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The discrete Fourier transform (DFT) of $\{x_0,\ldots,x_{N-1}\}\subset\mathbb C$ is defined on Wikipedia as $$ X_k\stackrel{def}{=}\sum_{n=0}^{N-1}x_n\cdot e^{-2\pi ikn/N},\quad k\in\mathbb Z. $$ However, sometimes the DFT is defined for all real frequencies $\omega\in\mathbb R$ as $$ X_\omega\stackrel{def}{=}\sum_{n=0}^{N-1}x_n\cdot e^{-i\omega n},\quad \omega\in\mathbb R. $$ I am trying to understand why the frequencies \begin{equation} \tag{*}\label{freq} \frac{2\pi k}N \end{equation} with $k\in\mathbb Z$ are important. Are frequencies \eqref{freq} more important than other frequencies? Is it sufficient to consider the DFT only for frequencies $\eqref{freq}$? It seems that these frequencies are sometimes called the canonical frequencies. Is this a commonly used term or are there other terms for these frequencies?

Any help is much appreciated!

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    Where have you seen it defined like that? – 2017-02-10
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    @Nitin For example, page 7 of [this paper](https://arxiv.org/abs/1305.2073). In any case, we can define it for any $\omega\in\mathbb R$ and I am trying to understand the purpose of choosing $2\pi k/N$ with $k\in\mathbb Z$. – 2017-02-13

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One reason to consider these frequencies is that the matrix $$\frac{1}{\sqrt{N}}\left[e^{-2\pi ikn/N}\right]_{k,n=0}^{N-1}$$ is unitary. This is nice for computation, since the inverse is just the conjugate transpose (not that you should ever form this matrix while computing the DFT). Also, it is clear that since the set of signals $\{(x_{0},\ldots,x_{N-1}):x_{i}\in\mathbb{C},0\leq i\leq N-1\}$ is a vector space, the rows/columns of this matrix form a Fourier basis for the space, corresponding to the complex exponential functions with frequencies $2\pi k/N$ (or $2\pi n/N,$ if you pick columns), which makes interpreting the $X_{k}$ easier (they are just the coefficients in the representation of the signal $x$ in terms of this basis).

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    Thank you very much for the answer (+1)! Does the term "Fourier basis" have a particular meaning, i.e. is it a basis that have some particular properties? – 2017-07-04
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    A Fourier basis is either the one formed by the rows/columns of the matrix above, or the real version of this, which is formed by taking the corresponding vectors of sines/cosines to make a basis (which are the imaginary/real parts of the rows/columns of the matrix above). The only reason for the "Fourier" modifier is that he is credited with idea behind using such a basis. In any other sense, they're just a basis (which is not to say that these bases don't have special properties). – 2017-07-05