Pseudo-uniform convexity of $H^{1}$ in several variables

A convergence theorem of D. J. Newman for the Hardy space ${H^1}$ is generalized to several complex variables. Specifically, in both ${H^1}$ of the polydisc and ${H^1}$ of the ball, weak convergence, together with convergence of norms, is shown to imply norm convergence. As in Newman’s work, approximation of ${L^1}$ by ${H^1}$ is also considered. It is shown that every function in ${L^1}$ of the torus, (or in ${L^1}$ of the boundary of the ball), has a best ${H^1}$-approximation which, in several variables, need not be unique.