Rational Approximation of Functions in Hardy Spaces

Hardy space decompositions of Lp(ℝn) for 0 < p < 1 with rational approximation

“…As is well known, rational approximation has a long history, and is naturally related to complex approximation [18]. As obtained in [5] through a rational approximation method, for a given real-valued function…”

“…Turning to the case of 0 < p ≤ 1, due to loss of integrability neither on the real line, nor on the unit circle, the Plemelj formula, or the Hilbert transformation are available. The references [2] and [5] study the Hardy space decomposition of L p (R) for 0 < p < 1. There is no Fourier transformation theory for functions in such L p (R) as they are even not distributions.…”

“…The methods in [5] are direct and constructive. The following Theorem A is obtained by Deng and Qian in [5]. Theorem A ( [5]) Suppose that 0 < p < 1 and f ∈ L p (R).…”

“…The following Theorem A is obtained by Deng and Qian in [5]. Theorem A ( [5]) Suppose that 0 < p < 1 and f ∈ L p (R). Then, there exist a positive constant A p and two sequences of rational functions {P k } and {Q k } such that…”

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This paper aims to obtain decompositions of higher dimensional L p (R n ) functions into sums of non-tangential boundary limits of the corresponding Hardy space functions on tubes for the index range 0 < p < 1. In the one-dimensional case, Deng and Qian [5] recently obtained such Hardy space decomposition result: for any function f ∈ L p (R), 0 < p < 1, there exist functions f 1 and f 2 such that f = f 1 + f 2 , where f 1 and f 2 are, respectively, the non-tangential boundary limits of some Hardy space functions in the upper-half and lower-half planes. In the present paper, we generalize the one-dimensional Hardy space decomposition result to the higher dimensions, and discuss the uniqueness issue of such decomposition.

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