I'm an electrical engineer with a background in digital signal processing. I know the one-dimensional Z-transform as a tool to describe discrete-time signals/sequences. For instance, if a discrete-time sequence $h(k), k \in \mathbb{Z}$ is given, its Z-transform is given by $$ H(z) = \sum_{k=-\infty}^{\infty} h(k) z^{-k}. $$ Neatly, the discrete-time Fourier transform (DTFT) can be found by substituting $z = e^{j \omega}$. I recently found that a Z-transform (and Laplace transform for the continuous case) can be defined as an expected value, but I have not understood this completely yet. Frequently, Z-transforms are represented by a rational function $$ H(z) = \frac{ A(z) }{B(z)} = H_0 \frac{ \prod_i \left( z - z_{0,i} \right) }{\prod_j \left( z - z_{\infty,j} \right)}, $$ where $z_{0,i}$ describe the zeros and $z_{\infty,j}$ describe the poles.
If I wanted to describe a complex-valued function on the unit circle, $g: \mathbb{S}^1 \rightarrow \mathbb{C}$, I could perfectly describe it via the zeros and poles of a Z-transform.
Now I would like to extend this concept in order to describe a function on the unit sphere, $f: \mathbb{S}^2 \rightarrow \mathbb{C}$, instead of on the unit circle. I have complex-valued data (measurements of the function) on a very dense grid on the sphere, where each point is specified by two angles, e.g., azimuth and elevation. My question is if there is a higher-dimensional version of the Z-transform which I could use for this purpose. Or how could I define it? I would like to obtain a representation using poles and zeros again because that seems to be suitable for my application. I suspect that the poles and zero would then lie inside or outside the unit sphere.
2-D Z-transforms are used in image processing. However, the two dimensions are separable for rectangular images. I don't that would be the case for the two-dimensional extension of the Z-transform I look for here because the dimensions on the sphere are not separable but "interconnected".
Thank you for your help.
Well, I suppose you could start with this formulation:
$$H(z, w) = \sum_{l=-\infty}^\infty \sum_{k=-\infty}^\infty h[k,l] z^{-k} w^{-l}$$
With the following substitutions for the unit sphere:
$$ z=e^{j\omega}$$ $$ w = e^{j\xi} \text{ or } w= e^{j\xi/2}$$
How were you actually going to specify the function on the unit sphere? I assume you need a function of 2 angles unless you have a high degree of symmetry.