Remarks on a paper of Gel'fand and Shilov on Fourier transforms
The principal result of this paper is the following: Let f be an entire function such that for every \(\epsilon >0\) but for no \(\epsilon <0\), \[ (i)\quad f(z)=O[\exp \{a+\epsilon)| y|^{\alpha}\}]\quad (| z| \to \infty) \] and \(f(t)=O[\exp \{(-b+\epsilon)| t|^{\alpha}\}]\) (\(| t| \to \infty)\) where \(a>0\), \(b>0\) and \(\alpha >1\). Then for every \(\epsilon >0\) but for no \(\epsilon <0\), \[ (ii)\quad \hat f(z)=O[\exp \{\beta^{-1}(\alpha b)^{-\beta /\alpha}+\epsilon \}| y|^{\beta}]\quad (| z| \to \infty) \] and \(\hat f(\)t)\(=O[\exp \{-\beta^{-1}(\alpha a)^{-\beta /\alpha}+\epsilon \}| t|^{\beta}\}]\) (\(| t| \to \infty)\), where \(\hat f\) denotes the Fourier transform of f. Conversely, every entire function satisfying condition (ii) for every \(\epsilon >0\) but for \(\epsilon <0\) is the Fourier transform of an entire function satisfying (i) for every \(\epsilon >0\) but for no \(\epsilon <0\). This result is a refinement of a classical result of \textit{I. M. Gel'fand} and \textit{G. E. Shilov} [Usp. Mat. Nauk 8, No. 6(58), 3--54 (1953; Zbl 0052.11605)] which has been an object of deep study by K. I. Babenko and S. Mandelbroit.
scientific article; zbMATH DE number 3879237
scientific article