In this short post, I am going to relate the relatively unknown notion of absement, the quantity
with some common transforms, in particular, with Fourier series and transforms more naturally.
The Fourier series for a periodic signal is
The complex forms reads
The coefficients for the real case read
and in the complex case
Here, the Fourier transforms reads
Key remark: the coefficient in the Fourier transforms is the average absement of .Indeed, the remaining coefficients are also the absement weighted with sinusoidal functions.
Exercise: given the Laplace transform
the discrete Fourier transformation
and the zeta transform
What is the relationship with the absement of the transformation coefficients? Interpret your results.