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Given a function $f:z\mapsto f(z)$ for a discrete set of points in the real interval $z\in[a,b]$ and the knowledge that $f$ is analytical along the real axis and that its Fourier transform is real valued along $\mathbb R$ (and thus $f(-z)=f^*(z)$), is it possible to reconstruct $f(z)$ for all $z\in\mathbb C$? If not, what approximations are possible?

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Are you familiar with analytical continuation? – Thomas Rot Mar 11 '11 at 8:38
@ThomasRot: doesn't that require knowledge along the whole real axis or some closed path in $\mathbb C$ already? – Tobias Kienzler Mar 11 '11 at 8:41
@Thomas you're right - I omitted the most important part, that $f$ is only known for some sampled points and not as an analytical expression – Tobias Kienzler Mar 11 '11 at 8:45
up vote 4 down vote accepted

No, it's not possible to reconstruct $f(z)$. You can augment your set of given function values by the corresponding points $-z$ to fulfill the symmetry condition, then pick an arbitrary sufficiently decaying even analytic function $g$, e.g. any Gaussian centered at $z=0$, and divide all the function values by the value of $g$ at that point. Then you can choose arbitrary polynomials that interpolate between the resulting values, and each such polynomial multiplied by $g$ will have the required properties. You have a lot of freedom of choice in $g$ and in the polynomials, so $f$ is very much not determined by these conditions.

Concerning the question what approximations are possible: For the polynomials, there's a natural choice, the unique polynomial of lowest degree that interpolates the function values divided by $g$. But there's no natural choice for $g$; different functions $g$ will lead to different interpolations for $f$, all with the required properties.

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If the set of points where $f$ is defined has an accumulation point, something might be possible though... – Thomas Rot Mar 11 '11 at 9:52
@Thomas Rot: I see now that I had implicitly assumed that only finitely many function values are given. I guess I looked at it as a computational problem (which the formulation "only known for some sampled points" in the OP's comment was suggestive of). If infinitely many function values are given, then a) there is necessarily an accumulation point, since $[a,b]$ is compact, b) my approach with polynomials wouldn't apply, and c) all derivatives at the accumulation point would be determined by the function values, so $f$ would be completely determined by analytic continuation. – joriki Mar 11 '11 at 10:11

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