How do I expand the Hermite Cubic Spline basis to the nth order?












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A useful basis for cubic polynomials are those used for Hermite interpolation:



$$h_{00}(t) = 2t^3-3t^2+1$$
$$h_{10}(t) = t^3-2t^2+t$$
$$h_{01}(t) = -2t^3+3t^2$$
$$h_{11}(t) = t^3-t^2$$



It is also possible to construct similar functions for interpolating with quintic polynomials. Is there a way to generate the polynomials for interpolating with $2k+1$th order polynomials, for $k in mathbb{Z}$










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    0












    $begingroup$


    A useful basis for cubic polynomials are those used for Hermite interpolation:



    $$h_{00}(t) = 2t^3-3t^2+1$$
    $$h_{10}(t) = t^3-2t^2+t$$
    $$h_{01}(t) = -2t^3+3t^2$$
    $$h_{11}(t) = t^3-t^2$$



    It is also possible to construct similar functions for interpolating with quintic polynomials. Is there a way to generate the polynomials for interpolating with $2k+1$th order polynomials, for $k in mathbb{Z}$










    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      A useful basis for cubic polynomials are those used for Hermite interpolation:



      $$h_{00}(t) = 2t^3-3t^2+1$$
      $$h_{10}(t) = t^3-2t^2+t$$
      $$h_{01}(t) = -2t^3+3t^2$$
      $$h_{11}(t) = t^3-t^2$$



      It is also possible to construct similar functions for interpolating with quintic polynomials. Is there a way to generate the polynomials for interpolating with $2k+1$th order polynomials, for $k in mathbb{Z}$










      share|cite|improve this question









      $endgroup$




      A useful basis for cubic polynomials are those used for Hermite interpolation:



      $$h_{00}(t) = 2t^3-3t^2+1$$
      $$h_{10}(t) = t^3-2t^2+t$$
      $$h_{01}(t) = -2t^3+3t^2$$
      $$h_{11}(t) = t^3-t^2$$



      It is also possible to construct similar functions for interpolating with quintic polynomials. Is there a way to generate the polynomials for interpolating with $2k+1$th order polynomials, for $k in mathbb{Z}$







      polynomials interpolation hermite-polynomials






      share|cite|improve this question













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      share|cite|improve this question










      asked Jan 8 at 20:39









      Fred FreyFred Frey

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      535






















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