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.

The four families

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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   and family   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   is  . 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 first family

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The generating curve of Padua points of the first family is

 

If we sample it as written above, we have:

 

where   when   is even or odd but   is even,   if   and   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   is odd.

The second family

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The generating curve of Padua points of the second family is

 

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

The third family

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The generating curve of Padua points of the third family is

 

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

The fourth family

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The generating curve of Padua points of the fourth family is

 

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

The interpolation formula

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

 

defined by

 

with   representing the normalized Chebyshev polynomial of degree   (that is,   and  , where   is the classical Chebyshev polynomial of first kind of degree  ).[3] For the four families of Padua points, which we may denote by  ,  , the interpolation formula of order   of the function   on the generic target point   is then

 

where   is the fundamental Lagrange polynomial

 

The weights   are defined as

 

References

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  1. ^ Caliari, Marco; Bos, Len; de Marchi, Stefano; Vianello, Marco; Xu, Yuan (2006), "Bivariate Lagrange interpolation at the Padua points: the generating curve approach", J. Approx. Theory, 143 (1): 15–25, arXiv:math/0604604, doi:10.1016/j.jat.2006.03.008
  2. ^ de Marchi, Stefano; Caliari, Marco; Vianello, Marco (2005), "Bivariate polynomial interpolation at new nodal sets", Appl. Math. Comput., 165 (2): 261–274, doi:10.1016/j.amc.2004.07.001
  3. ^ a b Caliari, Marco; de Marchi, Stefano; Vianello, Marco (2008), "Algorithm 886: Padua2D—Lagrange Interpolation at Padua Points on Bivariate Domains", ACM Transactions on Mathematical Software, 35 (3): 1–11, doi:10.1145/1391989.1391994
  4. ^ Bos, Len; de Marchi, Stefano; Vianello, Marco; Xu, Yuan (2007), "Bivariate Lagrange interpolation at the Padua points: the ideal theory approach", Numerische Mathematik, 108 (1): 43–57, arXiv:math/0604604, doi:10.1007/s00211-007-0112-z
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