A translation gives a change of phase, so if then (using a hat to denote the Fourier transform).
The Fourier transform commutes with rotations, so if then .
if is a scalar field in (3d space) with fourier transform and is a finite isometry (i.e a rotation and translation what is the fourier transform of
I keep wanting to write (i.e, a change of phase) but I aint so sure and I'm too dumb to prove it. Maybe I should say that doing this works for what I intend it for.
Thanks very much for your reply. There is too much detail to go through the whole problem, but the thing about rotations leaves me with a problem in that it won't work for what i want. I'm not saying you are wrong, I'm just stupid.
Can you prove it, or give me some pointers about how to. I would be gratefull. I have had a thought about it myself, and my argument is as follows.
with x --> x' a rotation or a translation. This isn't quite the same as what i thought in my first post, but it works. Unless I have done something completely wrong, I don't see what is wrong with this?
I'll use x and p to denote points in 3-dimensional space: x will be the variable for B-space (the space on wich the scalar field B is defined) and p will be the variable for -space (the space on which the Fourier trnsform is defined).
Let T:x→x' be an isometry of B-space. Then T consists of a rotation R followed by a translation x→x–a. If you are looking for a formula for the Fourier transform of B(Tx) that doesn't mention R and a explicitly, then I don't think you are going to find one, the reason being that the Fourier transform affects translations and rotations differently.
The Fourier transform of B is given by , where x.p denotes the inner product of x and p.
To find the effect of a translation, we have to calculate the Fourier transform of B'(x)=B(x–a). This is done by making the substitution y=x–a in the integral:
. . . . . .
To find the effect of a rotation R, we have to calculate the Fourier transform of B'(x)=B(Rx). This is again done by making a substitution, namely y=R(x). This time, we have to worry about how to express dy in terms of dx. For a general linear transformation R, the formula would be dy=Jdx, where J is the jacobian of the transformation, which is the determinant of R. Since R is a rotation, its determinant is 1, so in fact we just get dy=dx. We also need to use the fact that , where R^T is the transpose of R. This is also the inverse of R, so if then . If B'(x)=B(R(x)), the calculation then goes:
. . . . . .