Skip to main content
SpringerLink
Log in

Approximation and capacities

  • Chapter
  • First Online:
Linear and Complex Analysis Problem Book 3

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1574))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Vituškin A. G., Analytic capacity of sets in problems of approximation theory, Uspekhi Mat. Nauk 22 (1967), 141–199. (Russian)

    MathSciNet  Google Scholar 

  2. Verdera J., C m approximation by solutions of elliptic equations, and Calderon-Zygmund operators, Duke Math. J. 55 (1987), 157–187.

    Article  MathSciNet  MATH  Google Scholar 

  3. O’Farell A. G., Rational approximation in Lipschitz norms II, Proc. Royal Irish. Acad. 79 A (1979), 104–114.

    MathSciNet  Google Scholar 

  4. Tarkhanov N. N., Approximation in Sobolev spaces by solutions of elliptic systems, Dokl. Acad. Nauk SSSR 315 (1990), 1308–1313. (Russian).

    MathSciNet  MATH  Google Scholar 

  5. Tarkhanov N. N., Approximation on compact sets by solutions of systems with surjective symbols, Prepr. Inst. of Physics (Krasnoyarsk) #48 M (1989), 56 pp. (Russian)

    Google Scholar 

  6. Havin V. P., Approximation in the mean by analytic functions, Dokl. Acad. Nauk SSSR 178 (1968), 1025–1028. (Russian)

    MathSciNet  Google Scholar 

  7. Keldysh M. V., On the solvability and the stability of Dirichlet’s problem, Uspekhi Mat. Nauk 8 (1941), 171–231. (Russian)

    Google Scholar 

  8. Paramonov P. V., On harmonic approximation in the C1 norm, Mat. Sb. 181 (1990), 1341–1365. (Russian)

    MathSciNet  MATH  Google Scholar 

  9. Mateu, J., Orobitg, J., Lipschitz approximation by harmonic functions and some applications to spectral synthesis, Prepr. Univ. Auton. de Barcelona (1988), 31 pp.

    Google Scholar 

  10. Tarkhanov N. N., Uniform approximation by solutions of elliptic systems, Mat. Sb. 133 (1987), 356–381. (Russian)

    MathSciNet  MATH  Google Scholar 

References

  1. Lewis, J., Approximation of Sobolev functions in Jordan domains, Ark. Mat. 25 (1987), 255–264.

    Article  MathSciNet  MATH  Google Scholar 

References

  1. Clary S., Quasi-similarity and subnormal operators, Doct. Thesis Univ. Michigan, 1973.

    Google Scholar 

  2. Hastings W., A construction of Hilbert spaces of analytic functions, Proc. Amer. Math. Soc. 74 (1979), no. 2, 295–298.

    Article  MathSciNet  MATH  Google Scholar 

  3. Kriete T., On the structure of certain H2 (μ) spaces, Indiana Univ. Math. J. 28 (1979), no. 5, 757–773.

    Article  MathSciNet  MATH  Google Scholar 

  4. Brennan J. E., Approximation in the mean by polynomials on non-Caratheodory domains, Ark. Math. 15 (1977), 117–168.

    Article  MathSciNet  MATH  Google Scholar 

  5. Mergeljan S. N., On the completeness of systems of analytic functions, Uspekhi Mat. Nauk, 8 (1953), no 4, 3–63 (Russian); English translation in ser. 2, Amer. Math. Soc. Translations, 19 (1962), 109–166.

    MathSciNet  Google Scholar 

  6. Trent T., H 2 (μ) spaces and bounded evaluations, Doct. Thesis, Univ. Virginia, 1977.

    Google Scholar 

  7. Kirete T., Trutt D., On the Cesaro operator, Indiana Univ. Math. J. 24 (1974), 197–214.

    Article  MathSciNet  MATH  Google Scholar 

References

  1. Nikol’skii N. K., Selected problems of weighted approximation and spectral analysis, Trudy Mat. Inst. Steklov Akad. Nauk SSSR 120 (1974) (Russian); English transl. in Proc. Steklov Inst. Math. 120 (1974).

    Google Scholar 

  2. Vol’berg A. L., The logarithm of an almost analytic function is summable, Dokl. Akad. Nauk SSSR 265 (1982), 1297–1302 (Russian); English transl. in Soviet Math. Dokl. 26 (1982), 283–243.

    MathSciNet  MATH  Google Scholar 

  3. Hrušěëv S. V., The problems of simultaneous approximation and removal of singularities of Cauchy-type integrals, Trudy Mat. Inst. Steklov Akad. Nauk SSSR 130 (1978), 124–195 (Russian); English translation in Proc. Steklov Inst. Math. 130 (1979), 133–203.

    Google Scholar 

  4. Goluzin G. M., Geometric theory of functions of a complex variable, Transl. Math. Monographs, vol. 26, Amer. Math. Soc., Providence, 1969.

    MATH  Google Scholar 

  5. Vol’berg A. L., The simultaneous approximation of polynomials on the circle and in the interior of the disk, Zapiski Nauchn. Sem. LOMI 92 (1978), 60–84. (Russian)

    MathSciNet  MATH  Google Scholar 

References

  1. Hrušěëv S., The problem of simultaneous approximation and removal of singularities of Cauchytype integrals, Trudy Math. Inst. Steklov 130 (1978), 124–195 (Russian); English transl. in Proc. Steklov Inst. Math. 130 (1979), no. 4, 133–203.

    Google Scholar 

  2. Kriete T., MacCluer B., Mean-square approximation by polynomials on the unit disk, Trans. Amer. Math. Soc. 322 (1990), 1–34.

    Article  MathSciNet  MATH  Google Scholar 

  3. Miller T., Smith R., Nontangential limits of functions in some P2(μ) spaces, Indiana Univ. Math. J. 39 (1990), 19–26.

    Article  MathSciNet  Google Scholar 

  4. Olin R., Yang L., The commutant of multiplication by z on the closure of polynomials in Lt(μ), preprint.

    Google Scholar 

References

  1. Krein M. G., On an extrapolation problem of A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 46 (1945), 306–309. (Russian)

    MathSciNet  MATH  Google Scholar 

  2. Dym H., McKean H. P., Gaussian Processes, Function Theory and the Inverse Spectral Problem, Academic Press, New York, 1976.

    MATH  Google Scholar 

  3. Levinson N., Gap and Density Theorems, Colloquium Publ., 26, Amer. Math. Soc., New York, 1940.

    MATH  Google Scholar 

  4. Mandelbrojt S., Séries de Fourier et Classes Quasi-analytiques, Gauthier-Villars, Paris, 1935.

    MATH  Google Scholar 

  5. Redheffer R. M., Completeness of sets of complex exponentials, Adv. Math. 24 (1977), 1–62.

    Article  MathSciNet  MATH  Google Scholar 

  6. Koosis P., Sur l’approximation pondérée par des polynômes et par des sommes d’exponentielles imaginaires, Ann. Sci. Ec. Norm. Sup. 81 (1964), 387–408.

    MathSciNet  MATH  Google Scholar 

  7. Koosis P., Weighted polynomial approximation on arithmetic progressions of intervals or points, Acta Math. 116 (1966), 223–277.

    Article  MathSciNet  MATH  Google Scholar 

  8. Koosis P., Solution du problème de Bernstein sur les entiers, C. R. Acad. Sci. Paris, Ser. A 262 (1966), 1100–1102.

    MathSciNet  MATH  Google Scholar 

References

  1. Akhiezer N. I., The classical moment problem, Oliver and Boyd, 1965.

    Google Scholar 

  2. Berg C., Christensen J. P. R., Density questions in the classical theory of moments, Ann. Inst. Fourier (Grenoble) 31 (1981), no. 3, 99–114.

    Article  MathSciNet  MATH  Google Scholar 

  3. Hamburger H., Hermitian transformations of deficiency index (1,1), Jacobi matrices and un-determined moment problems, Amer. J. Math. 66 (1944), 489–522.

    Article  MathSciNet  MATH  Google Scholar 

  4. Koosis P., Measures orthogonales extrémales pour l’approximation ponderée par des polynômes, C. R. Acad. Sci. 311 (1990), 503–506.

    MathSciNet  MATH  Google Scholar 

References

  1. Anderson J. M., Barth K. F., Brannan D. A., Research Problems in Complex Analysis, Bull. London Math. Soc. 9 (1977), 152.

    MathSciNet  MATH  Google Scholar 

  2. Arakelyan N. U., Approximation complexe et propriétés des fonctions analytiques, Actes Congrès Intern. Math. 2 (1970), Gauthier-Villars/Paris, 595–600.

    Google Scholar 

  3. Brown L., Shields A. L., Approximation by analytic functions uniformly continuous on a set, Duke Math. Journal 42 (1975), 71–81.

    Article  MathSciNet  MATH  Google Scholar 

  4. Stray A., Uniform and asymptotic approximation, Math. Ann. 234 (1978), 61–68.

    Article  MathSciNet  MATH  Google Scholar 

  5. Stray A., Decomposition of approximable functions, Ann. of Math. (2) 120 (1984), no. 2, 225–235.

    Article  MathSciNet  MATH  Google Scholar 

References

  1. Lehto O., Virtanen K. I., Quasi-conformal Mappings in the Plane, Springer-Verlag, 1973.

    Google Scholar 

  2. Vitushkin A. G., The analytic capacity of sets in problems of approximation theory, Uspekhi Mat. Nauk 22 (1967), no. 5, 141–199 (Russian); English transl. in Russian Math. Surveys 22 (1967), 139–200.

    MathSciNet  MATH  Google Scholar 

  3. Zalcman L., Analytic Capacity and Rational Approximation, Lect. Notes in Math., 50, 1968.

    Google Scholar 

References

  1. Maimeskul V. V., To the question on the approximation of continuous functions by traces of generalized solutions of Beltrami equation, Theory of functions and approximations, Proc. of 2nd Saratov Winter School, chapter 3, Saratov University, 1986, 17–19. (Russian)

    Google Scholar 

References

  1. Carleman T., Sur un théorème de Weierstrass, Ark. Mat. Astronom. Fys. 4 (1927), no. 20B, 1–5.

    MATH  Google Scholar 

  2. Keldyš M. V., Lavrent’ev M. A., On a problem of Carleman, Dokl. Akad. Nauk SSSR 23 (1939), no. 8, 746–748. (Russian)

    Google Scholar 

  3. Mergeljan S. N., Uniform approximation to functions of a complex variable, Uspekhi Mat. Nauk 7 (1952), 31–123 (Russian); English transl. in Amer. Math. Soc. Translations 3 ser. 1 (1962), 294–391.

    Google Scholar 

  4. Arakeljan N. U., Uniform and tangential approximation by analytic functions, Izv. Akad. Nauk Armjan. SSR 3 (1968), 273–286 (Russian); English transl. in Amer. Math. Soc. Translations 122 ser. 2 (1984), 85–97.

    MathSciNet  Google Scholar 

  5. Nersesjan A. A., On uniform and tangential approximation by meromorphic functions, Izv. Akad. Nauk Armjan. SSR 7 (1972), no. 6, 405–412.

    MathSciNet  MATH  Google Scholar 

  6. Roth A., Meromorphe Approximationen, Comment. Math. Helv. 48 (1973), 151–176.

    Article  MathSciNet  MATH  Google Scholar 

  7. Roth A., Uniform and tangential approximations by meromorphic functions on closed sets, Canad. J. Math. 28 (1976), 104–111.

    Article  MathSciNet  MATH  Google Scholar 

  8. Nersesjan A. A., On Carleman sets, Izv. Akad. Nauk. Armjan. SSR 6 (1971), no. 6, 465–471 (Russian); English transl. in Amer. Math. Soc. Translations 122 ser. 2 (1984), 99–104.

    MathSciNet  Google Scholar 

  9. Boivin A., On Carleman approximation by meromorphic functions, Proceedings 8th Conference on Analytic Functions, Blazejewko, August, 1982.

    Google Scholar 

  10. Šaginjan A. A., Uniform and tangential harmonic approximation of continuous functions on arbitrary sets, Mat. Zametki 9 (1971), 131–142 (Russian); English transl. in Mat. Notes 9 (1971), 78–84.

    MathSciNet  Google Scholar 

  11. Gauthier P. M., Carleman approximation on unbounded sets by harmonic functions with Newtonian singularities, Proceedings 8th Conference on Analytic Functions, Blazejewko, August, 1982.

    Google Scholar 

  12. Labrèche M., De l’approximation harmonique uniforme, Doctoral Dissertation, Université de Montréal, 1982.

    Google Scholar 

References

  1. Alexander H., Explicit imbedding of the (punctured) disc into ℂ2, Comment. Math. Helv. 52 (1977), 539–544, MR 58, 1272.

    Article  MathSciNet  MATH  Google Scholar 

  2. Arakeljan N. U., Uniform and tangential approximation by analytic functions, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 3 (1968), 273–286, MR 43, 104 (Russian)

    MathSciNet  Google Scholar 

  3. Bagby T. A Runge theorem for harmonic functions on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988), 160–164.

    Article  MathSciNet  MATH  Google Scholar 

  4. Bagby T., Blanchet P., Uniform approximation by harmonic functions on closed subsets of Riemannian manifolds (to appear).

    Google Scholar 

  5. Bagby T., Gauthier P. M., Approximation by harmonic functions on closed subsets of Riemann surfaces, J. d’Analyse Math. 51 (1988) 259–284, MR 89j, 30064.

    Article  MathSciNet  MATH  Google Scholar 

  6. Boivin A., Carteman approximation on Riemann surface, Math. Ann. 275 (1986), 57–70, MR 87, 30054.

    Article  MathSciNet  MATH  Google Scholar 

  7. Carleman T., Sur un théorème de Weierstrass, Ark. Mat. Astronom. Fys. (1927), no. 20B, 1–5, JB 53, 237.

    MATH  Google Scholar 

  8. Gauthier P. M., Tangential approximation by entire functions and functions holomorphic in a disc, Izv. Akad. Nauk Arm. SSR Ser. Mat. 4 (1969), 319–326, MR 43, 1172.

    MathSciNet  MATH  Google Scholar 

  9. Gauthier P. M., Analytic approximation on closed subsets of open Riemann surfaces, Constructive Function Theory “77, Sofia, 1980, pp. 317–325.

    Google Scholar 

  10. Gauthier P. M., Hengartner W., Uniform approximation on closed sets by functions analytic on a Riemann surface, Approximation Theory (Z. Ciesielski and J. Musielak, eds.), Reidel, Dordrecht, 1975, pp. 63–70, MR 58, 6263.

    Google Scholar 

  11. Hitotumatu S., Some recent topics in several complex variables by the Japanese school, Proc. Rommanian-Finnish Seminar, Bucharest, 1971, MR 45, 8871.

    Google Scholar 

  12. Laufer H. B., Imbedding Annuli in ℂ2, J. Analyse Math. 26 (1973), 187–215, MR 49, 10915.

    Article  MathSciNet  MATH  Google Scholar 

  13. Narashiman R., Oral communication (1975).

    Google Scholar 

  14. Nesresjan A. A., Carleman sets, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 6 (1971), 465–471, MR 46, 66. (Russian)

    MathSciNet  Google Scholar 

  15. Scheinberg S., Uniform approximation by functions analytic on a Riemann surface, Ann. Math. 108 (257–298), MR 58, 17111.

    Google Scholar 

References

  1. Metzger T. A., On polynomial approximation in Aq(D), Proc. Amer. Math. Soc. 37 (1973), 468–470.

    MathSciNet  MATH  Google Scholar 

  2. Brennan, J., The integrability of the derivative in conformal mapping, J. London Math. Soc. 18 (1978), 261–272.

    Article  MathSciNet  MATH  Google Scholar 

  3. Carleson L., On the distortion of sets on a Jordan curve under conformal mapping, Duke Math. J. 40 (1973), 547–559.

    Article  MathSciNet  MATH  Google Scholar 

  4. McMillan J. E., Boundary behavior under conformal mapping, Proc. of the N. R. L. Conference on Classical Function Theory, Washington, D. C., 1970, pp. 59–76.

    Google Scholar 

  5. Lavrent’ev M. A., On some boundary problems in the theory of univalent functions, Math. Sb. 1(43) (1936), no. 6, 815–846 (Russian); English transl. in Amer. Math. Soc. Translations 32 ser. 2 (1963), 1–35.

    Google Scholar 

  6. McMillan J. E., Piranian G., Compression and expansion of boundary sets, Duke Math. J. 40 (1973), 599–605.

    Article  MathSciNet  MATH  Google Scholar 

  7. McMillan J. E., Boundary behavior of a conformal mapping, Acta Math. 123 (1969), 43–67.

    Article  MathSciNet  MATH  Google Scholar 

  8. Keldyš M. V., Sur l’approximation en moyenne quadratique des fonctions analytiques, Mat. Sb. 47 (1939), no. 5, 391–402.

    MathSciNet  MATH  Google Scholar 

  9. Mergeljan S. N., On the completeness of systems of analytic functions, Uspekhi Mat. Nauk 8 (1953), 3–63 (Russian); English transl. in Amer. Math. Soc. Translations 19 ser. 2 (1962), 109–166.

    MathSciNet  Google Scholar 

  10. Džrbašjan M. M., Metric theorems on completeness and the representation of analytic functions, Doctoral Dissertation, Erevan, 1948; Uspekhi Mat. Nauk. 5 (1950), 194–198. (Russian)

    MathSciNet  Google Scholar 

  11. Šaginjan A. L., On a criterion for the incompleteness of a system of analytic functions, Dokl. Akad. Nauk Armjan. SSR V (1946), no. 4, 97–100. (Russian)

    Google Scholar 

  12. Maz’ja V. G., Havin V. P., On approximation in the mean by analytic functions, Vestnik Leningrad Univ. Math. (1968), no. 13, 62–74. (Russian)

    MathSciNet  Google Scholar 

  13. Maz’ja V. G., Havin V. P., Use of (p, l)-capacity in problems of the theory of exceptional sets, Mat. Sb. 90 (1973), 558–591 (Russian); English transl. in Math. USSR Sbornik 19 (1973), 547–580.

    MathSciNet  Google Scholar 

  14. Brennan J., Invariant subspaces and weighted polynomial approximation, Ark. Mat. 11 (1973), 167–189.

    Article  MathSciNet  MATH  Google Scholar 

  15. Brennan J., Approximation in the mean by polynomials on non-Carathéodory domains, Ark. Mat. 15 (1977), 117–168.

    Article  MathSciNet  MATH  Google Scholar 

  16. Mel’nikov M. S., Sinanjan S. O., Questions in the theory of approximation of functions of one complex variable, Contemporary Problems of Mathematics, vol. 4, Itogi Nauki i Tekhniki, VINITI, Moscow, 1975, pp. 143–250 (Russian); English transl. in vol. 5, 1976, pp. 688–752.

    Google Scholar 

References

  1. Beardon A. F., Iteration of Rational Functions, Springer-Verlag, Berlin, 1991.

    Book  MATH  Google Scholar 

  2. Blanchard P., Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. 11 (1984), 85–141.

    Article  MathSciNet  MATH  Google Scholar 

  3. Carleson L., Makarov N. G., Some results connected with Brennan’s conjecture, preprint, 1992.

    Google Scholar 

  4. Eremenko A. E., Lyubich M. Yu., The dynamics of analytic transformations, Algebra i Analiz 1 (1989), 1–70 (Russian); English transl. in Leningrad Math. J. 1 (1990), 563–634.

    MathSciNet  MATH  Google Scholar 

  5. Eremenko A. E., Lower estimate in Littlewood’s conjecture on the mean spherical derivative of a polynomial and iteration theory, Proc. Amer. Math. Soc. 112 (1991), 713–715.

    Article  MathSciNet  MATH  Google Scholar 

  6. Falconer K., Fractal Geometry: Mathematical Foundations and Applications, Wiley, 1990.

    Google Scholar 

  7. Makarov N. G., Two remarks on integral means of the derivative of a univalent function, LOMI preprint (1985).

    Google Scholar 

  8. Makarov N. G., Conformal mappings and Hausdorff measures, Ark. Mat. 25 (1987), 41–85.

    Article  MathSciNet  MATH  Google Scholar 

  9. Milnor J., Dynamics in one complex variable: Introductory Lectures, preprint no. 5, Inst. Math. Sci. Stony Brook, 1990.

    Google Scholar 

  10. Pommerenke Ch., On the integral means of the derivative of a univalent function, J. London Math. Soc. 32 (1985), 254–258.

    Article  MathSciNet  MATH  Google Scholar 

  11. Pommerenke Ch., On the integral means of the derivative of a univalent function II, Bull. London Math. Soc. 17 (1985), 565–570.

    Article  MathSciNet  MATH  Google Scholar 

  12. Pommerenke Ch., The growth of the derivative of a univalent function, The Bieberbach Conjecture, Math. Surveys Monograph, vol. 21, 1986, pp. 143–152.

    Article  MathSciNet  MATH  Google Scholar 

References

  1. Keldysh M. V., Sur l’approximation en moyenne par polynômes des fonctions d’une variable complexe, Mat. Sb. 58 (1945), no. 1, 1–20.

    MATH  Google Scholar 

  2. Mergeljan S. N., On the completeness of systems of analytic functions, Uspekhi Mat. Nauk SSSR 8 (1953), no. 4, 3–63 (Russian); English transl. in Amer. Math. Soc. Translations 19 ser. 2 (1962), 109–166.

    MathSciNet  Google Scholar 

  3. Brennan J., Approximation in the mean by polynomials on non-Carathéodory domains, Ark. Mat. 15 (1977), 117–168.

    Article  MathSciNet  MATH  Google Scholar 

  4. Brennan J., Weighted polynomial approximation, quasianalyticity and analytic continuation, J. Reine Angew. Math. 357 (1985), 23–50.

    MathSciNet  MATH  Google Scholar 

  5. Maz’ja V. G., Havin V. P., Nonlinear potential theory, Uspekhi Mat. Nauk SSSR 27 (1972), no. 6, 67–138 (Russian); English transl. in Russian Math. Surveys 27 (1972), 71–148.

    MathSciNet  Google Scholar 

  6. Havin V. P., Approximation in the mean by analytic functions, Dokl. Akad. Nauk SSSR 178 (1968), no. 5, 1025–1028 (Russian); English transl. in Soviet Math. Dokl. 9 (1968), 245–248.

    MathSciNet  Google Scholar 

  7. Frostman O., Potentiel d’équilibre et capacité des ensembles, Meddel. Lunds Univ. Mat. Sem., (1935), no. 3, 1–118.

    MATH  Google Scholar 

  8. Carleson, L., On the distortion of sets on a Jordan curve under conformal mapping, Duke Math. J. 40 (1973), 547–559.

    Article  MathSciNet  MATH  Google Scholar 

  9. Brennan J., The integrability of the derivative in conformal mapping, J. London Math. Soc. 18 (1978), 261–272.

    Article  MathSciNet  MATH  Google Scholar 

  10. Brennan J., The integrability of the derivative of a conformal mapping, This Volume, Problem 12.9, 101–106.

    Google Scholar 

  11. Beurling A., Quasianalyticity and general distributions, Lecture Notes, Stanford Univ., 1961.

    Google Scholar 

References

  1. Brennan J., Weighted polynomial approximation and quasianalyticity for general sets, Centre De Recerca Matematica, Inst. D’Estudis Catalans, preprint.

    Google Scholar 

  2. Brennan J., Functions with rapidly decreasing negative Fourier coefficients, Lecture Notes in Math. Vol. 1275, 1987.

    Google Scholar 

  3. Vol’berg A., The logarithm of an almost analytic function is summable, Dokl. Akad. Nauk SSSR 265 (1982), 1297–1301. (Russian)

    MathSciNet  MATH  Google Scholar 

  4. Koosis P., The Logarithmic Integral I, Cambridge University Press, 1988; Vol. II, in press.

    Google Scholar 

  5. Borichev A. A., Vol’berg A., Unicity theorems for almost analytic functions, Algebra i Analiz 1 (1989), 146–177 (Russian); English transl. in Leningrad Math. J. 1 (1990), 157–191.

    MathSciNet  Google Scholar 

References

  1. Brennan J. E., Approximation in the mean by polynomials on non-Carathéodory domains, Ark. Mat. 15 (1977), 117–168.

    Article  MathSciNet  MATH  Google Scholar 

  2. Brennan J. E., Weighted polynomial approximation, quasianalyticity and analytic continuation, J. Reine. Angew Math. 357 (1985), 23–50.

    MathSciNet  MATH  Google Scholar 

  3. Brennan J. E., Weighted polynomial approximation and quasianalyticity for general sets, Centre De Recerca Matematica, Inst. D’Estudis Catalans, March 1992, preprint, No. 149.

    Google Scholar 

  4. Hruščëv S. V., The problem of simultaneous approximation and removal of singularities of Cauchy-type integrals, Trudy Mat. Inst. Steklov 130 (1978), 124–195 (Russian); English transl. in Proc. Steklov Inst. Math. 130 (1979), 133–203.

    MathSciNet  Google Scholar 

  5. Kriete T., On the structure of certain H2 (μ) spaces, Indiana Univ. Math. J. 28 (1979), 757–773.

    Article  MathSciNet  MATH  Google Scholar 

  6. Perez Gonzalez F., Stray A., Farrell and Mergeljan sets for Hp spaces, (0<p<1), Michigan Math. J. 36 (1989), 379–386.

    Article  MathSciNet  MATH  Google Scholar 

  7. Vol’berg A. L., Simultaneous approximation by polynomials on the circle and in the interior of the disk, Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov (LOMI) 92 (1979), 60–84. (Russian)

    MathSciNet  MATH  Google Scholar 

References

  1. Hedenmalm H., A factorization theorem for square area integrable analytic functions, J. Reine Angew. Math. 422 (1991), 45–68.

    MathSciNet  MATH  Google Scholar 

  2. Hedenmalm H., Factorization in weighted Bergman spaces, under preparation.

    Google Scholar 

References

  1. Hedenmalm H., A factorization theorem for square area integrable analytic functions, J. Reine Angew. Math. 422 (1991), 45–68.

    MathSciNet  MATH  Google Scholar 

  2. Hedenmalm H., Factorization in weighted Bergman spaces, under preparation.

    Google Scholar 

References

  1. Bagby T., Approximation in the mean by solutions of elliptic equations, Trans. Amer. Math. Soc.

    Google Scholar 

  2. Burenkov V. I., On the approximation of functions in the space W rp (Ω) by functions with compact support for an arbitrary open set Ω, Trudy Mat. Inst. Steklov Akad. Nauk SSSR 131 (1974), 51–63 (Russian); English transl. in Proc. Steklov Inst. Math. 131 (1974), 53–66.

    MathSciNet  Google Scholar 

  3. Choquet G., Forme abstraite du théorème de capacitabilité, Ann. Inst. Fourier 9 (1959), Grenoble, 83–89.

    Article  MathSciNet  MATH  Google Scholar 

  4. Hedberg L. I., Spectral synthesis in Sobolev spaces, and uniqueness of solutions of the Dirichlet problem, Acta Math. 147 (1981), 237–264.

    Article  MathSciNet  MATH  Google Scholar 

  5. Hedberg L. I., Wolff T. H., Thin sets in nonlinear potential theory, Stockholm, 1982, (Rep. Dept. of Math. Univ. of Stockholm, Sweden, ISSN 0348-7652 N 24).

    Google Scholar 

  6. Lindberg P., A constructive method for Lp-approximation by analytic functions, Ark. Mat. 20 (1982), 61–68.

    Article  MathSciNet  MATH  Google Scholar 

  7. Meyers N. G., A theory of capacities for functions in Lebesgue classes, Math. Scand. 26 (1970), 255–292.

    MathSciNet  MATH  Google Scholar 

  8. Polking J. C., Approximation in Lp by solutions of elliptic partial differential equations, Amer. J. Math. 94 (1972), 1231–1244.

    Article  MathSciNet  MATH  Google Scholar 

  9. Saak E. M., A capacitary criterion for a domain with stable Dirichlet problem for higher order elliptic equations, Mat. Sb. 100(142) (1976), no. 2(6), 201–209 (Russian); English transl. in Math. USSR Sbornik 29 (1976), 177–185.

    MathSciNet  Google Scholar 

  10. Vitushkin, A. G., The analytic capacity of sets in problems of approximation theory, Uspekhi Mat. Nauk 22 (1967), no. 6, 141–199 (Russian); English transl. in Russian Math. Surveys 22 (1967), 139–200.

    MathSciNet  MATH  Google Scholar 

References

  1. Sinanyan S. O., Approximation by analytic functions in the mean with respect to area, Math. Sb. 69 (1966), no. 4, 546–578. (Russian)

    MATH  Google Scholar 

  2. Mel’nikov M. S., Sinanyan S. O., Questions in the theory of approximations of functions of one complex variable, Contemporary Problems of Mathematics, Itogi Nauki i Tekniki, vol. 4, VINITI, 1975, pp. 143–250. (Russian)

    Google Scholar 

Reference

  1. Meyers N., Continuity properties of potentials, Duke Math. J. 42 (1975), 157–166.

    Article  MathSciNet  MATH  Google Scholar 

References

  1. Bliedtner J., Hansen W., Potential theory. An analytic and probabilistic approach to balayage, Springer, 1986.

    Google Scholar 

  2. Lysenko Yu. A., Pisarevskii B. M., Instability of harmonic capacity and approximations of continuous functions by harmonic functions, Mat. Sb. 76 (1968), 52–71 (Russian); English transl. in Math. USSR Sbornik 5 (1968), 53–72.

    MathSciNet  MATH  Google Scholar 

  3. Vitushkin A. G., The analytic capacity of sets in problems of approximation theory, Uspekhi Mat. Nauk 22 (1967), 141–199 (Russian); English transl. in Russian Math. Surveys 22 (1967), no. 6, 139–200.

    MathSciNet  MATH  Google Scholar 

References

  1. Carmona J. J., Mergelyan’s Approximation theorem for rational modules, J. Approx. Theory 44 (1985), 113–126.

    Article  MathSciNet  MATH  Google Scholar 

  2. Deny J., Systemes totaux de fonctions harmoniques, Ann. Inst. Fourier 1 (1949), 103–113.

    Article  MathSciNet  Google Scholar 

  3. Keldysh M. V., On the solubility and stability of the Dirichlet problem, Uspekhi Mat. Nauk 8 (1941), 171–231; Amer. Math. Soc. Trans. 51 (1966), no. 2, 1–73.

    Google Scholar 

  4. Mateu J., Verdera, J., BMO harmonic approximation in the plane and spectral synthesis for Hardy-Sobolev spaces, Rev. Math. Iberoamericana 4 (1988), 291–318.

    Article  MathSciNet  MATH  Google Scholar 

  5. Paramonov P., personal communication.

    Google Scholar 

  6. Trent T., Wang J., Uniform approximation by rational modules on nowhere dense sets, Proc. Amer. Math. Soc. 81 (1981), 62–64.

    Article  MathSciNet  MATH  Google Scholar 

  7. Verdera J., preprint.

    Google Scholar 

  8. Vitushkin A. G., Analytic capacity of sets in problems of approximation theory, Uspekhi Mat. Nauk 22 (1967), 141–199 (Russian); English transl. in Russian Math. Surveys 22 (1967), 139–200.

    MathSciNet  MATH  Google Scholar 

References

  1. Perron O., Die Lehre von den Kettenbrüchen, II, Stuttgart, 1957.

    Google Scholar 

  2. Baker G. A., Essentials of Padé Approximants, “AP”, New York, 1975.

    Google Scholar 

  3. Uchiyama S., Rational approximations to algebraic functions, Journal of the Faculty of Sciences Hokkaido University, Ser I XV (1961), no. 3, 4, 173–192.

    MathSciNet  MATH  Google Scholar 

  4. Gonchar A. A., A local condition of single-valuedness of analytic functions, Mat. Sb. 89 (1972). 148–164 (Russian); English transl. in Math. USSR Sbornik 18 (1972), 151–157.

    MathSciNet  Google Scholar 

  5. Gonchar A. A., On the convergence of Padé approximants, Mat. Sb. 92 (1973), 152–164 (Russian); English transl. in Math. USSR Sbornik 21 (1973), 155–166.

    MathSciNet  Google Scholar 

  6. Polya G., Untersuchungen über Lücken and Singularitäten von Potenzreihen, Math. Z. 29 (1929), 549–640.

    Article  MathSciNet  MATH  Google Scholar 

  7. Gonchar A. A., On the convergence of Padé approximants for some classes of meromorphic functions, Mat. Sb. 97 (1975), 605–627 (Russian); English transl. in Math. USSR Sbornik 26 (1975), 555–575.

    Google Scholar 

  8. Bieberbach L., Analytische Fortsetzung, Springer-Verlag, Berlin-Heidelberg, 1955.

    Book  MATH  Google Scholar 

  9. Walsh J. L., Interpolation and approximation by rational functions in the complex domain, second ed., vol. 20, AMS Coll. Publ., 1960.

    Google Scholar 

  10. Gonchar A. A., On the speed of rational approximation of some analytic functions, Mat. Sb. 105 (1978), 147–163 (Russian); English transl. in Math. USSR Sbornik 34 (1978), 131–145.

    MathSciNet  MATH  Google Scholar 

References

  1. Baker G. A., Essentials of Padé Approximants, Academic Press, New York, 1975.

    MATH  Google Scholar 

  2. de Montessus de Ballore R., Sur les fractions continues algébrique, Bull. Soc. Math. France 30 (1902), 28–36.

    MathSciNet  MATH  Google Scholar 

  3. Perron O., Die Lehre von den Kettenbrüchen, II, Teubner, Stuttgart, 1957.

    MATH  Google Scholar 

  4. Gragg W. B., On Hadamard’s theory of polar singularities, Padé Approximants and Their Applicants (Graves-Morris P. R., ed.), Academic Press, London, 1973, pp. 117–123.

    Google Scholar 

  5. Saff E. B., An extension of Montessus de Ballore’s theorem on the convergence of interpolation rational functions, J. Approx. Theory 6 (1972), 63–68.

    Article  MathSciNet  MATH  Google Scholar 

  6. Chisholm J. S. R., Graves-Morris P. R., Generalization of the theorem of de Montessus to two-variable approximants, Proc. Royal Soc. Ser. A 342 (1975), 341–372.

    Article  MathSciNet  MATH  Google Scholar 

  7. Karlsson J., Wallin H., Rational approximation by an interpolation procedure in several variables, Padé and rational approximation (Saff E. B. and Varga R. S., eds.), Academic Press, New York, 1977, pp. 83–100.

    Google Scholar 

  8. Gonchar A. A., On the convergence of generalized Padé approximants of meromorphic functions, Mat. Sb. 98 (1975), no. 4, 563–577 (Russian); English transl. in Math. USSR Sbornik 27 (1975), 503–514.

    Google Scholar 

  9. Chisholm J. S. R., N-variable rational approximants, Padé and rational approximation (Saff E. B. and Varga R. S., eds.), Academic Press, New York, 1977, pp. 23–42.

    Google Scholar 

  10. Gonchar A. A., A local condition for the single-valuedness of analytic functions of several variables, Mat. Sb. 93 (1974), no. 2, 296–313 (Russian); English transl. in Math. USSR Sbornik 22 (1974), 305–322.

    MathSciNet  Google Scholar 

  11. Graves-Morris P. R., Generalization of the theorem of de Montessus using Canterbury approximant, Padé and rational approximation (Saff E. B. and Verga R. S., eds.), Academic Press, New York, 1977, pp. 73–82.

    Google Scholar 

References

  1. Poreda S. J., A characterization of badly approximable functions, Trans. Amer. Math. Soc. 169 (1972), 249–256.

    Article  MathSciNet  MATH  Google Scholar 

  2. Gamelin T. W., Garnett J. B., Rubel L. A., Shields A. L., On badly approximable functions, J. Approx. Theory 17 (1976), 280–296.

    Article  MathSciNet  MATH  Google Scholar 

  3. Rudin W., Real and Complex analysis, New York, 1966.

    Google Scholar 

  4. Kronsstadt E., Private communication, September, 1977.

    Google Scholar 

  5. Luecking D. H., On badly approximable functions and uniform algebras, J. Approx. Theory 22 (1978), 161–176.

    Article  MathSciNet  MATH  Google Scholar 

  6. Rubel L. A., Shields A. L., Badly approximable functions and interpolation by Blaschke products, Proc. Edinburgh Math. Soc., 20 (1976), 159–161.

    Article  MathSciNet  MATH  Google Scholar 

References

  1. Alexander H., Projections of polynomial hulls, J. Funct. Anal. 3 (1973), 13–19.

    Article  MathSciNet  MATH  Google Scholar 

  2. Alexander H., On the area of the spectrum of an element of a uniform algebra, Complex Approximation (B. Aupetit, ed.), Birkhäuser, Basel, 1980, pp. 3–12.

    Google Scholar 

  3. Gamelin T., Khavinson D., The isoperimetric inequality and rational approximation, Amer. Math. Monthly 96 (1989), 18–30.

    Article  MathSciNet  MATH  Google Scholar 

  4. Khavinson D., Annihilating measures of the algebra R(X), J. Funct. Anal. 28 (1984), 175–193.

    Article  MathSciNet  MATH  Google Scholar 

  5. Khavinson D., A note on Töeplitz operators, Geometry of Banach Spaces (N. Kaltonand, E. Saab, eds.), Lecture Notes in Math., vol. 934, Springer-Verlag, 1986, pp. 89–95.

    Google Scholar 

  6. Khavinson D., Symmetry and uniform approximation by analytic functions, Proc. Amer. Math. Soc. 101 (1987), 475–483.

    Article  MathSciNet  MATH  Google Scholar 

  7. Khavinson, D., Lueking, D., On an extremal problem in the theory of rational approximation, J. Approx. Theory 50 (1987), 127–132.

    Article  MathSciNet  Google Scholar 

  8. Kosmodem’yanskii A. A., A converse of the mean value theorem for harmonic functions, Uspekhi Mat. Nauk 36 (1981), 175–176 (Russian); English transl. in Russian Math. Surveys 36 (1981), 159–160.

    MathSciNet  Google Scholar 

  9. Marrero-Rodriguez M. I., written communication, 1991.

    Google Scholar 

  10. Serrin J., A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318.

    Article  MathSciNet  MATH  Google Scholar 

  11. Weinberger H., Remark on the preceding paper of Serrin, Arch. Rational Mech. Anal. 43 (1971), 319–320.

    Article  MathSciNet  MATH  Google Scholar 

References

  1. Fisher S. D., Function Theory on Planar Domains, Wiley, New York, 1983.

    Google Scholar 

  2. Gamelin T. W., Uniform Algebras, Chelsea, New York, 1984.

    MATH  Google Scholar 

  3. Pólya G., Szegö G., Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematical Studies, vol. 27, Princeton University Press, 1951.

    Google Scholar 

  4. Tsuji M., Potential Theory in Modern Function Theory, Chelsea, New York, 1975.

    MATH  Google Scholar 

References

  1. Vitushkin A. G., On a problem of Rudin, Dokl. Akad. Nauk SSSR 213 (1973), no. 1, 14–15 (Russian); English transl. in Soviet Math. Dokl. 14 (1973), 1618–1619.

    MathSciNet  MATH  Google Scholar 

  2. Henkin G. M., Chirka E. M., Contemporary Problems of Mathematics, vol. 4, VINITI, Moscow, 1975, pp. 13–142 (Russian); English transl. in vol. 5, 1976, pp. 612–687.

    Google Scholar 

  3. Wells R. O., Function theory on differentiable submanifolds, Contributions to Analysis, a collection of papers dedicated to Lipman Bers, Academic Press, 1974, pp. 407–441.

    Google Scholar 

  4. Wermer, J., Polynomial approximation on an arc in ℂ3, Ann. Math. 62, N 2 (1955), 269–270.

    Article  MathSciNet  MATH  Google Scholar 

  5. Arens R., The maximal ideals of certain function algebras, Pacific J. Math. 8 (1958), 641–648.

    Article  MathSciNet  MATH  Google Scholar 

  6. Gamelin Th. W., Uniform Algebras, Prentice Hall, New Jersey, 1969.

    MATH  Google Scholar 

  7. Alexander H., Polynomial approximation and hulls in sets of finite linear measure in ℂn, Amer. J. Math. 93 (1971), no. 1, 65–74.

    Article  MathSciNet  Google Scholar 

References

  1. Ahlfors L. V., Beurling A., Conformal invariants and function-theoretic null sets, Acta Math. 83 (1950), 101–129.

    Article  MathSciNet  MATH  Google Scholar 

  2. Painlevé P., Sur les lignes singulières des fonctions analytiques, Ann. Fac. Sci. Toulouse 2 (1888).

    Google Scholar 

  3. Ahlfors L. V., Bounded analytic functions, Duke Math. J. 14 (1947), 1–11.

    Article  MathSciNet  MATH  Google Scholar 

  4. Vitushkin A. G., The analytic capacity of sets in problems of approximation theory, Uspekhi Mat. Nauk 22 (1967), no. 6, 141–199 (Russian); English transl. in Russian Math. Surveys 22 (1967), 139–200.

    MathSciNet  MATH  Google Scholar 

  5. Zalcman L., Analytic capacity and Rational Approximation, Lect. Notes Math., vol. 50, Springer, Berlin, 1968.

    MATH  Google Scholar 

  6. Collingwood E. P., Lohwater A. J., The Theory of Cluster Sets, Cambridge U. P., Cambridge, 1966.

    Book  MATH  Google Scholar 

  7. Besicovitch A., On sufficient conditions for a function to be analytic and on behavior of analytic functions in the neighborhood of non-isolated singular points, Proc. London Math. Soc. 32 (1931), no. 2, 1–9.

    Article  MathSciNet  Google Scholar 

  8. Vitushkin A. G., An example of a set of positive length, but zero analytic capacity Dokl. Akad. Nauk SSSR 127 (1959), 246–249. (Russian)

    MathSciNet  MATH  Google Scholar 

  9. Garnett J., Positive length but zero analytic capacity, Proc. Amer. Math. Soc. 24 (1970), 696–699.

    Article  MathSciNet  MATH  Google Scholar 

  10. Ivanov L. D., The variation of sets and functions, Nauka, Moscow, 1975.

    Google Scholar 

  11. Crofton M. W., On the Theory of Local Probability, Philos. Trans. Roy. Soc. 177 (1968), 181–199.

    MATH  Google Scholar 

  12. Sylvester J. J., On a funicular solution of Buffon’s “Problem of the needle” in its most general form, Acta Math. 14 (1891), 185–205.

    Article  MathSciNet  MATH  Google Scholar 

  13. Marstrand J. M., Fundamental geometrical properties of plane sets of fractional dimensions, Proc. London Math. Soc. 4 (1954), 257–302.

    Article  MathSciNet  MATH  Google Scholar 

  14. Denjoy A., Sur les fonctions analytiques uniformes à singularités discontinues, C. R. Acad. Sci. Paris 149 (1909), 258–260.

    MATH  Google Scholar 

  15. Havinson S. Ya., Analytic capacity of sets, joint nontriviality of various classes of analytic functions and the Schwarz lemma in arbitrary domains, Mat. Sb. 54 (1961), no. 1, 3–50 (Russian); English transl. in Amer. Math. Soc. Translations 43 ser. 2 (1964), 215–266.

    MathSciNet  Google Scholar 

  16. Ivanov L. D., On the analytic capacity of linear sets, Uspekhi Mat. Nauk 17 (1962), 143–144. (Russian)

    MathSciNet  MATH  Google Scholar 

  17. Davie A. M., Analytic capacity and approximation problems, Trans. Amer. Math. Soc. 171 (1972), 409–444.

    Article  MathSciNet  MATH  Google Scholar 

  18. Calderón A. P., Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. USA 74 (1977), 1324–1327.

    Article  MathSciNet  MATH  Google Scholar 

  19. Garabedian P. R., Schwarz’s lemma and the Szegö kernel function, Trans. Amer. Math. Soc. 67 (1949), 1–35.

    MathSciNet  MATH  Google Scholar 

  20. Havin V. P., Boundary properties of integrals of Cauchy type and harmonic conjugate functions in domains with rectifiable boundary, Mat. Sb. 68 (1965), 499–517 (Russian); English transl. in Amer. Math. Soc. Translations 74 ser. 2 (1968), 40–60.

    MathSciNet  Google Scholar 

  21. Havin V. P., Havinson S. Ya., Some estimates of analytic capacity, Dokl. Akad. Nauk SSSR 138 (1961), 789–792 (Russian); English transl. in Soviet Math. Dokl. 2 (1961), 731–734.

    MathSciNet  Google Scholar 

  22. Besicovitch A., On the fundamental geometric properties of linearly measurable plane sets of points, I, Math. Ann. 98 (1927), 422–464; II, Math. Ann. 115 (1938), 296–329.

    Article  MathSciNet  MATH  Google Scholar 

  23. Besicovitch A., On the fundamental geometric properties of linearly measurable plane sets of points, III, Math. Ann. 116 (1939), 349–357.

    Article  MathSciNet  MATH  Google Scholar 

  24. Uy N., Removable sets of analytic functions satisfying a Lipschitz condition, Ark. Mat. 17 (1979), 19–27.

    Article  MathSciNet  MATH  Google Scholar 

  25. Federer H., Geometric Measure Theory, Springer-Verlag, Berlin, 1969.

    MATH  Google Scholar 

  26. Marshall D. E., Painlevé null sets, Colloq. d’Analyse Harmonique et Complexe, Univ. Aix-Marseill I, Marseill, 1977.

    Google Scholar 

  27. Hruščëv S. V., A simple proof of a theorem on removable singularities of analytic functions satisfying a Lipschitz condition, Zapiski Nauchn. Sem. LOMI 113 (1981), 199–203 (Russian); English transl. in J. Soviet Math. 22 (1983), 1829–1832.

    Google Scholar 

References

  1. Zoretti L., Sur les fonctions analytiques uniformes qui possèdent un ensemble parfait discontinu de points singuliers, J. Math. Pures Appl. 6 N 1 (1905), 1–51.

    MATH  Google Scholar 

  2. Besicovitch A., On sufficient conditions for a function to be analytic and on behavior of analytic functions in the neighborhood of non-isolated singular points, Proc. London Math. Soc. 32 N 2 (1931), 1–9.

    Article  MathSciNet  Google Scholar 

  3. Denjoy A., Sur les fonctions analytiques uniformes à singularités discontinues, C. R. Acad. Sci. Paris 149 (1909), 258–260.

    MATH  Google Scholar 

  4. Havinson S. Ya., Analytic capacity of sets, joint nontriviality of various classes of analytic functions and the Schwarz lemma in arbitrary domains, Mat. Sb. 54 (1961), no. 1, 3–50 (Russian); English transl. in Amer. Math. Soc. Translations 43 ser. 2 (1964), 215–266.

    MathSciNet  Google Scholar 

  5. Ivanov L. D., On a conjecture of Denjoy, Uspekhi Mat. Nauk 18 (1964), 147–149. (Russian)

    Google Scholar 

  6. Davie A. M., Analytic capacity and approximation problems, Trans. Amer. Math. Soc. 171 (1972), 409–444.

    Article  MathSciNet  MATH  Google Scholar 

  7. Havin V. P., Havinson S. Ya., Some estimates of analytic capacity, Dokl. Akad. Nauk SSSR 138 (1961), 789–792 (Russian); English transl. in Soviet Math. Dokl. 2 (1961), 731–734.

    MathSciNet  Google Scholar 

  8. Havin V. P., Boundary properties of integrals of Cauchy type and harmonic conjugate functions in domains with rectifiable boundary, Mat. Sb. 68 (1965), 499–517 (Russian); English transl. in Amer. Math. Soc. Translations 74 ser. 2 (1968), 40–60.

    MathSciNet  Google Scholar 

  9. Calderón A. P. Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. USA 74 (1977), 1324–1327.

    Article  MathSciNet  MATH  Google Scholar 

  10. Marshall D. E., The Denjoy Conjecture, preprint, 1977.

    Google Scholar 

  11. Besicovitch A., On the fundamental geometric properties of linearly measurable plane sets of points, I, Math. Ann. 98 (1927), 422–464; II, Math. Ann. 115 (1938), 296–329.

    Article  MathSciNet  MATH  Google Scholar 

  12. Hayman W. K., Kennedy P. B., Subharmonic Functions, Vol. 1., Academic Press, London-New York, 1976.

    MATH  Google Scholar 

  13. Vitushkin A. G., An example of a set of positive length but zero analytic capacity, Dokl. Akad. Nauk SSSR 127 (1959), 246–249. (Russian)

    MathSciNet  MATH  Google Scholar 

  14. Garnett J., Positive length but zero analytic capacity, Proc. Amer. Math. Soc. 24 (1970), 696–699.

    Article  MathSciNet  MATH  Google Scholar 

  15. Vitushkin A. G., The analytic capacity of sets in problems of approximation theory, Uspekhi Mat. Nauk 22 (1967), no. 6, 141–199 (Russian); English transl. in Russian Math. Surveys 22 (1967), 139–200.

    MathSciNet  MATH  Google Scholar 

  16. Carleson L., Selected problems on exceptional sets, Van Nostrand Math. Stud., N 13, Van Nostrand, Toronto, 1967.

    MATH  Google Scholar 

  17. Forstman O., Potentiel d’équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions, Medded. Lunds. Univ. Mat. Sem. 3 (1935), 1–118.

    Google Scholar 

  18. Vitushkin A. G., On a problem of Denjoy, Izv. Akad. Nauk SSSR 28 (1964), no. 4, 745–756. (Russian)

    MathSciNet  MATH  Google Scholar 

  19. Val’skii R. É., Remarks on bounded functions representable by an integral of Cauchy-Stieltjes type, Sibirsk. Mat. Zh. 7 (1966), no. 2, 252–260 (Russian); English translation in Siberian Math. J. 7 (1966), 202–209.

    MathSciNet  MATH  Google Scholar 

References

  1. Ahlfors L. V., Bounded analytic functions, Duke Math. J. 14 (1947), 1–11.

    Article  MathSciNet  MATH  Google Scholar 

  2. Calderón A. P., Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. USA 74 (1977), 1324–1327.

    Article  MathSciNet  MATH  Google Scholar 

  3. Ivanov L. D., The variation of sets and functions, Nauka, Moscow, 1975. (Russian)

    Google Scholar 

  4. Garnett J., Analytic capacity and measure, Lect. Notes in Math., 297, Springer, Berlin, 1972.

    MATH  Google Scholar 

  5. Vitushkin A. G., The analytic capacity of sets in problems of approximation theory, Uspekhi Mat. Nauk 22 (1967), no. 6, 141–199 (Russian); English transl. in Russian Math. Surveys 22 (1967), 139–200.

    MathSciNet  MATH  Google Scholar 

  6. Mel’nikov M. S., Sinanjan S. O., Questions in the theory of approximation of functions of one complex variable, Contemporary Problems of Mathematics, vol. 4, Itogi Nauki i Tekhniki, VINITI, Moscow, 1975, pp. 143–250 (Russian); English transl. in vol. 5, 1976, pp. 688–752.

    Google Scholar 

  7. Zalcman L., Analytic capacity and rational approximation, Lect. Notes in Math., 50, Springer, Berlin, 1968.

    MATH  Google Scholar 

  8. Gamelin T. W., Uniform Algebras, Prentice Hall, N. J., 1969.

    MATH  Google Scholar 

  9. Davie A. M., Analytic capacity and approximation problems, Trans. Amer. Math. Soc 171 (1972), 409–444.

    Article  MathSciNet  MATH  Google Scholar 

  10. Mel’nikov M. S., An estimate of the Cauchy integral along an analytic curve, Mat. Sb. 71 (1966), no. 4, 503–514 (Russian); English transl. in AMS Translations 80 ser. 2 (1969), 243–255.

    MathSciNet  Google Scholar 

  11. Vitushkin A. G., Estimates of the Cauchy integral, Mat. Sb. 71 (1966), no. 4, 515–534 (Russian); English transl. in AMS Translations 80 ser. 2 (1969), 257–278.

    MathSciNet  Google Scholar 

  12. Shirokov N. A., On a property of analytic capacity, Vestnik Leningrad Univ., Mat. (1971), no. 19, 75–82. (Russian)

    MathSciNet  Google Scholar 

  13. Shirokov N. A., Some properties of analytic capacity, Vestnik Leningrad Univ., Mat. 1 (1972), no. 1, 77–86. (Russian)

    Google Scholar 

  14. Besicovitch A., On sufficient conditions for a function to be analytic and on behavior of analytic functions in the neighborhood of non-isolated singular points, Proc. London Math. Soc. 32 (1931), no. 2, 1–9.

    Article  MathSciNet  Google Scholar 

References

  1. Dolženko E. P., On the removal of singularities of analytic functions, Uspekhi Mat. Nauk 18 (1963), no. 4, 135–142 (Russian); English translation in Amer. Math. Soc. Translations 97 ser. 2 (1971), 33–41.

    Google Scholar 

  2. Calderón A. P., Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. USA 74 (1977), 1324–1327.

    Article  MathSciNet  MATH  Google Scholar 

  3. Rogers C. A., Hausdorff Measures, Cambridge University Press, Cambridge, 1970.

    MATH  Google Scholar 

  4. Garnett J., Positive length but zero analytic capacity, Proc. Amer. Math. Soc. 24 (1970), 696–699.

    Article  MathSciNet  MATH  Google Scholar 

References

  1. Eiderman V. Ya., On the comparison of Hausdorff measure and capacity, Algebra i Analiz 3 (1991), no. 6, 174–189 (Russian); English translation in St. Petersburg Math. J. 3(1992), no. 6, 1367–1381.

    MathSciNet  Google Scholar 

  2. Garnett J., Analytic capacity and measure, Lect. Notes in Math. 297 (1972), Springer, Berlin etc.

    MATH  Google Scholar 

References

  1. Peetre J., On the theory of L p,λ-spaces, J. Funct. Anal. 4 (1969), 71–87.

    Article  MathSciNet  Google Scholar 

  2. Harvey R., Polking J., A notion of capacity which characterizes removable singularities, Trans. Amer. Math. Soc. 169 (1972), 183–195.

    Article  MathSciNet  MATH  Google Scholar 

  3. Mel’nikov M. S., Sinanjan S. O., Questions in the theory of approximation of functions of one complex variable, Contemporary Problems of Mathematics, vol. 4, Itogi Nauki i Tekhniki, VINITI, Moscow, 1975, pp. 143–250 (Russian); English transl. in vol. 5, 1976, pp. 688–752.

    Google Scholar 

  4. Král J., Analytic capacity, Proc. Conf. “Elliptische Differentialgleichungen”, Rostock, 1977.

    Google Scholar 

  5. John F., Nirenberg L., On functions of bounded mean oscillations, Comm. Pure Appl. Math. 14 (1961), 415–426.

    Article  MathSciNet  MATH  Google Scholar 

  6. Vitushkin A. G., An example of a set of positive length, but zero analytic capacity, Dokl. Akad. Nauk SSSR 127 (1959), 246–249. (Russian)

    MathSciNet  MATH  Google Scholar 

  7. Calderón A. P., Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. USA 74 (1977), 1324–1327.

    Article  MathSciNet  MATH  Google Scholar 

  8. Davie A. M., Analytic capacity and approximation problems, Trans. Amer. Math. Soc. 171 (1972), 409–444.

    Article  MathSciNet  MATH  Google Scholar 

  9. Havin V. P., Havinson S. Ya., Some estimates of analytic capacity, Dokl. Akad. Nauk SSSR 138 (1961), 789–792 (Russian); English transl. in Soviet Math. Dokl. 2 (1961), 731–734.

    MathSciNet  Google Scholar 

  10. Havin V. P., Boundary properties of integrals of Cauchy type and harmonic conjugate functions in domains with rectifiable boundary, Mat. Sb. 68 (1965), 499–517 (Russian); English transl. in Amer. Math. Soc. Translations 74 ser. 2, (1968), 40–60.

    MathSciNet  Google Scholar 

  11. Garnett J., Analytic capacity and measure, Lect. Notes Math., 297, Springer, Berlin, 1972.

    MATH  Google Scholar 

  12. Ivanov L. D., On a conjecture of Denjoy, Uspekhi Mat. Nauk, 18 (1964), 147–149. (Russian)

    Google Scholar 

  13. Pommerenke Ch., Über die analytische Kapazität, Arch. Math. 11 (1960), 270–277.

    Article  MathSciNet  MATH  Google Scholar 

  14. Fuka J., Král J., Analytic capacity and linear measure, Czechoslovak Math. J. 28 (103) (1978), no. 3, 445–461.

    MathSciNet  MATH  Google Scholar 

References

  1. Vitushkin A. G., The analytic capacity of sets in problems of approximation theory, Uspekhi Mat. Nauk 22 (1967), 141–199 (Russian); English transl. in Russian Math. Surveys 22 (1967), 139–200.

    MathSciNet  MATH  Google Scholar 

  2. Mattila P., Smooth maps, null-sets for integral geometric measure and analytic capacity, Ann. Math. 123 (1986), 303–309.

    Article  MathSciNet  MATH  Google Scholar 

  3. Murai T., Comparison between analytic capacity and the Buffon needle probability, Amer. Math. Soc. Trans. 304 (1987), 501–514.

    Article  MathSciNet  MATH  Google Scholar 

  4. Jones P., Murai T., Positive analytic capacity but zero Buffon needle probability, Pacific J. Math. 133 (1988), 99–114.

    Article  MathSciNet  MATH  Google Scholar 

References

  1. Ahlfors L., Bounded analytic functions, Duke Math. J. 14 (1947), 1–11.

    Article  MathSciNet  MATH  Google Scholar 

  2. Garnett J., Analytic Capacity and Measure, Lecture Notes in Math., 297, Springer-Verlag, Berlin, 1972.

    MATH  Google Scholar 

  3. Murai T., Analytic capacity for arcs, ICM90 Kyoto, Springer-Verlag, Tokyo, 1991.

    MATH  Google Scholar 

  4. Murai T., Analytic capacity (Theory of the Szegö kernel function), Sūgaku, 1991 (to appear).

    Google Scholar 

References

  1. Gamelin T. W., Uniform algebras, Prentice-Hall, 1969.

    Google Scholar 

  2. Goluzin G. M., Geometric theory of functions of a complex variable, Amer. Math. Soc., 1969.

    Google Scholar 

  3. Mel’nikov M. S., Sinanjan S. O., Questions in the theory of approximation of functions of a one complex variable, Contemporary Problems of Mathematics, vol. 4, Itogi Nauki i Tekhniki, VINITI, Moscow, 1975, pp. 143–250 (Russian); English transl. in vol. 5, 1976, pp. 688–752.

    Google Scholar 

  4. Shirokov N. A., On a property of analytic capacity, Vestnik Leningrad Univ., Mat. 1 (1971), 75–82. (Russian)

    MathSciNet  Google Scholar 

  5. Tsuji M., Potential theory in modern function theory, Maruzen, 1959.

    Google Scholar 

  6. Vitushkin A. G., The analytic capacity of sets in problems of approximation theory, Uspekhi Mat. Nauk 22 (1967), 141–199 (Russian); English transl. in Russian Math. Surveys 22 (1967), 139–200.

    MathSciNet  MATH  Google Scholar 

  7. Zalcman L., Analytic capacity and rational approximation, Lecture Notes in Math., vol. 50, Springer-Verlag, 1968.

    Google Scholar 

References

  1. Wiener N., The Dirichlet problem, J. Math. and Phys. 3 (1924), 127–146.

    Article  MATH  Google Scholar 

  2. Wiener N., Certain notions in potential theory, J. Math. and Phys. 3 (1924), 24–51.

    Article  MATH  Google Scholar 

  3. Landis E. M., Second-order equations of elliptic and parabolic type, Nauka, M., 1971. (Russian)

    Google Scholar 

  4. Maz’ja V. G., On the behavior near the boundary of solutions of the Dirichlet problem for the biharmonic operator, Dokl. Akad. Nauk SSSR 18 (1977), no. 4, 15–19 (Russian); English transl. in Soviet Math. Dokl. 18 (1977), 1152–1155.

    Google Scholar 

  5. Maz’ja V. G., Regularity at the boundary of solutions of elliptic equations and conformal mapping, Dokl. Akad. Nauk SSSR 152 (1963), no. 6, 1297–1300 (Russian); English transl. in Soviet Math. Dokl. 152 (1963), 1547–1551.

    MathSciNet  Google Scholar 

  6. Maz’ja V. G., Behavior near the boundary of solutions of the Dirichlet problem for a second-order elliptic equation in divergent form, Mat. Zametki 2 (1967), 209–220 (Russian); English transl. in Math. Notes 2 (1967), 610–617.

    MathSciNet  Google Scholar 

  7. Maz’ja V. G., Continuity at a boundary point of solutions to quasi-linear elliptic equations, Vestnik Leningrad Univ., Math. 25 (1970), 42–55; Correction, Vestnik Leningrad Univ., Math. 1 (1972), 160. (Russian)

    MathSciNet  Google Scholar 

  8. Gariepy R., Ziemer W. P., A regularity condition at the boundary for solutions of quasilinear elliptic equations, Arch. Rat. Mech. Anal. 67 (1977), no. 1, 25–39.

    Article  MathSciNet  MATH  Google Scholar 

  9. Krol I. N., Maz’ja V. G., On the absence of continuity and Hölder continuity of solutions of quasilinear elliptic equations near a nonregular boundary, Trudy Moskovsk. Mat. Obshch. 26 (1972), 73–93 (Russian); English transl. in Trans. Moscow Math. Soc.

    MathSciNet  Google Scholar 

  10. Hedberg L., Non-linear potentials and approximation in the mean by analytic functions, Math. Z. 129 (1972), 299–319.

    Article  MathSciNet  MATH  Google Scholar 

  11. Adams D. R., Meyers N., Thinness and Wiener criteria for non-linear potentials, Indiana Univ. Math. J. 22 (1972), 169–197.

    Article  MathSciNet  MATH  Google Scholar 

  12. Maz’ja V. G., Donchev T., On Wiener regularity at a boundary point for a polyharmonic operator, C.R. Acad. Bulgare Sci. 36 (1983), no. 2, 177–179 (Russian); English transl. in Amer. Math. Soc. Translations, ser. 2 137 (1987), 53–55.

    MathSciNet  Google Scholar 

  13. Maz’ja V. G., Behaviour of solutions to the Dirichlet problem for the biharmonic operator at the boundary point, Equadiff IV, Lect. Notes Math. 703 (1979), 250–262.

    Article  MathSciNet  Google Scholar 

  14. Kondrat’ev V. A., Boundary problems for elliptic equations in domains with conical or angular points, Trudy Moscow Mat. Obsc. 16 (1967), 209–292 (Russian); English transl. in Trans. Moscow Math. Soc. 16 (1967), 227–313.

    MathSciNet  MATH  Google Scholar 

  15. Maz’ja V. G., Nazarov S. A., Plamenevskii B. A., Absence of De Georgi-type theorems for strongly elliptic equations, Zapiski Nauch. Sem. LOMI 115 (1982), 156–168 (Russian); English transl. in J. Soviet Math. 28 (1985), 726–734.

    Google Scholar 

  16. Maz’ja V. G., Nazarov S. A., Plamenevskii B. A., On homogeneous solutions of the Dirichlet problem in the exterior of a slender cone, Dokl. Akad. Nauk SSSR 266 (1982), no. 2, 281–284 (Russian); English transl. in Soviet Math. Dokl. 26 (1982), 320–323.

    MathSciNet  Google Scholar 

  17. Maz’ja V. G., Plamenevskii B. A., The maximum principle for the biharmonic equations in a region with conical points, Izv. Vyssh. Uchebn. Zaved. Mat. 25 (1981), no. 2, 52–59 (Russian); English transl. in Soviet Math. (Iz. VUZ) 25 (1981), 61–70.

    Google Scholar 

  18. Maz’ja V. G., Plamenevskii B. A., 99–120, Dinamika Sploshnoi Sredy, Vyp. 50, Novosibirsk, 1981 (Russian); English transl. in Amer. Math. Soc. Translations, ser. 2 vol. 123, 1984, pp. 109–123.

    Google Scholar 

References

  1. Adams D. R., On the exceptional sets for spaces of potentials, Pac. J. Math. 52 (1974), 1–5.

    Article  MathSciNet  MATH  Google Scholar 

  2. Adams D. R., Lectures on Lp-potential theory, Ume» Univ. Reports, 1981.

    Google Scholar 

  3. Adams D. R., Meyers, N. G., Bessel potentials: Inclusion relations among classes of exceptional sets, Ind. U. Math. J. 221 (1973), 873–905.

    Article  MathSciNet  MATH  Google Scholar 

  4. Aronszajn N., Mulla F., Szeptycki P., On spaces of potentials connected with Lp classes, Ann. Inst. Fourier 13 (1962), 211–306.

    Article  MathSciNet  MATH  Google Scholar 

  5. Hedberg L. I., Wolff T., Thin sets in nonlinear potential theory, Ann. Inst. Fourier 33 (1983).

    Google Scholar 

  6. Jawerth B., The trace of Sobolev and Besov spaces, 0<p<1, Studia Math. 62 (1978), 65–71.

    MathSciNet  MATH  Google Scholar 

  7. Jawerth B., Some observations on Besov and Lizorkin-Triebel spaces, Math. Scand. 40 (1977), 94–104.

    MathSciNet  MATH  Google Scholar 

  8. Peetre J., New Thoughts on Besov Spaces, Duke Univ. Press, 1976.

    Google Scholar 

  9. Stein E., Singular Integrals and Differentiability Properties of Functions, Princeton U. Press, 1970.

    Google Scholar 

References

  1. Adams D. R., The classification problem for the capacities associated with the Besov and Triebel-Lizorkin spaces, Approx. and Function Spaces, Banach Center Pub., vol. 22, PWN Polish Scientific Publishers, Warsaw, 1989, pp. 9–24.

    MATH  Google Scholar 

  2. Adams D. R., A note on the Choquet integral with respect to Hausdorff capacity, Function Spaces and Applications, Lecture Notes in Mathematics, 1302, Springer-Verlag, 1988, pp. 115–124.

    Article  Google Scholar 

  3. Jawerth B., Perez C., Welland G., The positive cone in Triebel-Lizorkin spaces and the relation among potential and maximal operators, Contemporary Math. (M. Milman, ed.), Amer. Math. Soc., Providence, 1989.

    Google Scholar 

  4. Ahern P. Exceptional sets for holomorphic functions, Mich. Math. J. 35 (1988), 29–41.

    Article  MathSciNet  MATH  Google Scholar 

  5. Strichartz R. S., H p Sobolev spaces, preprint.

    Google Scholar 

  6. Orobitg J., Personal communication, July 1988.

    Google Scholar 

  7. Netrusov Yu. V., Metric estimates for the capacities of sets in Besov spaces, Trudy Math. Inst. Steklov 190 (1989), 159–185 (Russian); English transl. in Proc. Steklov Inst. Math. 190 (1992), 167–192.

    MathSciNet  MATH  Google Scholar 

References

  1. Netrusov Yu. V., Singularities of functions from Besov and Lizorkin-Triebel spaces, Trudy Mat. Instituta AN SSSR 187 (1989), 162–178 (Russian); English transl. in Proc. Steklov Inst. Math.

    MathSciNet  Google Scholar 

  2. Netrusov Yu. V., Estimates of capacities assosiated with Besov spaces, Zapiski nauchn. semin. LOMI 201 (1992), 124–156 (Russian); English transl. in J. Soviet Math.

    MATH  Google Scholar 

References

  1. Wolff T., A note on interpolation spaces, Lecture Notes in Math., 908, Springer-Verlag, 1982.

    Google Scholar 

  2. De Vore R., Scherer K., Interpolation of linear operators on Sobolev spaces, Ann. Math. 109 (1979), 583–599.

    Article  MathSciNet  MATH  Google Scholar 

References

  1. Schulze B.-W., Wildenhain G., Methoden der Potentialtheorie für elliptische Differentialgleichungen beliebiger Ordnung, Akademie-Verlag, Berlin, 1977.

    Book  MATH  Google Scholar 

  2. Havin V. P., Approximation in the mean by analytic functions, Dokl. Akad. Nauk SSSR 178 (1968), 1025–1028 (Russian); English transl. in Soviet Math. Dokl. 9 (1968), 245–248.

    MathSciNet  Google Scholar 

  3. Bagby T., Quasi topologies and rational approximation, J. Funct. Anal. 10 (1972), 259–268.

    Article  MathSciNet  MATH  Google Scholar 

  4. Beurling A., Deny J., Dirichlet spaces, Proc. Nat. Acad. Sci. 45 (1959), 208–215.

    Article  MathSciNet  MATH  Google Scholar 

  5. Hedberg L. I., Two approximation problems in function spaces, Ark. Mat. 16 (1978), 51–81.

    Article  MathSciNet  MATH  Google Scholar 

  6. Triebel H., Boundary values for Sobolev-spaces with weights. Density of D(Ω), Ann. Sc. Norm. Sup. Pisa 3 (1973), no. 27, 73–96.

    MathSciNet  MATH  Google Scholar 

References

  1. Hedberg L. I., Spectral synthesis in Sobolev spaces, and uniqueness of solutions of the Dirichlet problem, Acta Math. 147 (1981), 237–264.

    Article  MathSciNet  MATH  Google Scholar 

  2. Hedberg L. I., Wolff T. H., Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 4, 161–187.

    Article  MathSciNet  MATH  Google Scholar 

  3. Kolsrud T., A uniqueness theorem for higher order elliptic partial differential equations, Math. Scand. 51 (1982), 323–332.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Mathematics, University of Kentucky, 40506, Lexington, KY, USA

    J. Brennan

  2. Dept. of Mathematics, Michigan State University, 48824, East Lansing, MI, USA

    A. Volberg

  3. St. Petersburg branch, Steklov Math. Institute, Fontanka 27, 191011, St. Petersburg, Russia

    A. Volberg

  4. Dept. of Math. and Mechanics, St. Petersburg State University, Staryi Peterhof, 198904, St. Petersburg, Russia

    V. P. Havin

Authors
  1. J. Brennan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Volberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. P. Havin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Victor P. Havin Nikolai K. Nikolski

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer-Verlag

About this chapter

Cite this chapter

Brennan, J., Volberg, A., Havin, V.P. (1994). Approximation and capacities. In: Havin, V.P., Nikolski, N.K. (eds) Linear and Complex Analysis Problem Book 3. Lecture Notes in Mathematics, vol 1574. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0101062

Download citation

  • DOI: https://doi.org/10.1007/BFb0101062

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-57871-0

  • Online ISBN: 978-3-540-48368-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation