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

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.

The four families

We can see the Padua point as a "sampling" of a parametric curve, called generating curve can be defined as

. The cardinality 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, and are a special case of Lissajous curves.

The first family

The generating curve of Padua points of the first family is

If we sample it as written above, we have:

are both odd

with

is odd.

The second family

The generating curve of Padua points of the second family is

is odd.

The third family

The generating curve of Padua points of the third family is

is odd.

The fourth family

The generating curve of Padua points of the fourth family is

is odd.

The interpolation formula

The explicit representation of their fundamental Lagrange polynomial is based on the reproducing kernel, of the space equipped with the inner product

defined by

representing the normalized Chebyshev polynomial is the classical Chebyshev polynomial of first kind). is then

is the fundamental Lagrange polynomial

are defined as

References

External links

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