On a property of the Fourier-cosine transform (Q1116394)

The authors show that the Fourier-cosine transform maps functions of the form \(t\mapsto \phi (1-s \tanh^ 2t)\cosh^{-\nu} t,\) with \(\phi\) an entire analytic function and Re \(\nu\) \(>0\), bijectively onto the functions \(x\mapsto \Gamma ((\nu +ix))\Gamma ((\nu -ix))\psi (x).\) Here \(\psi\) is an even and entire analytic function of sub-exponential growth, i.e. \(\sup_{z\in {\mathbb{C}}}e^{-\epsilon | z|}| \psi (z)| <\infty\) for all \(\epsilon >0\). The proof is based on estimates of real numbers \(c_{j,n}\), \(0\leq j\leq n<\infty\) which satisfy certain recurrence relations.