Approximation in the mean by analytic functions

Hedberg, Lars Inge

doi:10.1090/S0002-9947-1972-0432886-6

Approximation in the mean by analytic functions
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Trans. Amer. Math. Soc. 163 (1972), 157-171 Request permission

Abstract:

Let $E$ be a compact set in the plane, let ${L^p}(E)$ have its usual meaning, and let $L_a^p(E)$ be the subspace of functions analytic in the interior of $E$. The problem studied in this paper is whether or not rational functions with poles off $E$ are dense in $L_a^p(E)$ (or in ${L^p}(E)$ in the case when $E$ has no interior). For $1 \leqq p \leqq 2$ the problem has been settled by Bers and Havin. By a method which applies for $1 \leqq p < \infty$ we give new results for $p > 2$ which improve earlier results by Sinanjan. The results are given in terms of capacities.

References

Thomas Bagby, $L_{p}$ approximation by analytic functions, J. Approximation Theory 5 (1972), 401–404. MR 348116, DOI 10.1016/0021-9045(72)90006-8

Quasi topologies and rational approximation

Lipman Bers, An approximation theorem, J. Analyse Math. 14 (1965), 1–4. MR 178287, DOI 10.1007/BF02806376
James E. Brennan, Invariant subspaces and rational approximation, J. Functional Analysis 7 (1971), 285–310. MR 0423059, DOI 10.1016/0022-1236(71)90036-x
A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139. MR 52553, DOI 10.1007/BF02392130

Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen

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Lennart Carleson, Mergelyan’s theorem on uniform polynomial approximation, Math. Scand. 15 (1964), 167–175. MR 198209, DOI 10.7146/math.scand.a-10741
Lennart Carleson, Selected problems on exceptional sets, Van Nostrand Mathematical Studies, No. 13, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0225986
Jacques Deny, Sur la convergence de certaines intégrales de la théorie du potentiel, Arch. Math. (Basel) 5 (1954), 367–370 (French). MR 66513, DOI 10.1007/BF01898378
Nicolaas du Plessis, A theorem about fractional integrals, Proc. Amer. Math. Soc. 3 (1952), 892–898. MR 51909, DOI 10.1090/S0002-9939-1952-0051909-2
Bent Fuglede, On generalized potentials of functions in the Lebesgue classes, Math. Scand. 8 (1960), 287–304. MR 159023, DOI 10.7146/math.scand.a-10612
Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
A. A. Gončar, On the uniform approximation of continuous functions by harmonic functions, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 1239–1250 (Russian). MR 0159012
A. A. Gončar, On the approximation of continuous functions by harmonic functions, Dokl. Akad. Nauk SSSR 154 (1964), 503–506 (Russian). MR 0159173
A. A. Gončar, On the property of instability of harmonic capacity, Dokl. Akad. Nauk SSSR 165 (1965), 479–481 (Russian). MR 0190366
V. P. Havin, Approximation by analytic functions in the mean, Dokl. Akad. Nauk SSSR 178 (1968), 1025–1028 (Russian). MR 0224830
V. G. Maz′ja and V. P. Havin, Approximation in the mean by analytic functions, Vestnik Leningrad. Univ. 23 (1968), no. 13, 62–74 (Russian, with English summary). MR 0235131
S. Ya. Havinson, Extremal problems for certain classes of analytic functions in finitely connected regions, Amer. Math. Soc. Transl. (2) 5 (1957), 1–33. MR 0083573, DOI 10.1090/trans2/005/01
Lars Inge Hedberg, Weighted mean approximation in Carathéodory regions, Math. Scand. 23 (1968), 113–122 (1969). MR 257377, DOI 10.7146/math.scand.a-10902
N. S. Landkof, Osnovy sovremennoĭ teorii potentsiala, Izdat. “Nauka”, Moscow, 1966 (Russian). MR 0214795
Ju. A. Lysenko and B. M. Pisarevskiĭ, The instability of harmonic capacity and the approximation of continuous functions by harmonic functions, Mat. Sb. (N.S.) 76 (118) (1968), 52–71 (Russian). MR 0234179
S. N. Mergeljan, On the completeness of systems of analytic functions, Amer. Math. Soc. Transl. (2) 19 (1962), 109–166. MR 0131561
S. O. Sinanjan, The uniqueness property of analytic functions on closed see without interior points, Sibirsk. Mat. Ž. 6 (1965), 1365–1381 (Russian). MR 0192063
S. O. Sinanjan, Approximation by analytical functions and polynomials in the mean with respect to the area, Mat. Sb. (N.S.) 69 (111) (1966), 546–578 (Russian). MR 0209491
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894
A. G. Vituškin, Analytic capacity of sets in problems of approximation theory, Uspehi Mat. Nauk 22 (1967), no. 6 (138), 141–199 (Russian). MR 0229838
Hans Wallin, A connection between $\alpha$-capacity and $L^{p}$-classes of differentiable functions, Ark. Mat. 5 (1963/65), 331–341 (1963/65). MR 217326, DOI 10.1007/BF02591134
Lawrence Zalcmann, Analytic capacity and rational approximation, Lecture Notes in Mathematics, No. 50, Springer-Verlag, Berlin-New York, 1968. MR 0227434
William P. Ziemer, Extremal length as a capacity, Michigan Math. J. 17 (1970), 117–128. MR 268401