Waves and Fourier sums |

A **Fourier sum** may be written

(6) |

(7) |

Observe that the top row of the matrix evaluates the polynomial at , a point where also . The second row evaluates , where is some base frequency. The third row evaluates the Fourier transform for , and the bottom row for . The matrix could have more than four rows for more frequencies and more columns for more time points. I have made the matrix square in order to show you next how we can find the inverse matrix. The size of the matrix in (8) is . If we choose the base frequency and hence correctly, the inverse matrix will be

Multiplying the matrix of (9) with that of (8), we first see that the diagonals are +1 as desired. To have the off diagonals vanish, we need various sums, such as and , to vanish. Every element (, for example, or ) is a unit vector in the complex plane. In order for the sums of the unit vectors to vanish, we must ensure that the vectors pull symmetrically away from the origin. A uniform distribution of directions meets this requirement. In other words, should be the -th root of unity, i.e.,

The lowest frequency is zero, corresponding to the top row of
(8).
The next-to-the-lowest frequency we find by setting in
(10) to
.
So
; and
for (9) to be inverse to (8),
the frequencies required are

Waves and Fourier sums |

2013-01-06