Given a function $f(t): \mathbb{R}\to\mathbb{R}^n$, can 2D, or $n$D discrete Fourier transforms be used on $f(t)$ to perform frequency analysis?

Question

$\DeclareMathOperator{\R}{\mathbb{R}}$Frequency analysis is often performed on wave forms (1D DFT = discrete Fourier transform), and images (2D DFT), where the function in question often takes the form:

$f(t): \R \to \R$

$f(x,y): \R^2 \to \R$

$f(x_1, x_2, \ldots, x_n): \R^n \to \R$

However, note that in all 3 cases $f$ maps to a single real value. If, however, f takes the form:

$f(t): \R \to \R^n$

... it isn't clear to me whether the Fourier transform can be used to perform any kind of frequency analysis that would provide any information across dimensions.

If the dimensions are spatially correlated, eg, the sample $f_1(t_0)$ is physically adjacent to $f_2(t_0)$, and circularly adjacent to $f_n(t_0)$, then intuitively it would make sense to use the 2D DFT to perform the desired analysis. This interpretation, in essence, transforms f into the form $f(t,x): \R^2 \to \R$.

However, if no such relationship can be imposed on the set $\lbrace f_i\rbrace$, does there exist an analog of the FFT/DFT for this sort of problem?

To put this another way: if $f(t): \R \to \R^n$, and $f(t)$ is transformed in a similar fashion -- eg, $f(t,i): \R^2 \to \R$, where $i$ indexes into the dimensions of $f(t)$ -- is there a generalized approach to Fourier analysis that can make use of the index variable without making the assumption that $i=1$ and $i=2$ have any spatial relationship?

A vector-valued function can be used in the expression for computing a Fourier Transform, but unfortunately that results in computing the Fourier transform of each component of the vector-valued function without making use of information available in other dimensions of the range. In other words, if $f(t): \R \to \R^3$, then $F\lbrace f(t)\rbrace = (F\lbrace f_1(t)\rbrace, F\lbrace f_2(t)\rbrace, F\lbrace f_3(t)\rbrace)$. The question isn't whether this is possible, but whether more can be done than just this level of analysis.

A somewhat recent paper by Thomas Batard may answer the question, but I don't have the expertise to know whether it does. His paper, A metric approach to nD images edge detection with Clifford algebras, demonstrates a technique for performing analysis on color images, where the mapping might take a form similar to $(c,m,y,k) = f(x,y): \R^2 \to \R^4$.

As I have time, I would like to study Clifford algebras, but if this is a good path for me to go down it would give me more incentive to do so earlier rather than later.

Answer 1

It is reasonably straightforward to generalize traditional Fourier analysis to vector-valued functions $f$, because the transform process only requires the operations of adding and taking scalar multiples (or suitable limits thereof, like integrals). You end up with a vector-valued Fourier transform function $\check{f}$, and if you have a preferred basis like in your examples, then you can get the components of $\check{f}$ simply by transforming the components of $f$. I do not know what this has to do with Clifford algebras.

I should point out that the dimension of the DFT is defined as the dimension of the source space, not the target.