Polynomial is the simplest form of an interpolant. we can construct an unique polynomial of degree n that passes through (n+1) distinct data points.
\(P_n(x) =\sum_{i=0}^{n} y_i l_i(x) \)
here n in \( P_n(x) \) denotes the degree of the polynomial and
\(l_i(x) =\frac{x-x_0}{x_i-x_0} \frac{x-x_1}{x_i-x_1}…. \frac{x-x_i-1}{x_i-x_i-1}.\frac{x-x_i+1}{x_i-x_i+1} ….\frac{x-x_n}{x_i-x_n} \newline \newline l_i(x) = \prod_{j=0 , j \neq i}^{n} \frac {x-x_i}{x_i-x_j} , i =0,1,2,3….n \)
here \( l_i (x) for i = 0,1,2….n \) are called cardinal functions.
now let us consider an example
let n=1 then
\( P_1(x)=y_0 l_0 + y_1 l_1 \) is the interpolant for the straight line
where
\( l_0 (x) = \frac{x-x_1}{x_0-x_1} \) and \( l_1(x)= \frac{x-x_0}{x_1-x_0} \)
now let us consider n=2 ,then the equation becomes
\( P_2(x) = y_0 l_0(x) +y_ 1 l_1(x) +y_2 l_2(x) \) is the interpolant for the parabola (polynomial of degree 2)
here
\( l_0(x) = \frac{(x-x_1)(x-x_2)}{x_0-x_1)(x_0-x_2)} \newline \) ,
\( l_1(x) = \frac{(x-x_0)(x-x_2)}{x_1-x_0)(x_1-x_2)} \),
\( l_0(x) = \frac{(x-x_0)(x-x_1)}{x_2-x_0)(x_2-x_1)} \),
Cardinal fucntions are polynomials of degree n and have the property
\( l_i (x_i)= \Bigg\{ 0 \ if \ i \neq j \hspace{0.5cm} and \hspace{0.5cm} 1 \ if \ \ i =j \Bigg\} =\delta_ij \) ,
here the \( \delta_ij \) is known as kronecker delta
Now to verify that the interpolating polynomial passes through the given data points ,we will substitue \( x = x_j\) in \( p_n(x) = \sum_{i=0}^{n} y_i l_i(x) \) , and use \( l_i (x_i)= \Bigg\{ 0 \ if \ i \neq j \hspace{0.5cm} and \hspace{0.5cm} 1 \ if \ \ i =j \Bigg\} =\delta_ij \)
then we get ,
\( P_n(x_j) = \sum_{i=0}^{n} y_i l_i (x_j) = \sum_{i=0}^{n} y_i \delta_ij = y_j \)