Sometimes, when working in higher-dimensional vector spaces, one runs into higher-dimensional delta distributions whose support is a linear subspace. A fairly common example from $\Bbb R^2$ would be, for instance, $\delta(x+y)$, as well as $\delta(x)$ and $\delta(y)$, all of which are basically lines in $\Bbb R^2$. These have sometimes been called "delta line functions."
These distributions are separable in the sense that we have $\delta(x, y) = \delta(x) \delta(y)$, where $\delta(x,y)$ is the point distribution whose support is the origin $(0,0)$, and the $\delta(x)$ and $\delta(y)$ are line distributions along the y and x axes.
We can generalize this to $\Bbb R^n$, where if $f$ is a linear functional, then $\delta(f(\mathbf{v}))$ is a delta hyperplane distribution which has support along the hyperplane that is in the nullspace of $f$. By taking the product of different hyperplane distributions, we can get arbitrary delta subspace distributions. If $M$ is a matrix whose rows are $f_1$, $f_2$, etc, we will use the notation $\delta(M\cdot\mathbf{v})$ for this, which is interpreted $\delta(f_1(v)) \delta(f_2(v)) ... \delta(f_m(v))$.
(I think this is equivalent to saying $\delta(||M\cdot\mathbf{v}||)$, where $||...||$ is any norm, but I'm not quite sure about the scaling. Question: is it?)
It is not difficult to see that the Fourier transform of any delta subspace distribution will be another subspace distribution whose support is the orthogonal complement of the original subspace. However, where I am getting thrown off here is in the scaling. Given some subspace of $\Bbb R^n$, there are many different bases for it, and I am interested in particular in those which are not orthonormal or even orthogonal. This matters even in the one-dimensional case, where we have $\delta(2x) = \delta(x)/2$, for instance. Now we are looking at arbitrary linear transformations instead. How do things scale when changing basis, and how do they scale when taking the Fourier transform? (I would imagine the Fourier transform divides by the $\ell_2$ norm of the exterior product of the rows, or the square of the $\ell_2$ norm, or something like that.)
Questions:
Suppose that we have some delta subspace distribution given by matrix $M$, i.e. what I termed $\delta(M\cdot \mathbf v)$ above. It is easy to see that the Fourier transform will be of the form $k \cdot \delta(\ker(M) \cdot \mathbf{\omega})$, where $\ker(M)$ is the null space of $M$, and $k$ is some constant that depends on $M$. What is a closed-form expression for $k$? I am sure it's going to be something like: the reciprocal of the norm of the exterior product of the rows of $M$ or something like that, e.g. the reciprocal of the square root of the Gram determinant, but it'd be nice to know for sure.
In general, the intent here with these delta subspace distributions is that the inner product of any function with one of them is the integral on the subspace spanned by the support of the distribution, and for non-orthonormal bases this result will be scaled accordingly in some way that depends on the volume of the parallelotope traced out by the basis vectors. Given, this have I correctly stated the separability criterion, that these distributions are just the product of the individual hyperplane distributions?