Hardy space decomposition of on the unit circle:

Abstract

In this paper we consider Hardy space decomposition of where stands for the open unit disc, and is its boundary. Hardy spaces decompositions for and for are, as classical results, available in the literature. For the basic tools are the Plemelj formula and the boundedness of the Hilbert transformation. For neither on the real line, nor on the unit circle, a Plemelj formula, or Hilbert transformation are available. In a recent paper Deng and Qian obtain Hardy spaces decomposition for on the real line by means of rational approximation. In the present paper by using rational functions, we achieve the same goal for for the range The work on the unit circle exposes the particular features of the kind of decomposition in the compact situation adaptable to higher dimensions.