Laplace transform representations and Paley-Wiener theorems for functions on vertical strips
The author works on a very classical problem, but nevertheless succeeds in obtaining very meaningful results. Let \(F(z)\) be a holomorphic function on a strip \(\{a<\text{Re}(z)< b\}\). The problem is to determine when this function can be represented as a bilateral Laplace transform \[ F(z)= \int^\infty_{-\infty} e^{-zt} f(t)\,dt \] in some reasonable sense. The main result is a Paley-Wiener type theorem which gives sufficient conditions for the existence of \(f(t)\) for functions \(F\) belonging to a certain weighted Bergman space. The author considers the representation problem also in the case of bounded functions on half-planes and extends a result of \textit{Ch. K. Batty} and \textit{M. D. Blake} [C. R. Acad. Sci., Paris, Sér. I, Math. 330, No. 2, 71--75 (2000; Zbl 0945.44001)]. The paper ends with a discussion of the uniqueness of these representations. If the function \(F(z)\) is represented by \(f_1\) and \(f_2\) on two disjoint strips, but extends analytically between them, when is \(f_1= f_2\)? The author shows that this question is related to the uniqueness of the Cauchy problem for the heat equation with complex space variable.
scientific article; zbMATH DE number 5703146
scientific article