Approximating continuous functions by holomorphic and harmonic functions

Bishop, Christopher J.

doi:10.1090/S0002-9947-1989-0961619-2

Approximating continuous functions by holomorphic and harmonic functions
HTML articles powered by AMS MathViewer

by Christopher J. Bishop PDF

Trans. Amer. Math. Soc. 311 (1989), 781-811 Request permission

Abstract:

If $\Omega$ is a Widom domain in the plane (e.g., finitely connected) and $f$ is any bounded harmonic function on $\Omega$ which is not holomorphic, then we prove the algebra ${H^\infty }(\Omega )[f]$ contains all the uniformly continuous functions on $\Omega$. The basic tools are the solution of the $\overline \partial$ equation with ${L^\infty }$ estimates and some estimates on the level sets of functions in BMOA.

References

Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
Sheldon Axler and Pamela Gorkin, Algebras on the disk and doubly commuting multiplication operators, Trans. Amer. Math. Soc. 309 (1988), no. 2, 711–723. MR 961609, DOI 10.1090/S0002-9947-1988-0961609-9
Sheldon Axler and Allen Shields, Algebras generated by analytic and harmonic functions, Indiana Univ. Math. J. 36 (1987), no. 3, 631–638. MR 905614, DOI 10.1512/iumj.1987.36.36034
Albert Baernstein II, Analytic functions of bounded mean oscillation, Aspects of contemporary complex analysis (Proc. NATO Adv. Study Inst., Univ. Durham, Durham, 1979) Academic Press, London-New York, 1980, pp. 3–36. MR 623463
B. T. Batikyan and S. A. Grigoryan, Uniform algebras containing $A(K)$, Uspekhi Mat. Nauk 40 (1985), no. 2(242), 169–170 (Russian). MR 786092
Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
Christopher J. Bishop, Constructing continuous functions holomorphic off a curve, J. Funct. Anal. 82 (1989), no. 1, 113–137. MR 976315, DOI 10.1016/0022-1236(89)90094-3
C. J. Bishop, L. Carleson, J. B. Garnett, and P. W. Jones, Harmonic measures supported on curves, Pacific J. Math. 138 (1989), no. 2, 233–236. MR 996199
Andrew Browder and John Wermer, Some algebras of functions on an arc, J. Math. Mech. 12 (1963), 119–130. MR 0144223
Lennart Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547–559. MR 141789, DOI 10.2307/1970375
Lennart Carleson, On $H^{\infty }$ in multiply connected domains, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 349–372. MR 730079
Sun Yung A. Chang, A characterization of Douglas subalgebras, Acta Math. 137 (1976), no. 2, 82–89. MR 428044, DOI 10.1007/BF02392413

Approximation by holomorphic functions on smooth manifolds in

78

7

A. M. Davie, T. W. Gamelin, and J. Garnett, Distance estimates and pointwise bounded density, Trans. Amer. Math. Soc. 175 (1973), 37–68. MR 313514, DOI 10.1090/S0002-9947-1973-0313514-8
C. J. Earle and A. Marden, On Poincaré series with application to $H^{p}$ spaces on bordered Riemann surfaces, Illinois J. Math. 13 (1969), 202–219. MR 237766
Frank Forelli, Bounded holomorphic functions and projections, Illinois J. Math. 10 (1966), 367–380. MR 193534
H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625
Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
T. W. Gamelin and J. Garnett, Constructive techniques in rational approximation, Trans. Amer. Math. Soc. 143 (1969), 187–200. MR 249639, DOI 10.1090/S0002-9947-1969-0249639-4
T. W. Gamelin and John Garnett, Pointwise bounded approximation and Dirichlet algebras, J. Functional Analysis 8 (1971), 360–404. MR 0295085, DOI 10.1016/0022-1236(71)90002-4
John Garnett, Analytic capacity and measure, Lecture Notes in Mathematics, Vol. 297, Springer-Verlag, Berlin-New York, 1972. MR 0454006
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
John B. Garnett and Peter W. Jones, The corona theorem for Denjoy domains, Acta Math. 155 (1985), no. 1-2, 27–40. MR 793236, DOI 10.1007/BF02392536
W. K. Hayman and Ch. Pommerenke, On analytic functions of bounded mean oscillation, Bull. London Math. Soc. 10 (1978), no. 2, 219–224. MR 500932, DOI 10.1112/blms/10.2.219
Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
K. Hoffman and I. M. Singer, Maximal algebras of continuous functions, Acta Math. 103 (1960), 217–241. MR 117540, DOI 10.1007/BF02546357
L. Hörmander and J. Wermer, Uniform approximation on compact sets in $C^{n}$, Math. Scand. 23 (1968), 5–21 (1969). MR 254275, DOI 10.7146/math.scand.a-10893
Peter W. Jones, Carleson measures and the Fefferman-Stein decomposition of $\textrm {BMO}(\textbf {R})$, Ann. of Math. (2) 111 (1980), no. 1, 197–208. MR 558401, DOI 10.2307/1971197
Peter W. Jones, $L^{\infty }$ estimates for the $\bar \partial$ problem in a half-plane, Acta Math. 150 (1983), no. 1-2, 137–152. MR 697611, DOI 10.1007/BF02392970
Peter W. Jones and Donald E. Marshall, Critical points of Green’s function, harmonic measure, and the corona problem, Ark. Mat. 23 (1985), no. 2, 281–314. MR 827347, DOI 10.1007/BF02384430
Linda Keen, Canonical polygons for finitely generated Fuchsian groups, Acta Math. 115 (1965), 1–16. MR 183873, DOI 10.1007/BF02392200
Sun Yung A. Chang, A characterization of Douglas subalgebras, Acta Math. 137 (1976), no. 2, 82–89. MR 428044, DOI 10.1007/BF02392413

On a theorem of M. A. Lavren’ev

77

3

S. N. Mergelyan, On the representation of functions by series of polynomials on closed sets, Doklady Akad. Nauk SSSR (N.S.) 78 (1951), 405–408 (Russian). MR 0041929

Approximation by polynomials in two complex variables

246

A. G. O’Farrell and K. J. Preskenis, Approximation by polynomials in two diffeomorphisms, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 1, 105–107. MR 722862, DOI 10.1090/S0273-0979-1984-15203-0
Robert F. Olin, Functional relationships between a subnormal operator and its minimal normal extension, Pacific J. Math. 63 (1976), no. 1, 221–229. MR 420324
Ch. Pommerenke, On the Green’s function of Fuchsian groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 2 (1976), 409–427. MR 0466534
Kenneth John Preskenis, Approximation on disks, Trans. Amer. Math. Soc. 171 (1972), 445–467. MR 312123, DOI 10.1090/S0002-9947-1972-0312123-3
Kenneth John Preskenis, Another view of the Weierstrass theorem, Proc. Amer. Math. Soc. 54 (1976), 109–113. MR 390779, DOI 10.1090/S0002-9939-1976-0390779-6
Walter Rudin, Spaces of type $H^{\infty }+C$, Ann. Inst. Fourier (Grenoble) 25 (1975), no. 1, vi, 99–125 (English, with French summary). MR 377520
Donald Sarason, Algebras of functions on the unit circle, Bull. Amer. Math. Soc. 79 (1973), 286–299. MR 324425, DOI 10.1090/S0002-9904-1973-13144-1
David A. Stegenga, A geometric condition which implies BMOA, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 427–430. MR 545283
David A. Stegenga and Kenneth Stephenson, A geometric characterization of analytic functions with bounded mean oscillation, J. London Math. Soc. (2) 24 (1981), no. 2, 243–254. MR 631937, DOI 10.1112/jlms/s2-24.2.243
Carl Sundberg, A constructive proof of the Chang-Marshall theorem, J. Functional Analysis 46 (1982), no. 2, 239–245. MR 660188, DOI 10.1016/0022-1236(82)90037-4
N. Th. Varopoulos, BMO functions and the $\overline \partial$-equation, Pacific J. Math. 71 (1977), no. 1, 221–273. MR 508035

A remark on

and bounded harmonic functions

73

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
A. L. Vol′berg, Constructive proof of the Marshall-Chang theorem, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 141 (1985), 149–153, 190 (Russian, with English summary). Investigations on linear operators and the theory of functions, XIV. MR 788894
John Wermer, On algebras of continuous functions, Proc. Amer. Math. Soc. 4 (1953), 866–869. MR 58877, DOI 10.1090/S0002-9939-1953-0058877-9
J. Wermer, Approximation on a disk, Math. Ann. 155 (1964), 331–333. MR 165386, DOI 10.1007/BF01354865
J. Wermer, Polynomially convex disks, Math. Ann. 158 (1965), 6–10. MR 174968, DOI 10.1007/BF01370392
John Wermer, Banach algebras and several complex variables, 2nd ed., Graduate Texts in Mathematics, No. 35, Springer-Verlag, New York-Heidelberg, 1976. MR 0394218
Harold Widom, The maximum principle for multiple-valued analytic functions, Acta Math. 126 (1971), 63–82. MR 279311, DOI 10.1007/BF02392026