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It may be necessary to transform our representation to one in a new basis, say, ``\{\psi_m(x)\}_{m\ge0}``:
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```math
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f(x) \sim \sum_{m=0}^{\infty} g_m \psi_m(x).
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f(x) = \sum_{m=0}^{\infty} g_m \psi_m(x).
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```
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In many cases of interest, both representations are of finite length``n`` and we seek a fast method (faster than ``\mathcal{O}(n^2)``) to transform the original coefficients ``f_{\ell}`` to the new coefficients ``g_m``.
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In many cases of interest, both representations have finite dimension``n`` and we seek a fast method (faster than ``\mathcal{O}(n^2)``) to transform the original coefficients ``f_{\ell}`` to the new coefficients ``g_m``.
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A similar problem arises when we wish to evaluate ``f`` at a set of points ``\{x_m\}_{m=0}^n``. We wish to transform coefficients of ``f`` to values at the set of points in fewer than ``\mathcal{O}(n^2)`` operations.
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