I am reading this paper and I am a little confused by the proof of Theorem 2.5.
If I understand them correctly, they claim, that real-valued, bandlimited L^p functions are holomorphic. This seems utterly wrong to me, since the only real-valued functions that are holomorphic are the constant ones.
Is there another definiton of the word holomorphic in this context?
Take $f \in L^1$ then $\hat{f}(\xi) = \int_{-\infty}^\infty e^{-2i \pi \xi x} f(x)dx$ is continuous. Assume $\text{supp }(\hat{f}) \subset [a,b]$ and let $$g(z) = \int_a^b e^{2i \pi \xi z} \hat{f}(\xi)d\xi$$ Clearly this last integral is entire in $z$ (holomorphic on the whole complex plane) and by the Fourier inversion theorem $\|g - f\|_{L^1} = 0$ and $g = f$ a.e.
Take $f \in L^p, p \ge 1$ then there is $n \in \mathbb{Z}_{\ge 1}$ such that $\forall \epsilon > 0$, $f(x) e^{-\epsilon x^{2n}} \in L^1$ (*) so that $f(x) e^{-\epsilon x^{2n}}$ is a.e. equal to an entire function which means $f(x)$ is a.e. equal to an entire function.
(*) not sure of this part : maybe you'll need something more subtle such as an entire function with arbitrary fast growth rate