- Padua points
In

polynomial interpolation of two variables, the**Padua points**are the first known example (and up to now the only one) of "unisolvent" point set (that is, the interpolating polynomial is unique) with "minimal growth" of their Lebesgue constant, proved to be O(log^{2}"n")citation

first1 = M. Caliari

last1 = L. Bos

first2 = S. De Marchi,

last2 = M. Vianello

first3 = Y.

last3 = Xu

title = Bivariate Lagrange interpolation at the Padua points: the generating curve approach

journal = J. Approx. Theory

volume = 143

issue = 1

pages = 15-25

year = 2006] .Their name is due to theUniversity of Padua , where they were originally discoveredcitation

first1 = S. De Marchi

last1 = M. Caliari

first2 = M.

last2 = Vianello

title = Bivariate polynomial interpolation at new nodal sets

journal = Appl. Math. Comput.

volume = 165

issue = 2

pages = 261-274

year = 2005] .The points are defined in the domain $scriptstyle\; [-1,1]\; imes\; [-1,1]\; subset\; mathbb\{R\}^2$. It is possible to use the points with four orientations, obtained with subsequent 90-degree rotations: in this way, what we get are 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 $n$ and family $s$ can be defined as:$ext\{Pad\}\_n^s=lbracemathbf\{xi\}=(xi\_1,xi\_2)\; brace=leftlbracegamma\_sleft(frac\{kpi\}\{n(n+1)\}\; ight),k=0,ldots,n(n+1)\; ight\; brace.$

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 $[-1,1]\; ^2$. The

cardinality of the set $scriptstyle\; ext\{Pad\}\_n^s$, obtained by $scriptstyle\; |\; ext\{Pad\}\_n^s|$ is $N=frac\{(n+1)(n+2)\}\{2\}$. Moreover, for each family of Padua points, two points lie on consecutive vertices of the square $[-1,1]\; ^2$, $2n-1$ points lie on the edges of the square, and the remaining points lie on the self-intersections of the generating curve inside the squarecitation

first1 = S. De Marchi

last1 = M. Caliari

first2 = M.

last2 = Vianello

title = Algorithm 886: Padua2D: Lagrange interpolation at Padua points on bivariate domains

journal = ACM T. Math. Software

year = 2008

volume = 35

issue = 3] citation

first1 = M. Vianello

last1 = L. Bos

first2 = Y.

last2 = Xu

title = Bivariate Lagrange interpolation at the Padua points: the ideal theory approach

journal = Numer. Math.

year = 2007

volume = 108

issue = 1

pages = 43-57] .The four generating curves are "closed" parametric curves in the interval $[0,2pi]$, and are a special case of Lissajous curves.

**The first family**The generating curve of Padua points of the first family is

:$gamma\_1(t)=\; [-cos((n+1)t),-cos(nt)]\; ,quad\; tin\; [0,pi]\; .$

If we sample it as written above, we have:

:$ext\{Pad\}\_n^1=lbracemathbf\{xi\}=(mu\_j,eta\_k),\; 0le\; jle\; n;\; 1le\; klelfloorfrac\{n\}\{2\}\; floor+1+delta\_j\; brace,$where $delta\_j=0$ when $n$ is even or odd but $j$ is even, $delta\_j=1$if $n$ and $k$ are both odd

with

:$mu\_j=cosleft(frac\{jpi\}\{n\}\; ight),\; eta\_k=egin\{cases\}cosleft(frac\{(2k-2)pi\}\{n+1\}\; ight)\; jmbox\{\; odd\}\; \backslash cosleft(frac\{(2k-1)pi\}\{n+1\}\; ight)\; jmbox\{\; even.\}end\{cases\}$

From this we can understand that the Padua points of first family will have two vertices on the bottom if $n$ is even, or on the left if $n$ is odd.

**The second family**The generating curve of Padua points of the second family is

:$gamma\_2(t)=\; [-cos(nt),-cos((n+1)t)]\; ,quad\; tin\; [0,pi]\; ,$

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

**The third family**The generating curve of Padua points of the third family is

:$gamma\_3(t)=\; [cos((n+1)t),cos(nt)]\; ,quad\; tin\; [0,pi]\; ,$

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

**The fourth family**The generating curve of Padua points of the fourth family is

:$gamma\_4(t)=\; [cos(nt),cos((n+1)t)]\; ,quad\; tin\; [0,pi]\; ,$

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

**The interpolation formula**The explicit representation of their fundamental

Lagrange polynomial is based on thereproducing kernel $scriptstyle\; K\_n(mathbf\{x\},mathbf\{y\})$, $scriptstyle\; mathbf\{x\}=(x\_1,x\_2)$ and $scriptstyle\; mathbf\{y\}=(y\_1,y\_2)$, of the space $scriptstylePi\_n^2(\; [-1,1]\; ^2)$ equipped with theinner product :$langle\; f,g\; angle\; =frac\{1\}\{pi^2\}\; int\_\{\; [-1,1]\; ^2\}\; f(x\_1,x\_2)g(x\_1,x\_2)frac\{dx\_1\}\{sqrt\{1-x\_1^2frac\{dx\_2\}\{sqrt\{1-x\_2^2$

defined by

:$K\_n(mathbf\{x\},mathbf\{y\})=sum\_\{k=0\}^nsum\_\{j=0\}^k\; hat\; T\_j(x\_1)hat\; T\_\{k-j\}(x\_2)hat\; T\_j(y\_1)hat\; T\_\{k-j\}(y\_2)$

with $scriptstyle\; hat\; T\_j$ representing the normalized

Chebyshev polynomial of degree $j$ (that is, $scriptstyle\; hat\; T\_0=T\_0$, $scriptstyle\; hat\; T\_p=sqrt\{2\}T\_p$ where $scriptstyle\; T\_p(cdot)=cos(parccos(cdot))$ is the classical Chebyshev polynomial "of first kind" of degree $p$). For the four families of Padua points, that we may denote by $scriptstyle\; ext\{Pad\}\_n^s=lbracemathbf\{xi\}=(xi\_1,xi\_2)\; brace$, $s=lbrace\; 1,2,3,4\; brace$, the interpolation formula of order $n$ of the function $scriptstyle\; fcolon\; [-1,1]\; ^2\; omathbb\{R\}^2$ on the generic target point $scriptstyle\; mathbf\{x\}in\; [-1,1]\; ^2$ is then:$mathcal\{L\}\_n^s\; f(mathbf\{x\})=sum\_\{mathbf\{xi\}in\; ext\{Pad\}\_n^s\}f(mathbf\{xi\})L^s\_\{mathbfxi\}(mathbf\{x\})$

where $scriptstyle\; L^s\_\{mathbfxi\}(mathbf\{x\})$ is the fundamental Lagrange polynomial

:$L^s\_\{mathbfxi\}(mathbf\{x\})=w\_\{mathbfxi\}(K\_n(mathbfxi,mathbf\{x\})-T\_n(xi\_i)T\_n(x\_i)),quad\; s=1,2,3,4,quad\; i=2-(smod\; 2).$

The weights $scriptstyle\; w\_\{mathbfxi\}$ are defined as

:$w\_\{mathbfxi\}=frac\{1\}\{n(n+1)\}cdotegin\{cases\}frac\{1\}\{2\}\; ext\{\; if\; \}mathbfxi\; ext\{\; is\; a\; vertex\; point\}\backslash 1\; ext\{\; if\; \}mathbfxi\; ext\{\; is\; an\; edge\; point\}\backslash 2\; ext\{\; if\; \}mathbfxi\; ext\{\; is\; an\; interior\; point.\}end\{cases\}$

**References**

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