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

[1] Their name is due to the University of Padua, where they were originally discovered.

[2] The points are defined in the domain

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.

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

can be defined as 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

The cardinality of the set

Pad

{\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

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

, and are a special case of Lissajous curves.

The generating curve of Padua points of the first family is If we sample it as written above, we have: where

are both odd with From this follows that the Padua points of first family will have two vertices on the bottom if

is even, or on the left if

The generating curve of Padua points of the second family is which leads to have vertices on the left if

The generating curve of Padua points of the third family is which leads to have vertices on the top if

The generating curve of Padua points of the fourth family is which leads to have vertices on the right if

The explicit representation of their fundamental Lagrange polynomial is based on the reproducing kernel

equipped with the inner product defined by with

representing the normalized Chebyshev polynomial of degree

( ⋅ ) = cos ⁡ ( p arccos ⁡ ( ⋅ ) )

is the classical Chebyshev polynomial of first kind of degree

[3] For the four families of Padua points, which we may denote by

Pad

{\displaystyle \operatorname {Pad} _{n}^{s}=\lbrace \mathbf {\xi } =(\xi _{1},\xi _{2})\rbrace }

, the interpolation formula of order

on the generic target point

is the fundamental Lagrange polynomial The weights