On Fourier transforms of functions of the R. Nevanlinna class in the half-plane

Abstract:

Let $f$ be a function holomorphic in the upper half-plane and belonging to the Nevanlinna class $N(\mathbb {C}_+)$. Assume that \[ \limsup \limits _{y \to +\infty } \frac {\ln |f(iy)|}{y} \le 0 \] and that the boundary values of $f$ on the real axis lie in $L^1(\mathbb {R})$. It is shown that if $\vert \widehat {f}(x)\vert \le \frac {1}{\lambda (|x|)}$, $x\in {\mathbb {R}_-}$, where $\widehat {f}$ is the Fourier transform of $f$ and $\lambda$ is a logarithmically convex positive function on ${\mathbb {R}_+}$, then the condition $\int _{1}^{+\infty }\frac {\ln \lambda (x)}{x^{3/2}} dx=+\infty$ implies that $\widehat {f}(x)=0$ for all $x\in {\mathbb {R}_-}$. Conversely, if one of the conditions listed above fails, then there exists $f\in N(\mathbb {C}_+) \cap L^1(\mathbb {R})$ with $\widehat {f}(x)\ne 0$, $x\in {\mathbb {R}_-}$.