Count Joseph Louis Lagrange, French mathematician was born in 1736 and died in 1813.
We look, in this paragraph, an expression of the polynomial with degree at most \(\mathfrak{n}\) taking the same values as a given function in \(\mathfrak{n} + 1\) given two, two distincts points, then to study the committed error by trying to minimize it.
We look \(\left( {\mathfrak{n} + 1} \right)\) polynomials \({L_0}, \ldots ,{L_n}\) all with degree at most \(\mathfrak{n}\), verifying the equalities :
Let \(i\) a natural integer element of given \(\left[\kern-0.15em\left[ {0,n}
\right]\kern-0.15em\right]\). The polynomial \({L_i}\) has a degree at most \(\mathfrak{n}\) and admits the \(\mathfrak{n}\) complex two, two distincts \({x_j}\), \(\mathfrak{j} \ne i\), as roots. Then, necessairely, it exist a constant \(C\) such as
\({L_i} = C\prod\limits_{i \ne j} {(X - {x_j})} \) .
The equality\({L_i}\left( {{x_i}} \right) = 1\) gives then \(C = \frac{1}{{\prod\limits_{j \ne i} {\left( {{x_i} - {x_j}} \right)} }}\) and then necessairely \({L_i} = \prod\limits_{j \ne i} {\left( {\frac{{X - {x_\mathfrak{j}}}}{{{x_i} - {x_\mathfrak{j}}}}} \right)} \) .
Reciprocally, if for all \(i \in \left[\kern-0.15em\left[ {0,n}
\right]\kern-0.15em\right],\)\({L_i} = \prod\limits_{\mathfrak{j} \ne i} {\left( {\frac{{X - {x_\mathfrak{j}}}}{{{x_{\dot i}} - {x_\mathfrak{j}}}}} \right)} \) then \({L_i}\) est well defined because the \({x_{\dot j}}\) are two, two distincts, with degree \(\mathfrak{n}\) exactly and finally the polynomials \({L_i}\) verify clairly the duality equalities: $\forall \left( i,~\mathfrak{j} \right)\in {{\left[\!\left[ 0,n \right]\!\right]}^{2}},\text{ }{{L}_{i}}\left( {{x}_{j}} \right)={{\delta }_{i,j}}$.
Let \(n\) a natural integer then \({x_0},{x_1},...,{x_n}\), \(\left( {n + 1} \right)\) given complex two, two distincts.
It exists one and only one set, noted \({({L_i})_{0 \le i \le \mathfrak{n}}}\), with \(\left( {n + 1} \right)\) polynomials of degree at most \(n\) verifying :
We look, in this paragraph, an expression of the polynomial with degree at most \(\mathfrak{n}\) taking the same values as a given function in \(\mathfrak{n} + 1\) given two, two distincts points, then to study the committed error by trying to minimize it.
1) Definition of polynomials \({L_k}\)
Let \(\mathfrak{n}\) be a natural integer then \({x_0},{x_1}, \ldots ,{x_n}\) \(\left( {\mathfrak{n} + 1} \right)\) complex two, two distincts.We look \(\left( {\mathfrak{n} + 1} \right)\) polynomials \({L_0}, \ldots ,{L_n}\) all with degree at most \(\mathfrak{n}\), verifying the equalities :
$\forall \left( i,~j \right)\in {{\left[\!\left[ 0,n \right]\!\right]}^{2}},~{{L}_{i}}\left( {{x}_{j}} \right)={{\delta }_{i,j}},$
(where \({\delta _{i,\mathfrak{j}}}\) is the kronecker symbol defined by \({\delta _{i,\mathfrak{j}}} = \left\{ \begin{array}{l}0 \ {\rm{ si }} \ i \ne j\\1 \ {\rm{ si }} \ i = j\end{array} \right.\) ).Let \(i\) a natural integer element of given \(\left[\kern-0.15em\left[ {0,n}
\right]\kern-0.15em\right]\). The polynomial \({L_i}\) has a degree at most \(\mathfrak{n}\) and admits the \(\mathfrak{n}\) complex two, two distincts \({x_j}\), \(\mathfrak{j} \ne i\), as roots. Then, necessairely, it exist a constant \(C\) such as
\({L_i} = C\prod\limits_{i \ne j} {(X - {x_j})} \) .
The equality\({L_i}\left( {{x_i}} \right) = 1\) gives then \(C = \frac{1}{{\prod\limits_{j \ne i} {\left( {{x_i} - {x_j}} \right)} }}\) and then necessairely \({L_i} = \prod\limits_{j \ne i} {\left( {\frac{{X - {x_\mathfrak{j}}}}{{{x_i} - {x_\mathfrak{j}}}}} \right)} \) .
Reciprocally, if for all \(i \in \left[\kern-0.15em\left[ {0,n}
\right]\kern-0.15em\right],\)\({L_i} = \prod\limits_{\mathfrak{j} \ne i} {\left( {\frac{{X - {x_\mathfrak{j}}}}{{{x_{\dot i}} - {x_\mathfrak{j}}}}} \right)} \) then \({L_i}\) est well defined because the \({x_{\dot j}}\) are two, two distincts, with degree \(\mathfrak{n}\) exactly and finally the polynomials \({L_i}\) verify clairly the duality equalities: $\forall \left( i,~\mathfrak{j} \right)\in {{\left[\!\left[ 0,n \right]\!\right]}^{2}},\text{ }{{L}_{i}}\left( {{x}_{j}} \right)={{\delta }_{i,j}}$.
Let \(n\) a natural integer then \({x_0},{x_1},...,{x_n}\), \(\left( {n + 1} \right)\) given complex two, two distincts.
It exists one and only one set, noted \({({L_i})_{0 \le i \le \mathfrak{n}}}\), with \(\left( {n + 1} \right)\) polynomials of degree at most \(n\) verifying :
$\forall \left( i,~j \right)\in {{\left[\!\left[ 0,n \right]\!\right]}^{2}},~{{L}_{i}}\left( {{x}_{i}} \right)={{\delta }_{i,j}}.$
Furthermore : $\forall \left( i,~j \right)\in {{\left[\!\left[ 0,n \right]\!\right]}^{2}},\text{ }{{L}_{i}}=\prod\limits_{j\ne i}{\left( \frac{X-{{x}_{j}}}{{{x}_{{\dot{i}}}}-{{x}_{j}}} \right)}$.
