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?

  • $\begingroup$ What particular interpolation filter? $\endgroup$ – J. M. is a poor mathematician Jul 17 '12 at 12:24
  • $\begingroup$ @J.M. From a cubic convolution kernel. I didn't post too much about the derivation, because I'm interested in the abstract form of the result (i.e. other ways to get there, if any). $\endgroup$ – datageist Jul 17 '12 at 16:11
  • 1
    $\begingroup$ Something something cubic spline, yes? Apparently a cardinal spline with tension $c=-1$. $\endgroup$ – Rahul Jan 6 '13 at 23:52
  • $\begingroup$ Something like that, IIRC. $\endgroup$ – datageist Jan 7 '13 at 5:15

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.

  • $\begingroup$ Thanks. I'm actually more interested in properties they may have in general (besides just potential orthogonality). $\endgroup$ – datageist Jul 16 '12 at 2:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.