Fast Discrete Polynomial Transforms with Applications to Data Analysis for Distance Transitive Graphs

@article{Driscoll1997FastDP,
  title={Fast Discrete Polynomial Transforms with Applications to Data Analysis for Distance Transitive Graphs},
  author={James R. Driscoll and Dennis M. Healy and Daniel N. Rockmore},
  journal={SIAM J. Comput.},
  year={1997},
  volume={26},
  pages={1066-1099}
}
Let $\poly = \{P_0,\dots,P_{n-1}\}$ denote a set of polynomials with complex coefficients. Let $\pts = \{z_0,\dots,z_{n-1}\}\subset \cplx$ denote any set of {\it sample points}. For any $f = (f_0,\dots,f_{n-1}) \in \cplx^n$, the {\it discrete polynomial transform} of $f$ (with respect to $\poly$ and $\pts$) is defined as the collection of sums, $\{\fhat(P_0),\dots,\fhat(P_{n-1})\}$, where $\fhat(P_j) = \langle f,P_j \rangle = \sum_{i=0}^{n-1} f_iP_j(z_i)w(i)$ for some associated weight function… 

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