Norm convergence of partial sums of $H^1$ functions

Abstract

A classical observation of Riesz says that truncations of a general ${\sum }_{n=0}^{\infty }{a}_n{z}^n$ in the Hardy space $H^1$ do not converge in $H^1$. A substitute positive result is proved: these partial sums always converge in the Bergman norm $A^1$. The result is extended to complete Reinhardt domains in $\C^n$. A new proof of the failure of $H^1$ convergence is also given.

Polynomial density argument added. Several typos corrected