Certain classes of analytic functions in tube domains T C = ℝ n + i C in n -dimensional complex space, where C is an open connected cone in ℝ n , are studied. We show that the functions have a boundedness property in the strong topology of the space of tempered distributions g ′ . We further give a direct proof that each analytic function attains the Fourier transform of its spectral function as distributional boundary value in the strong (and weak) topology of g ′ .