These notes are based on Gunther Uhlmann’s lectures for MATH 581 taught at the University of Washington in Autumn 2009.
An index to all of the notes is available here.
We have studied the X-Ray transform on very restricted domains — and — and found an inversion formula there. Now we want to move to our next basic question: is the inversion stable?
As with the Radon transform, we will see that it is stable: small errors in the X-Ray transform lead to small errors in the reconstructed function, and vice versa, when errors are measured in an appropriate norms.
Sobolev spaces provide one family of norms that will work, but to use these we need to extend the domain of to distributions, and prove the inversion formula on the broader domain.
1. The X-Ray Transform of Distributions
As usual, we define on distributions by duality, and most of the results are tautologies.
Definition 1 For , define by its action on test functions
Proposition 2 is linear and continuous.
We define similarly.
Definition 3 For , define by its action on test functions
Proposition 4 is linear and continuous.
We can also define and for tempered distributions but we will see that these domains are more natural for the problem at hand because we want to study domains where we can use our inversion formula. But the inversion formula involves a fractional power of the Laplacian, which is not tempered!
2. Powers of the Laplacian, Sobolev Spaces
For define powers of the negative Laplacian using the fourier Transform as follows.
for , so that the right hand side is locally integrable.
Proposition 5 for and .
If then , so
proving the result. Now we need to consider the case . We will proceed by splitting the integral into a high-frequency and low-frequency part.
Where is some smooth function — your choice — that is identically 1 on a neighborhood of . But then
If you chose well, this shows that , and thus (with a slightly larger constant)
completing the proof.
So now we have defined on and defined on . You might wonder why we did not define these on tempered distributions; the definitions are straightforward after all.
The answer is that these operators appear alongside in the inversion formula and this operator is not tempered. For some ,try applying to a test function :
But this is not well defined because . For now, we will restrict our attention to domains where the inversion formula makes sense.
We’ll close this post with the obvious theorem. The Proof is left as an exercise.
Theorem 6 (X-Ray Inversion I and II For Distributions) The following identities hold
(A pdf version of these notes is available here.)