Is it ever possible for two distinct polynomials to interpolate the same n data points? If so, under what conditions? If not, why?
We know that an interpolating polynomial can have different representations which interpolate the same n data points (depending on the set of basis used). However, these different representations yields the same polynomial when simplified (which explains why they interpolate same data points). What I want to know is if it is possible for distinct polynomials to interpolate the same n data points. Can someone help me with this, please?
You can always add a polynomial that is zero at all the sample points, which can be expressed as $q(x)(x-x_1)…(x-x_n)$.
You need to demand minimal degree to get the uniqueness of the interpolation polynomial.