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

2015/03/29 by Guantie Deng, Deng, Guantie, Tao Qian +1
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #math.CV

paper · pdf · doi:10.48550/arxiv.1503.08417

arxiv created 2015/03/29 · arxiv updated 2015/03/31

Abstract

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

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