Find the polynomial function

Question

Anybody knows how to find the polynomial function with evaluated values, where if the degree is $n$ I have $n+1$ values of the function like $f(0) = a_0, f(1) = a_1, \ldots, f(n) = a_n$.

Answer 1

Use $$p_i(x)=\frac {\prod_{i\neq j} (x-x_i)} {\prod_{i\neq j} (x_j - x_i)}$$ as a basis for the vector space, and notice that $p_i(x_j)=\delta_{ij}$.