Banach space valued Hardy functions $H^{p}, \; 0 < p \leq \infty,$ are defined with the functions having domain in tubes $T^{C} = \mathbb{R}^{n}+\I C \subset \mathbb{C}^{n}$; $H^{2}$ functions with values in Hilbert space are characterized as Fourier-Laplace transforms of functions which satisfy a certain norm growth property. These $H^{2}$ functions are shown to equal a Cauchy integral when the base $C$ of the tube $T^{C}$ is specialized. For certain Banach spaces and certain bases $C$ of the tube $T^{C}$, all $H^{p}$ functions, \; $1 \leq p \leq \infty$, are shown to equal the Poisson integral of $L^{p}$ functions, have boundary values in $L^{p}$ norm on the distinguished boundary $\mathbb{R}^{n}+\I \{ \overline{0} \}$ of the tube $T^{C}$, and have pointwise growth properties. For $H^{2}$ functions with values in Hilbert space we show the existence of $L^{2}$ boundary values on the topological boundary $\mathbb{R}^{n}+\I\, \partial C$ of the tube $T^{C}$.