Chebyshev Circles Blueprint: Connection to discrete Fourier transform

3.1.1 Connection to discrete Fourier transform

The vanishing sums in Lemmas 3.1.1 and 3.1.2 reflect the orthogonality of the discrete Fourier basis. Define the discrete Fourier transform of a sequence \((a_0, \ldots , a_{N-1})\) as

\begin{equation} \hat{a}_m = \sum _{k=0}^{N-1} a_k \mathrm{e}^{-2\pi i mk/N}. \end{equation}

Our lemmas show that the constant sequence \(a_k = \mathrm{e}^{i\theta }\) (rotated unit phasor) has Fourier coefficients \(\hat{a}_m = 0\) for \(0 {\lt} m {\lt} N\) and \(\hat{a}_0 = N \mathrm{e}^{i\theta }\). This concentration of energy in the DC component (zero frequency) is characteristic of constant signals.

Similarly, for the sequence \(b_k = \mathrm{e}^{im(\theta + 2\pi k/N)}\) (complex exponential at frequency \(m\)), the projection onto the real axis (cosine) integrates to zero when summed uniformly over the roots of unity. This is the discrete analog of

\begin{equation} \int _0^{2\pi } \cos (m\theta ) \, d\theta = 0 \quad \text{for } m \neq 0. \end{equation}