Analytic Functions Related to the Distributions of Exponential Growth

Abstract

We study the relationship between certain classes of analytic functions in tubes, the distributions of exponential growth $\mathcal{K}'_p ,p \geqq 1$, and the Fourier transform spaces $K'_p ,p \geqq 1$, of such distributions. Representations of the analytic functions are obtained in terms of the Fourier–Laplace transform of distributions in $\mathcal{K}'_p $, and when the analytic functions are considered as elements in $K'_p $, we obtain representations of them in terms of the Fourier transform in $K'_p $, of certain elements in $\mathcal{K}'_p $. In every case the distributions which yield these representations are analyzed. Further, we obtain strong boundedness properties of the analytic functions when considered as elements of $K'_p $, and certain of our analytic functions are shown to have distributional boundary values in the strong (and weak) topology of $K'_p $, Our results are motivated by analysis of V. S. Vladimirov who has considered similar problems to those that we study but for spaces of analytic functions that are properly contained in the spaces that we define in this paper. The spectral functions of Vladimirov which are associated with his analytic functions are distributions in $\mathcal{S}'$ which are defined by continuous functions of power increase in $\mathbb{R}^n $ while the distributions which correspond to spectral functions in this paper are distributions in $\mathcal{K}'_p $ which are defined by continuous functions of exponential growth in $\mathbb{R}^n $, a more general situation than that of Vladimirov.