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 ) {\displaystyle 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 ] × [ − 1 , 1 ] ⊂ R 2 {\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
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 {\displaystyle n} and family s {\displaystyle s} can be defined as
Pad n s = { ξ = ( ξ 1 , ξ 2 ) } = { γ s ( k π n ( n + 1 ) ) , k = 0 , … , n ( n + 1 ) } . {\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 [ − 1 , 1 ] 2 {\displaystyle [-1,1]^{2}} . The cardinality of the set Pad n s {\displaystyle \operatorname {Pad} _{n}^{s}} is | Pad n s | = ( n + 1 ) ( n + 2 ) 2 {\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 [ − 1 , 1 ] 2 {\displaystyle [-1,1]^{2}} , 2 n − 1 {\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. The four generating curves are closed parametric curves in the interval [ 0 , 2 π ] {\displaystyle [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
γ 1 ( t ) = [ − cos ( ( n + 1 ) t ) , − cos ( n t ) ] , t ∈ [ 0 , π ] . {\displaystyle \gamma _{1}(t)=[-\cos((n+1)t),-\cos(nt)],\quad t\in [0,\pi ].}
If we sample it as written above, we have:
Pad n 1 = { ξ = ( μ j , η k ) , 0 ≤ j ≤ n ; 1 ≤ k ≤ ⌊ n 2 ⌋ + 1 + δ j } , {\displaystyle \operatorname {Pad} _{n}^{1}=\lbrace \mathbf {\xi } =(\mu _{j},\eta _{k}),0\leq j\leq n;1\leq k\leq \lfloor {\frac {n}{2}}\rfloor +1+\delta _{j}\rbrace ,}
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