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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
; | | Date: |
29 Mar 2015 | | Abstract: | Subsequent to our recent work on Fourier spectrum characterization of Hardy
spaces $H^p(mathbb{R})$ for the index range $1leq pleq infty,$ in this
paper we prove further results on rational Approximation, integral
representation and Fourier spectrum characterization of functions in the Hardy
spaces $H^p(mathbb{R}), 0 < pleq infty,$ 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 $H^p,$
$0<pleq2.$ 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 $0<pleq 1.$ | | Source: | arXiv, 1503.8417 | | Services: | Forum | Review | PDF | Favorites | |
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