Is there a name for these polynomials? Given $t \in \mathbb{R}[0,1]$, consider the following set of polynomials:
$$        
\left[-{\left(t - 1\right)}^{2} t, {\left(t - 1\right)} {\left(t^{2} - t
- 1\right)}, -{\left(t^{2} - t - 1\right)} t, {\left(t - 1\right)}
t^{2}\right].
$$
They show up as the coefficients of an interpolation filter. I've put them in a form that reminds me of Bernstein polynomials.  They sum to unity, but don't appear to be orthogonal on $[0,1]$. They might be with respect to some weight function (or other interval).
Other than digging through lists of various types of polynomials, I'm at a loss for terminology to use in searching for further information. Does anyone recognize these as being part of some larger class, regardless of their connection to interpolation?

If you are interested in finding an orthonormal set of polynomials of the set of polynomials you have, then you can use Gram–Schmidt process to get it. Make sure your polynomials are linearly independent on the interval.
