Geodesic
On Fourier transforms of functions of Nevanlinna class in a half-plane
Algebra i analiz, Tome 20 (2008) no. 4, pp. 218-240.

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Let $f$ be a function holomorphic in the upper half-plane and belonging to the Nevanlinna class $N(\mathbb{C}_+)$. Assume that $$ \varlimsup_{y\to+\infty}\frac{\ln|f(iy)|}{y}\le 0 $$ 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 $\widehat{f}(x)\ne 0$, $x\in{\mathbb{R}_-}$.
Keywords: Function of bounded characteristic, Fourier transform. 
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F. A. Shamoyan. On Fourier transforms of functions of Nevanlinna class in a half-plane. Algebra i analiz, Tome 20 (2008) no. 4, pp. 218-240. http://geodesic.mathdoc.fr/item/AA_2008_20_4_a7/

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