Some theorems concerning holomorphic Fourier transforms
Let \(a>0\), \(\alpha >1\), \(\alpha^{-1}+\beta^{-1}=1\), \(K(\beta,x)=\beta^{-1}(\alpha x)^{-\beta /\alpha}\), let \(\hat f\) denote the Fourier transform of f, and let \(\| \hat f(x+iy)\|_{x,2}\) stand for the \(L_ 2(-\infty,\infty)\) norm of \(\hat f(x+iy)\) with respect to x (i.e. \(\hat f(x+iy)\|_{x,2}\) is a function of y). The principal result of this paper is that the following propositions are equivalent: (a) \(f(t)\exp (a| t|^{\alpha})\) is in \(L_ 2(-\infty,\infty)\), (b) \(\hat f\) is an entire function and \(| y|^{(\beta /2)-1}\exp [-2K(\beta,a)| y|^{\beta}](\| \hat f(x+iy)\|_{x,2})^ 2\) is in \(L_ 1(- \infty,\infty)\). A somewhat similar result involving uniform norms had been previously obtained by the author [J. Math. Anal. Appl. 102, 102-112 (1984; Zbl 0552.30019)].
scientific article; zbMATH DE number 4042506