Rational Approximation, Hardy Space - Decomposition of Functions in $L_p, p<1$: Further Results in Relation to Fourier Spectrum Characterization of Hardy Spaces

TL;DR

This paper advances the understanding of Hardy spaces $H^p(R)$ for $0<p extless 1$, focusing on rational approximation, boundary decomposition, and Fourier spectrum characterization, with new density and representation results.

Contribution

It proves the density of rational functions with a single pole in $H^p$ spaces and provides a boundary decomposition for functions in $L^p(R)$ for $0<p<1$, along with new integral and spectral characterizations.

Findings

Rational functions with a single pole are dense in $H^p$ for $0<p<\infty$.

Functions in $L^p(R)$ for $0<p<1$ can be decomposed into boundary limits of Hardy space functions.

New integral representations and Fourier spectrum characterizations for $H^p$ functions are established.

Abstract

Subsequent to our recent work on Fourier spectrum characterization of Hardy spaces $H^p(\mathbb{R})$ for the index range $1\leq p\leq \infty,$ in this paper we prove further results on rational Approximation, integral representation and Fourier spectrum characterization of functions in the Hardy spaces $H^p(\mathbb{R}), 0 < p\leq \infty,$ with particular interest in the index range $ 0< p \leq 1.$ We show that the set of rational functions in $ H^p(\mathbb{C}_{+1}) $ with the single pole $-i$ is dense in $ H^p(\mathbb{C}_{+1}) $ for $0<p<\infty.$ Secondly, for $0<p<1$, through rational function approximation we show that any function $f$ in $L^p(\mathbb{R})$ can be decomposed into a sum $g+h$, where $g$ and $h$ are, in the $L^p(\mathbb{R})$ convergence sense, the non-tangential boundary limits of functions in, respectively, $ H^p(\mathbb{C}_{+1})$ and $H^{p}(\mathbb{C}_{-1}),$ where $H^p(\mathbb{C}_k)\ (k=\pm 1) $ are the Hardy spaces in the half plane $ \mathbb{C}_k=\{z=x+iy: ky>0\}$. We give Laplace integral representation formulas for functions in the Hardy spaces $H^p,$ $0<p\leq2.$ Besides one in the integral representation formula we give an alternative version of Fourier spectrum characterization for functions in the boundary Hardy spaces $H^p$ for $0<p\leq 1.$