Editing Discrete Fourier transform (section)

===Trigonometric interpolation polynomial===
The [[trigonometric interpolation polynomial]]
:<math>p(t) = \begin{cases}
  \displaystyle\frac{1}{N}
  \left[ \begin{alignat}{3}
  X_0 + X_1 e^{i 2\pi t} + \cdots
     &+ X_{\frac{N}{2}-1} e^{i 2\pi\big(\!\frac{N}{2}-1\!\big) t} &\\
     &+ X_{\frac{N}{2}} \cos(N\pi t) &\\
     &+ X_{\frac{N}{2}+1} e^{-i 2\pi\big(\!\frac{N}{2}-1\!\big) t} &+ \cdots + X_{N-1} e^{-i 2\pi t}
  \end{alignat} \right]
  & N\text{ even} \\
  \displaystyle\frac{1}{N}
  \left[ \begin{alignat}{3}
  X_0 + X_1 e^{i 2\pi t} + \cdots
     &+ X_{\frac{N-1}{2}} e^{i 2\pi\frac{N-1}{2} t} &\\
     &+ X_{\frac{N+1}{2}} e^{-i 2\pi\frac{N-1}{2} t} &+ \cdots + X_{N-1} e^{-i 2\pi t}
  \end{alignat} \right]
  & N\text{ odd}
\end{cases}</math>

where the coefficients ''X''<sub>''k''</sub> are given by the DFT of ''x''<sub>''n''</sub> above, satisfies the interpolation property <math>p(n/N) = x_n</math> for <math>n = 0, \ldots, N-1</math>.

For even ''N'', notice that the [[Nyquist frequency|Nyquist component]] <math display="inline">\frac{X_{N/2}}{N} \cos(N\pi t)</math> is handled specially.

This interpolation is ''not unique'': aliasing implies that one could add ''N'' to any of the complex-sinusoid frequencies (e.g. changing <math>e^{-it}</math> to <math>e^{i(N-1)t}</math>) without changing the interpolation property, but giving ''different'' values in between the <math>x_n</math> points.  The choice above, however, is typical because it has two useful properties.  First, it consists of sinusoids whose frequencies have the smallest possible magnitudes: the interpolation is [[bandlimited]]. Second, if the  <math>x_n</math> are real numbers, then <math>p(t)</math> is real as well.

In contrast, the most obvious trigonometric interpolation polynomial is the one in which the frequencies range from 0 to <math>N-1</math> (instead of roughly <math>-N/2</math> to <math>+N/2</math> as above), similar to the inverse DFT formula. This interpolation does ''not'' minimize the slope, and is ''not'' generally real-valued for real <math>x_n</math>; its use is a common mistake.