  In this article, we extend a result of L. Loomis and W. Rudin, regarding
boundary behavior of positive harmonic functions on the upper half space
$\R_+^{n+1}$. We show that similar results remain valid for more general
approximate identities. We apply this result to prove a result regarding
boundary behavior of nonnegative eigenfunctions of the Laplace-Beltrami
operator on real hyperbolic space $\mathbb H^n$. We shall also prove a
generalization of a result regarding large time behavior of solution of the
heat equation proved in \cite{Re}. We use this result to prove a result
regarding asymptotic behavior of certain eigenfunctions of the Laplace-Beltrami
operator on real hyperbolic space $\mathbb H^n$.
