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Interpolation theory in Cn: A suryey

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References

  1. Y. ABE, A necessary condition for the existence of peak functions, Mem. Fac. Sci. Kyushu Univ. Ser. A 37 (1983), 1–8.

    MathSciNet  MATH  Google Scholar 

  2. R.F. BASENER, Peak points, barriers and pseudoconvex boundary points, Proc. Am. Math. Soc 65 (1977), 89–92.

    Article  MathSciNet  MATH  Google Scholar 

  3. E. BEDFORD and J.E. FORNAESS, A construction of peak functions on weakly pseudoconvex domains, Ann. Math. II Ser. 107 (1978), 555–568.

    Article  MathSciNet  MATH  Google Scholar 

  4. E. BISHOP, A general Rudin-Carleson theorem, Proc. Am. Math. Soc. 13 (1962), 140–143.

    Article  MathSciNet  MATH  Google Scholar 

  5. T. BLOOM, C peak functions for pseudoconvex domains of strict type, Duke Math. J. 45 (1978), 133–147.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. BRUNA and J. M. ORTEGA, Interpolation by holomorphic functions smooth to the boundary in the unit ball, preprint.

    Google Scholar 

  7. D. BURNS and E.L. STOUT, Extending functions from submanifolds of the boundary, Duke Math. J. 43 (1976), 391–404.

    Article  MathSciNet  MATH  Google Scholar 

  8. L. CARLESON, Sets of uniqueness for functions regular in the unit circle, Acta Math. 87 (1952), 325–345.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. del CASTILLO, On interpolation, peak and zero set on a weakly pseudoconvex domain, Proceedings of 7th Spanish-Portuguese Conference on Math, Part II, Publ. Sec. Mat. Univ. Autonoma Barcelona 21 (1980), 175–176.

    MathSciNet  Google Scholar 

  10. D. CATLIN, Global regularity of the \(\bar \partial \)-Neumann problem, Complex Analysis of Several Variables, Proc. Symp. Pure Math. 41, Am. Math. Soc., Providence, 1984, pp. 39–49.

    Chapter  Google Scholar 

  11. J. CHAUMAT and A.-M. CHOLLET, Ensembles pics pour A(D), Ann. Inst. Fourier (Grenoble) 29 (3) (1979), 171–200.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. CHAUMAT and A.-M. CHOLLET, Caractér et propriétés des ensembles localement pics de A(D), Duke Math. J. 47 (1980), 763–787.

    Article  MathSciNet  MATH  Google Scholar 

  13. J. CHAUMAT and A.-M. CHOLLET, Ensembles pics pour A(D) non globalement inclus dans une variété intégrale, Math. Ann. 258 (1982), 243–252.

    Article  MathSciNet  MATH  Google Scholar 

  14. A.-M. CHOLLET, Ensembles de zéros à la frontière de fonctions analytiques dans des domaines strictement pseudo-convexes, Ann. Inst. Fourier (Grenoble) 26 (1) (1976), 51–80.

    Article  MathSciNet  MATH  Google Scholar 

  15. A.-M. Chollet, Ensembles de zéros, ensembles pics pour A(D) et A(D), Complex Analysis (Québec), Progress in Math. 4, Birkhaüser, Boston, 1980, pp. 57–66.

    Google Scholar 

  16. R.R. COIFMAN and G. WEISS, Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes, Lect. Notes in Math. 242, Springer-Verlag, Berlin, 1971.

    Book  MATH  Google Scholar 

  17. J. P. D'Angelo, Finite-type conditions for real hypersurfaces in Cn, preprint.

    Google Scholar 

  18. A.M. DAVIE and B.K. ØKSENDAL, Peak interpolation sets for some algebras of analytic functions, Pac. J. Math. 41 (1972), 81–87.

    Article  MathSciNet  MATH  Google Scholar 

  19. T. DUCHAMP and E.L. STOUT, Maximum modulus sets, Ann. Inst. Fourier (Grenoble) 31 (3) (1981), 37–69.

    Article  MathSciNet  MATH  Google Scholar 

  20. F. FORELLI, Measures orthogonal to polydisc algebras, J. Math. Mech 17 (1968), 1073–1086.

    MathSciNet  MATH  Google Scholar 

  21. J.E. FORNAESS, Peak points on weakly pseudoconvex domains, Math. Ann. 227 (1977), 173–175.

    Article  MathSciNet  MATH  Google Scholar 

  22. J.E. FORNAESS and B.S. HENRIKSEN, Characterization of global peak sets for A(D), Math. Ann. 259 (1982), 125–130.

    Article  MathSciNet  MATH  Google Scholar 

  23. J.E. FORNAESS and S.G. KRANTZ, Continuously varying peaking functions, Pac. J. Math. 83 (1979), 341–347.

    Article  MathSciNet  MATH  Google Scholar 

  24. J.E. FORNAESS and A. NAGEL, The Mergelyan property for weakly pseudoconvex domains, Man. Math. 22 (1977), 199–208.

    Article  MathSciNet  MATH  Google Scholar 

  25. J.E. FORNAESS and N. ØVRELID, Finitely generated ideals in A(ω), Ann. Inst. Fourier (Grenoble) 33 (1983), 77–86.

    Article  MathSciNet  MATH  Google Scholar 

  26. I. GLICKSBERG, Recent Results in Function Algebras, Regional Conf. Series 11, Am. Math. Soc., Providence, 1972.

    MATH  Google Scholar 

  27. J. GLOBEVNIK, Peak sets for polydisc algebras, Mich. Math. J. 29 (1982), 221–227.

    Article  MathSciNet  MATH  Google Scholar 

  28. J. GLOBEVNIK, Norm preserving interpolation sets for polydisc algebras, Math. Proc. Cam. Philos. Soc. 91 (1982), 291–303.

    Article  MathSciNet  MATH  Google Scholar 

  29. M. HAKIM and N. SIBONY, Quelques conditions pour l'existence de fonctions pics dans des domaines pseudoconvexes, Duke Math. J. 44 (1977), 399–406.

    Article  MathSciNet  MATH  Google Scholar 

  30. M. HAKIM and N. SIBONY, Ensembles pics dans des domaines strictement pseudoconvexes, Duke Math. J. 45 (1978), 601–617.

    Article  MathSciNet  MATH  Google Scholar 

  31. G.M. HENKIN and A.E. TUMANOV, Interpolation submanifolds of pseudoconvex manifolds, Tr. Am. Math. Soc. 115 (1980), 59–69.

    MATH  Google Scholar 

  32. B.S. HENRIKSEN, A peak set of Hausdorff dimension 2n-1 for the algebra A(D) in the boundary of a domain D with C-boundary in Cn, Math. Ann. 259 (1982), 271–277.

    Article  MathSciNet  MATH  Google Scholar 

  33. A. IORDAN, Peak sets in weakly pseudoconvex domains, Math. Z. 188 (1985), 171–188.

    Article  MathSciNet  MATH  Google Scholar 

  34. A. IORDAN, Ensembles de module maximal dans des domaines pseudoconvexes, C. R. Acad. Sci. Paris Sér I 300 (1985), 655–656.

    MathSciNet  MATH  Google Scholar 

  35. A. IORDAN, Peak sets in pseudoconvex doamins with the (NP) property, Math. Ann. 272 (1985), 231–236.

    Article  MathSciNet  MATH  Google Scholar 

  36. T. JIMBO, Peak sets on the boundary of a weakly pseudoconvex domains, Math. Jap. 29 (1984), 51–55.

    MathSciNet  MATH  Google Scholar 

  37. J.-M. LABONDE, thesis, Université de Paris-Sud, Centre d'Orsay, 1985.

    Google Scholar 

  38. A. NAGEL, Smooth zero sets and interpolation sets for some algebras of holomorphic functions on strictly pseudoconvex domains, Duke Math. J. 43 (1976), 323–348.

    Article  MathSciNet  MATH  Google Scholar 

  39. A. NAGEL, Cauchy transforms of measures and a characterization of smooth peak interpolation sets for the ball algebra, Rocky Mt. J. Math. 9 (1979), 299–305.

    Article  MathSciNet  MATH  Google Scholar 

  40. A. NAGEL and W. RUDIN, Local boundary behavior of bounded holomorphic functions, Can. J. Math. 30 (1978), 583–592.

    Article  MathSciNet  MATH  Google Scholar 

  41. A. V. NOELL, Properties of peak sets in weakly pseudoconvex domains in ℂ2, Math. Z. 186 (1984), 99–116.

    Article  MathSciNet  MATH  Google Scholar 

  42. A. V. NOELL, Interpolation in weakly pseudoconvex domains in ℂ2, Math. Ann. 270 (1985), 339–348.

    Article  MathSciNet  MATH  Google Scholar 

  43. A. V. NOELL, Differentiable peak-interpolation on bounded domains with smooth boundary, Bull. Lond. Math. Soc. 17 (1985), 134–136.

    Article  MathSciNet  MATH  Google Scholar 

  44. A. V. NOELL, Peak points in boundaries not of finite type, Pac. J. Math. 123 (1986), 385–390.

    Article  MathSciNet  MATH  Google Scholar 

  45. W. RUDIN, Function Theory in Polydiscs, W. A. Benjamin, New York, 1969.

    MATH  Google Scholar 

  46. W. RUDIN, Peak-interpolation sets of class ℂ1, Pac. J. Math. 75 (1978), 267–279.

    Article  MathSciNet  MATH  Google Scholar 

  47. W. RUDIN, Holomorphic Lipschitz functions in balls, Comment. Math. Helv. 53 (1978), 143–147.

    Article  MathSciNet  MATH  Google Scholar 

  48. R. SAERENS, Interpolation manifolds, Ann. Sc. Norm. Sup. Pisa Cl. Sci. IV Ser. 11 (1984), 177–211.

    MathSciNet  MATH  Google Scholar 

  49. R. SAERENS and E. L. STOUT, Differentiable interpolation on the polydisc, Complex Variables 2 (1984), 271–282.

    Article  MathSciNet  MATH  Google Scholar 

  50. E. M. STEIN, Singular integrals and estimates for the Cauchy-Riemann equations, Bull. Am. Math. Soc. 79 (1973), 440–445.

    Article  MathSciNet  MATH  Google Scholar 

  51. B. STENSØNES, Zero sets for A functions, preprint.

    Google Scholar 

  52. E. L. STOUT, The Theory of Uniform Algebras, Tarrytown-on-Hudson, New York, 1971.

    MATH  Google Scholar 

  53. E. L. STOUT, The dimension of peak-interpolation sets, Proc. Am. Math. Soc. 86 (1982), 413–416.

    Article  MathSciNet  MATH  Google Scholar 

  54. B. A. TAYLOR and D.L. WILLIAMS, The peak sets of Am, Proc. Am. Math. Soc. 24 (1970), 604–605.

    MathSciNet  MATH  Google Scholar 

  55. A. E. TUMANOV, A peak set for the disc algebra of metric dimension 2.5 in the three-dimensional unit sphere, Math. USSR Izv. 11 (1977), 353–359.

    Article  MATH  Google Scholar 

  56. R. E. VALSKII, On measures orthogonal to analytic functions in ℂn, Soviet Math. Dokl. 12 (1971), 808–812.

    Google Scholar 

  57. N. T. VAROPOULOS, Ensembles pics et ensembles d'interpolation pour les algèbres uniformes, C. R. Acad. Sci. Paris Sér. A 272 (1971), 866–867.

    MathSciNet  MATH  Google Scholar 

  58. J. VERDERA, A remark on zero and peak sets on weakly pseudoconvex domains, Bull. Lond. Math. Soc. 16 (1984), 411–412.

    Article  MathSciNet  MATH  Google Scholar 

  59. B. M. WEINSTOCK, Zero-sets of continuous holomorphic functions on the boundary of a strongly pseudoconvex domain, J. Lond. Math. Soc. 18 (1978), 484–488.

    Article  MathSciNet  MATH  Google Scholar 

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Saerens, R. (1987). Interpolation theory in Cn: A suryey. In: Krantz, S.G. (eds) Complex Analysis. Lecture Notes in Mathematics, vol 1268. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097302

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