Tag: variables

Notation: I will use the following conventions for discrete Fourier transforms (DFT) and discrete time Fourier transforms (DTFT):
$$mathcal{D}_N(x_j)(k) := sum_{j=0}^{N-1} e^{-2pi i j k} x_j$$
$$mathcal{D}^{-1}_N(y_k)(j) := frac{1}{N}sum_{k=0}^{N-1} e^{2pi i j k} y_k$$
$$mathcal{F}(x_j)(k) := sum_{j=-infty}^infty e^{-2pi i j k} x_j$$
$$mathcal{F}^{-1}(y(k))(j) := int_0^1 e^{2pi i j k} y(k)$$
Additionally, $Omega_N$ will denote the set of $N^{th}$ roots of unity.

Let $f$ be a polynomial of degree $N$ with coefficients $c_j$, $j=0,1,dots,N$. Consider the list of all coefficients as a vector $c$. By taking a DFT of $c$ you can show that $$c_j = mathcal{D}_N^{-1}left(fleft(e^{-2pi i k/N}right)right)(j)tag{1}$$ And analogously with a DTFT $$c_j = mathcal{F}^{-1}left(fleft(e^{-2pi i k}right)right)(j)tag{2}$$

Actually the latter formula works perfectly well for analytic functions $f$, where $c$ is the the vector of Taylor coefficients, or with meromorphic functions and $c$ the Laurant coefficients.

I realized recently that this is basically just Cauchy’s integral formula with a change of variables. To see this, expand out the inverse DTFT:
$$begin{align*}
c_j & = int_0^1 e^{2pi i k j} f(e^{-2pi i k}) dk \
& = oint_{S^1} z^{-j} f(z) frac{dz}{2pi i z} \
& = frac{1}{j!} frac{d^j f(0)}{dz^j}
end{align*}$$
Where the last line is from Cauchy’s formula.

So far this is all familiar. However, from this perspective we see that eq. (1) can also be interpreted as a “discrete” version of Cauchy’s integral formula, valid only for polynomials. Rewriting it slightly, we have
$$begin{align*}
frac{1}{j!}frac{d^j f(0)}{dz^j} = c_j & = frac{1}{N} sum_k e^{2pi i j k} f(e^{-2pi i k/N})\
& = frac{1}{N} sum_{zin Omega_N} z^{-j} f(z) tag{3}\
end{align*}$$

This circle of ideas seems like it would be useful for e.g. efficiently determining coefficients of a “black box” polynomial that you can only evaluate. Or more generally any situation where you would want to use Cauchy’s formula with a polynomial, this might be a convenient alternative. However, I’ve never seen any of this before in textbooks or other literature.

Question: What, if anything, are eqs. (1) and (3) useful for? Citations to literature welcome.

As I wrote this, I remembered vaguely something about “z-transforms”. After looking this up, it seems very closely related, though not quite identical. I still haven’t found anything exactly like eqs. (1) or (3).