build the interpolation polynomial of degree higher than n-1

Question

I know that degree of the interpolation polynomial with n nodes cannot be more than n-1. I am asking about THE unique interpolation polynomial, what is the reason for it to have degree at-most-n-1 and not n. Can I build the interpolation polynomial of degree higher than n-1 for n points?

Answer 1

Interpolation polynomials can have higher degree. But then we cannot any longer talk about the interpolation polynomial. There is only one polynomial of degree at-most-$(n-1)$ going through any given set of $n$ points, but there are infinitely many degree-$n$ polynomials, and even more on higher degrees (for certain notions of "more").

For instance, given two points $(0,0)$ and $(1,1)$, the interpolation polynomial $x$ is a straight line between the points. If we go up to second degree, however, we have loads of other options, such as $x^2$, or $-3x^2+4x$, and so on.