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 ${\displaystyle O(\log ^{2}n)}$.^[1] Their name is due to the University of Padua, where they were originally discovered.^[2]

The points are defined in the domain ${\displaystyle [-1,1]\times [-1,1]\subset \mathbb {R} ^{2}}$. 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

Padua points of the first family and of degree 5, plotted with their generating curve.

Padua points of the first family and of degree 6, plotted with their generating curve.

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 ${\displaystyle n}$ and family ${\displaystyle s}$ can be defined as

${\displaystyle {\text{Pad}}_{n}^{s}=\lbrace \mathbf {\xi } =(\xi _{1},\xi _{2})\rbrace =\left\lbrace \gamma _{s}\left({\frac {k\pi }{n(n+1)}}\right),k=0,\ldots ,n(n+1)\right\rbrace .}$

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 ${\displaystyle [-1,1]^{2}}$. The cardinality of the set ${\displaystyle \operatorname {Pad} _{n}^{s}}$ is ${\textstyle |\operatorname {Pad} _{n}^{s}|={\frac {(n+1)(n+2)}{2}}}$. Moreover, for each family of Padua points, two points lie on consecutive vertices of the square ${\displaystyle [-1,1]^{2}}$, ${\displaystyle 2n-1}$ points lie on the edges of the square, and the remaining points lie on the self-intersections of the generating curve inside the square.^[3][4]

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

The interpolation formula

The explicit representation of their fundamental Lagrange polynomial is based on the reproducing kernel ${\displaystyle K_{n}(\mathbf {x} ,\mathbf {y} )}$, ${\displaystyle \mathbf {x} =(x_{1},x_{2})}$ and ${\displaystyle \mathbf {y} =(y_{1},y_{2})}$, of the space ${\displaystyle \Pi _{n}^{2}([-1,1]^{2})}$ equipped with the inner product

${\displaystyle \langle f,g\rangle ={\frac {1}{\pi ^{2}}}\int _{[-1,1]^{2}}f(x_{1},x_{2})g(x_{1},x_{2}){\frac {dx_{1}}{\sqrt {1-x_{1}^{2}}}}{\frac {dx_{2}}{\sqrt {1-x_{2}^{2}}}}}$

defined by

${\displaystyle K_{n}(\mathbf {x} ,\mathbf {y} )=\sum _{k=0}^{n}\sum _{j=0}^{k}{\hat {T}}_{j}(x_{1}){\hat {T}}_{k-j}(x_{2}){\hat {T}}_{j}(y_{1}){\hat {T}}_{k-j}(y_{2})}$

with ${\displaystyle {\hat {T}}_{j}}$ representing the normalized Chebyshev polynomial of degree ${\displaystyle j}$ (that is, ${\displaystyle {\hat {T}}_{0}=T_{0}}$ and ${\displaystyle {\hat {T}}_{p}={\sqrt {2}}T_{p}}$, where ${\displaystyle T_{p}(\cdot )=\cos(p\arccos(\cdot ))}$ is the classical Chebyshev polynomial of first kind of degree ${\displaystyle p}$).^[3] For the four families of Padua points, which we may denote by ${\displaystyle \operatorname {Pad} _{n}^{s}=\lbrace \mathbf {\xi } =(\xi _{1},\xi _{2})\rbrace }$, ${\displaystyle s=\lbrace 1,2,3,4\rbrace }$, the interpolation formula of order ${\displaystyle n}$ of the function ${\displaystyle f\colon [-1,1]^{2}\to \mathbb {R} ^{2}}$ on the generic target point ${\displaystyle \mathbf {x} \in [-1,1]^{2}}$ is then

${\displaystyle {\mathcal {L}}_{n}^{s}f(\mathbf {x} )=\sum _{\mathbf {\xi } \in \operatorname {Pad} _{n}^{s}}f(\mathbf {\xi } )L_{\mathbf {\xi } }^{s}(\mathbf {x} )}$

where ${\displaystyle L_{\mathbf {\xi } }^{s}(\mathbf {x} )}$ is the fundamental Lagrange polynomial

${\displaystyle L_{\mathbf {\xi } }^{s}(\mathbf {x} )=w_{\mathbf {\xi } }(K_{n}(\mathbf {\xi } ,\mathbf {x} )-T_{n}(\xi _{i})T_{n}(x_{i})),\quad s=1,2,3,4,\quad i=2-(s\mod 2).}$

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

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

References

External links

• List of publications related to the Padua points and some interpolation software.