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Rational approximation of functions in Hardy spaces. (English) Zbl 1353.30032

Let \(H^p(C_k)\) (\(k=\pm 1\)) be the Hardy space in the half plane \(C_k=\{z=x+iy:ky>0\}\). J. B. Garnett [Bounded analytic functions. New York etc.: Academic Press (1981; Zbl 0469.30024)] showed that the class \(U_N\) of analytic functions satisfying some smoothness conditions are dense in \(H^p(C_{+1})\) for \(0<p<\infty\) and \(pN>1\), where \(N\) is a positive integer. In this paper the class \(U_N\) is replaced by the family of rational functions \(R_N(\alpha)\) and it is shown that \(R_N(\alpha)\) is contained in the class \(U_N\) for some \(\mathrm{Im\,}\alpha>0\), where \(\alpha\) is a complex number. In particular, it is proved that \(R_N(i)\) and \(R_N(-i)\) are dense in \(H^p(C_{+1})\) and \(H^p(C_{-1})\), respectively. The functions in \(L^p(R)\), \(0<p<1\), are decomposed into the sum of corresponding Hardy space functions in \(H^p_{+1}(R)\) and in \(H^p_{-1}(R)\) through rational atoms. For the uniqueness of the above decomposition a straightforward proof of the results of A. B. Aleksandrov [Math. USSR, Sb. 35, 301–316 (1979; Zbl 0426.30005)] and J. A. Cima and W. T. Ross [The backward shift on the Hardy space. Providence, RI: American Mathematical Society (AMS) (2000; Zbl 0952.47029)] is given by using the Fourier transform of tempered distributions. In the latter, the Fourier spectrum results obtained for \(1\leq p<\infty\) in [T. Qian, J. Integral Equations Appl. 17, No. 2, 159–198 (2005; Zbl 1086.30035)] are extended to \(0<p<1\).

MSC:

30E10 Approximation in the complex plane
41A20 Approximation by rational functions
30H10 Hardy spaces
46E15 Banach spaces of continuous, differentiable or analytic functions
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[2] Cima, J.A., Ross, W.T.: The Backward Shift on the Hardy Space Mathematical Surveys and Monographs, vol. 79. American Mathematical Society, Providence (2000) · Zbl 0952.47029 · doi:10.1090/surv/079
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