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

Guantie Deng; Tao Qian

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arXiv · 1503.08417

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

Guantie Deng·Tao Qian

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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 $0 0\}$. 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 $0<p\leq 1.$

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Guantie Deng, Tao Qian. 2015-03-29. Rational Approximation, Hardy Space - Decomposition of Functions in $L_p, p<1$: Further Results in Relation to Fourier Spectrum Characterization of Hardy Spaces. https://arxiv.org/abs/1503.08417

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