)

$$

with $$\hat T_j$$ representing the normalized Chebyshev polynomial of degree $$j$$ (that is, $$\hat T_0=T_0$$ and $$\hat T_p=\sqrt{2}T_p$$, where $$T_p(\cdot)=\cos(p\arccos(\cdot))$$ is the classical Chebyshev polynomial of first kind of degree $$p$$). For the four families of Padua points, which we may denote by $$\operatorname{Pad}_n^s=\lbrace\mathbf{\xi}=(\xi_1,\xi_2)\rbrace$$, $$s=\lbrace 1,2,3,4\rbrace$$, the interpolation formula of order $$n$$ of the function $$f\colon [-1,1]^2\to\mathbb{R}^2$$ on the generic target point $$\mathbf{x}\in [-1,1]^2$$ is then



\mathcal{L}_n^s f(\mathbf{x})=\sum_{\mathbf{\xi}\in\operatorname{Pad}_n^s}f(\mathbf{\xi})L^s_{\mathbf\xi}(\mathbf{x}) $$

where $$L^s_{\mathbf\xi}(\mathbf{x})$$ is the fundamental Lagrange polynomial


 * $$L^s_{\mathbf\xi}(\mathbf{x})=w_{\mathbf\xi}(K_n(\mathbf\xi,\mathbf{x})-T_n(\xi_i)T_n(x_i)),\quad s=1,2,3,4,\quad i=2-(s\mod 2).

$$

The weights $$w_{\mathbf\xi}$$ are defined as



w_{\mathbf\xi}=\frac{1}{n(n+1)}\cdot \begin{cases} \frac{1}{2}\text{ if }\mathbf\xi\text{ is a vertex point}\\ 1\text{ if }\mathbf\xi\text{ is an edge point}\\ 2\text{ if }\mathbf\xi\text{ is an interior point.} \end{cases} $$