Factoring Fourier transforms with zeros in a strip

Abstract:

$f$ is the Fourier transform of an infinitely differentiable function of compact support on ${\mathbf {R}}$ if, and only if, $f$ is entire and of exponential type with $\left | {f\left ( x \right )} \right | = O\left ( {|x{|^{ - N}}} \right )$ for each $N > 0$ as $|x| \to \infty$ for real $x$. In some sense, such an $f$ has its zeros close to the real axis and has positive density of zeros $F$ with $n\left ( r \right ) = Dr + o\left ( r \right )$. It is shown here that if the zeros of $f$ are in a strip parallel to the real axis and if $n\left ( r \right ) = Dr + O\left ( 1 \right )$, then $f$ is the product of two such transforms with zero densities $D/2$ and indicators one-half of the indicator of $f$. There is a factorable $f$ in $\widehat {\mathcal {D}}\left ( {\mathbf {R}} \right )$ with zeros on a line and not satisfying the stricter density condition. Analogous results hold for transforms of distributions of compact support on ${\mathbf {R}}$. The study was motivated by the outstanding problem of Ehrenpreis that asks if $\mathcal {D}\left ( {\mathbf {R}} \right ) * \mathcal {D}\left ( {\mathbf {R}} \right ) = \mathcal {D}\left ( {\mathbf {R}} \right )$.