Let $d>2$. Let $M$ be a 2-dimensional submanifold of $\mathbb{R}^d$. For instance (and this is the type of example I primarily care about) we could have $M$ being the set of scalar multiples of a smooth curve that is such that this results in a 2-dimensional manifold.

How can I tell from the properties of $M$ whether or not the following is true:

Suppose $\mu_1$ and $\mu_2$ are compactly supported measures on $\mathbb{R}^d$, with Fourier transforms (characteristic functions) $\phi_1$ and $\phi_2$ respectively. If we suppose that $\forall \xi \in M$ we have $\phi_1(\xi)=\phi_2(\xi)$ then actually $\mu_1=\mu_2$.

In other words, I am asking for which dimensions $d$ and manifolds $M$ do we have that frequency information along $M$ determines functions that we know a priori are better than analytic: i.e. the fourier transform of bounded random variables with values in $\mathbb{R}^d$. In the case of the scalar multiples of a curve, this question comes down to asking when we can get away with much less than checking all projections in the Cramer Wold device. (Which states that to check the equality in distribution of random vectors, one need only check the projections are equal in distribution.)

We can also view this as a question about a special subclass of analytic variety, arising from analytic Fourier transforms.

Ideas and/or references are appreciated. I would be especially interested in knowing about existing work along these lines because my search has turned up nothing. Also helpful would be if this problem has a name.


An interesting article may be A sharp form of the Cramér-Wold theorem by Cuesta-Albertos, Fraiman and Ransford, which can be found here: klick

Their Theorem 3.1 states that under a moment condition a measures is well-defined by the values of the Fourier-transform on a set which "is not contained in any projective hypersurface."


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