applications - Given a function f(t): t -> R^n, can 2D, or nD DFTs be used on f(t) to perform frequency analysis?

Frequency analysis is often performed on wave forms (1D DFT), and images (2D DFT), where the function in question often takes the form:

$f(t): Re mapsto Re$

$f(x,y): Re^2 mapsto Re$

$f(x_1, x_2, ldots, x_n): Re^n mapsto Re$

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

$f(t): Re mapsto Re^n$

... 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 $f(t,x): Re^2 mapsto Re$.

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

To put this another way: if $f(t): Re mapsto Re^n$, and $f(t)$ is transformed in a similar fashion -- eg, $f(t,i): Re^2 mapsto Re$, 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 $f(t): Re mapsto Re^3$, then $Flbrace f(t)rbrace = (Flbrace f_1(t)rbrace, Flbrace f_2(t)rbrace, Flbrace f_3(t)rbrace)$. 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 $(c,m,y,k) = f(x,y): Re^2 mapsto Re^4$.

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.