Finding the basis functions given the boundary values and values of derivatives at the boundary

Question

Given an interval $I=[a,b]$ we define $$P_3(I):=\{v:I\rightarrow\mathbb{R}\mid v \text{ is a polynomial of degree} \leq 3 \text{ i.e } \\v=a_3x^3+a_2x^2+a_1x+a_0 \text{ for } a_i\in\mathbb{R}\}.$$ How can we determine the basis functions $b_i(x)$ such that $$v(x)=v(a)b_0(x)+v'(a)b_1(x)+v(b)b_2(x)+v'(b)b_3(x)$$ if $v(a),v(b),v'(a),v'(b)$ are already given.

Answer 1

This is called the Lagrange form of Hermite interpolation, $$ p(x)=\sum_{i=0}^n f_i\; l_{i,0}(x)+\sum_{i=0}^n f_i^\prime\; l_{i,1}(x) $$ The standard Lagrange basis polynomials $l_i(x)$ of degree $n$ are used to construct $2n+1$ basis polynomials $l_{i,0}(x)$ and $l_{i,1}(x)$ such that $$ \begin{align} l_{i,1}(x_j)& =0 & l_{i,0}(x_j)& =\delta_{ij}\\ l_{i,1}^\prime(x_j)& =\delta_{ij} & l_{i,0}^\prime(x_j)& =0 \end{align} $$ Let us choose $l_{i,1}(x)=(x-x_i)l_i^2(x)$ and less obviously $l_{i,0}(x)= l_i^2(x)-2l_i^\prime(x_i)(x-x_i)l_i^2(x)$. Verify that the conditions are satisfied.

The above and even your 2-point case can be found in the Ch6.3 of Kincaid&Cheney's "Numerical Analysis".