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

TL;DR

The paper proves that partial sums of $H^1$ functions do not converge in $H^1$ norm but do converge in the Bergman norm, extending the result to higher dimensions and providing a new proof of non-convergence.

Contribution

It establishes the convergence of partial sums in the Bergman norm for $H^1$ functions and extends classical results to complex Reinhardt domains.

Findings

Partial sums of $H^1$ functions do not converge in $H^1$ norm.

Partial sums always converge in the Bergman norm $A^1$.

Extension of results to Reinhardt domains in $ extbf{C}^n$.

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.