Rational Approximation of Functions in Hardy Spaces

Abstract

Subsequent to our recent work on Fourier spectrum characterization of Hardy spaces $H^p({\mathbb {R}})$ for the index range $1\le p\le \infty ,$ in this paper we prove further results on rational Approximation, integral representation and Fourier spectrum characterization of functions for the Hardy spaces $H^p({\mathbb {R}}), 0 < p\le \infty ,$ with particular interest in the index range $0< p \le 1.$ 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 $H^p,$ $0<p\le 2.$ 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 $0<p\le 1$.