Geodesic
Approximation by rational functions, and an analogue of the M. Riesz theorem on conjugate functions for $L^p$-spaces with $p\in(0,1)$
Sbornik. Mathematics, Tome 35 (1979) no. 3, pp. 301-316 
A. B. Aleksandrov. Approximation by rational functions, and an analogue of the M. Riesz theorem on conjugate functions for $L^p$-spaces with $p\in(0,1)$. Sbornik. Mathematics, Tome 35 (1979) no. 3, pp. 301-316. http://geodesic.mathdoc.fr/item/SM_1979_35_3_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper the solution to some problems concerning rational approximation in the $L^p$-metric ($p\in(0,1)$) is given. The following is a typical problem: to describe the closure in the space $L^p[-1,1]$ of the linear hull of the Cauchy family $\{1/(x-a)\}_{a\in[-1,1]}.$ In the paper it is shown that this closure consists of all functions $f\in L^p[-1,1]$ for which there exists a functon $\tilde f$, analytic in $\mathbf C\setminus[-1,1]$, decreasing to zero at infinity, and such that $f(x)=\lim_{y\to0+}\tilde f(x+iy)=\lim_{y\to0+}\tilde f(x-iy)$ for almost all $x\in[-1,1]$. Bibliography: 6 titles.

[1] K. Gofman, Banakhovy prostranstva analiticheskikh funktsii, IL, Moskva, 1963

[2] I. I. Privalov, Granichnye svoistva analiticheskikh funktsiya, Gostekhizdat, Moskva–Leningrad, 1950

[3] I. Stein, Singulyarnye integraly i differentsialnye svoistva funktsii, izd-vo “Mir”, Moskva, 1973 | MR

[4] P. L. Duren, B. W. Romberg, A. L. Shields, “Linear functionals on $H^p$-spaces with $0

1$”, J. reine und angew. Math., 238 (1969), 32–60 | MR | Zbl

[5] R. R. Coifman, “A real variable characterization of $H^p$”, Studia Math., 51:3 (1974), 269–274 | MR | Zbl

[6] K. de Leeuw, “The failure of spectral analysis in $L^p$ for $0

$”, Bull. Amer. Math. Soc., 82:1 (1976), 111–114 | DOI | MR | Zbl