Skip to main content

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 156))

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover 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. A.B. Aleksandrov, A-integrability of boundary values of harmonic functions, Mat. Zametki 30 (1981), no. 1, 59–72, 154. MR 83j:30039

    MATH  MathSciNet  Google Scholar 

  2. _____, Essays on nonlocally convex Hardy classes, Complex analysis and spectral theory (Leningrad, 1979/1980), Springer, Berlin, 1981, pp. 1–89. MR 84h:46066

    Google Scholar 

  3. _____, Invariant subspaces of shift operators. An axiomatic approach, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 113 (1981), 7–26, 264, Investigations on linear operators and the theory of functions, XI. MR 83g:47031

    MATH  Google Scholar 

  4. _____, Multiplicity of boundary values of inner functions, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 22 (1987), no. 5, 490–503, 515. MR 89e:30058

    MATH  MathSciNet  Google Scholar 

  5. _____, Inner functions and related spaces of pseudocontinuable functions, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 170 (1989), no. Issled. Linein. Oper. Teorii Funktsii. 17, 7–33, 321. MR 91c:30063

    Google Scholar 

  6. A. Aleman and J. Cima, An integral operator on Hp and Hardy’s inequality, J. Anal. Math. 85 (2001), 157–176. MR 1 869 606

    MathSciNet  Google Scholar 

  7. A. Aleman and A. Siskakis, An integral operator on Hp, Complex Variables Theory Appl. 28 (1995), no. 2, 149–158. MR 2000d:47050

    MathSciNet  Google Scholar 

  8. N.K. Bary, A treatise on trigonometric series. Vols. I, II, Authorized translation by Margaret F. Mullins. A Pergamon Press Book, The Macmillan Co., New York, 1964. MR 30 #1347

    Google Scholar 

  9. S. Bell, The Cauchy transform, potential theory, and conformal mapping, CRC Press, Boca Raton, FL, 1992. MR 94k:30013

    Google Scholar 

  10. S.V. Bočkarev, Existence of a basis in the space of functions analytic in the disc, and some properties of Franklin’s system, Mat. Sb. (N.S.) 95(137) (1974), 3–18, 159. MR 50 #8036

    MathSciNet  Google Scholar 

  11. G. Boole, On the comparison of transcendents with certain applications to the theory of definite integrals, Phil. Trans. Royal Soc. 147 (1857), 745–803.

    Google Scholar 

  12. P. Bourdon and J.A. Cima, On integrals of Cauchy-Stieltjes type, Houston J. Math. 14 (1988), no. 4, 465–474. MR 90h:30095

    MathSciNet  Google Scholar 

  13. L. Brown and A.L. Shields, Cyclic vectors in the Dirichlet space, Trans. Amer. Math. Soc. 285 (1984), no. 1, 269–303. MR 86d:30079

    MathSciNet  Google Scholar 

  14. J.A. Cima and A. Matheson, Cauchy transforms and composition operators, Illinois J. Math. 42 (1998), no. 1, 58–69. MR 98k:42028

    MathSciNet  Google Scholar 

  15. J.A. Cima, A. Matheson, and W.T. Ross, The backward shift on the space of Cauchy transforms, to appear, Proc. Amer. Math. Soc.

    Google Scholar 

  16. J.A. Cima and W.T. Ross, The backward shift on the Hardy space, American Mathematical Society, Providence, RI, 2000. MR 2002f:47068

    Google Scholar 

  17. J.A. Cima and A. Siskakis, Cauchy transforms and Cesàro averaging operators, Acta Sci. Math. (Szeged) 65 (1999), no. 3–4, 505–513. MR 2000m:47043

    MathSciNet  Google Scholar 

  18. D. Clark, One dimensional perturbations of restricted shifts, J. Analyse Math. 25 (1972), 169–191. MR 46 #692

    MATH  MathSciNet  Google Scholar 

  19. B. Davis, On the distributions of conjugate functions of nonnegative measures, Duke Math. J. 40 (1973), 695–700. MR 48 #2649

    Article  MATH  MathSciNet  Google Scholar 

  20. _____, On the weak type (1, 1) inequality for conjugate functions, Proc. Amer. Math. Soc. 44 (1974), 307–311. MR 50 #879

    MATH  MathSciNet  Google Scholar 

  21. _____, On Kolmogorov’s inequalities \(\tilde f_p\) ≤ Cp f1, 0 < p < 1, Trans. Amer. Math. Soc. 222 (1976), 179–192. MR 54 #10967

    MATH  MathSciNet  Google Scholar 

  22. F. Delbaen, Weakly compact operators on the disc algebra, J. Algebra 45 (1977), no. 2, 284–294. MR 58 #2304

    MathSciNet  Google Scholar 

  23. J. Diestel, Sequences and series in Banach spaces, Springer-Verlag, New York, 1984. MR 85i:46020

    Google Scholar 

  24. R.G. Douglas, H.S. Shapiro, and A.L. Shields, Cyclic vectors and invariant subspaces for the backward shift operator., Ann. Inst. Fourier (Grenoble) 20 (1970), no. fasc. 1, 37–76. MR 42 #5088

    MathSciNet  Google Scholar 

  25. P.L. Duren, Theory of Hp spaces, Academic Press, New York, 1970. MR 42 #3552

    Google Scholar 

  26. P. Fatou, Séries trigonométriques et séries de Taylor, Acta Math. 30 (1906), 335–400.

    MATH  Google Scholar 

  27. O. Frostman, Sur les produits de Blaschke, Kungl. Fysiografiska Sällskapets i Lund Förhandlingar [Proc. Roy. Physiog. Soc. Lund] 12 (1942), no. 15, 169–182. MR 6,262e

    MATH  MathSciNet  Google Scholar 

  28. J.B. Garnett, Analytic capacity and measure, Springer-Verlag, Berlin, 1972, Lecture Notes in Mathematics, Vol. 297. MR 56 #12257

    Google Scholar 

  29. _____, Bounded analytic functions, Academic Press Inc., New York, 1981. MR 83g:30037

    Google Scholar 

  30. V.V. Golubev, Single valued analytic functions with perfect sets of singular points, Ph.D. thesis, Moscow, 1917.

    Google Scholar 

  31. G.M. Goluzin, Geometric theory of functions of a complex variable, AmericanMathematical Society, Providence, R.I., 1969. MR 40 #308

    Google Scholar 

  32. G.H. Hardy and J.E. Littlewood, Some more theorems concerning Fourier series and Fourier power series, Duke Math. J. 2 (1936), 354–382.

    Article  MathSciNet  Google Scholar 

  33. V.P. Havin, On analytic functions representable by an integral of Cauchy-Stieltjes type, Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr. 13 (1958), no. 1, 66–79. MR 20 #1762

    MATH  MathSciNet  Google Scholar 

  34. _____, Analytic representation of linear functionals in spaces of harmonic and analytic functions which are continuous in a closed region, Dokl. Akad. Nauk SSSR 151 (1963), 505–508. MR 27 #2636

    MATH  MathSciNet  Google Scholar 

  35. _____, The spaces H∞ and L1/H 01 , Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 39 (1974), 120–148, Investigations on linear operators and the theory of functions, IV. MR 50 #965

    MATH  MathSciNet  Google Scholar 

  36. S.Ya. Havinson, On an extremal problem of the theory of analytic functions, Uspehi Matem. Nauk (N.S.) 4 (1949), no. 4(32), 158–159. MR 11,508e

    MathSciNet  Google Scholar 

  37. _____, On some extremal problems of the theory of analytic functions, Moskov. Gos. Univ. Učenye Zapiski Matematika 148(4) (1951), 133–143. MR 14,155f

    MATH  MathSciNet  Google Scholar 

  38. H. Helson and G. Szegö, A problem in prediction theory, Ann. Mat. Pura Appl. (4 51 (1960), 107–138. MR 22 #12343

    MathSciNet  Google Scholar 

  39. E. Hewitt and K. Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, Springer-Verlag, New York, 1965. MR 32 #5826

    Google Scholar 

  40. K. Hoffman, Banach spaces of analytic functions, Dover Publications Inc., New York, 1988, Reprint of the 1962 original. MR 92d:46066

    Google Scholar 

  41. B. Hollenbeck and I. Verbitsky, Best constants for the Riesz projection, J. Funct. Anal. 175 (2000), no. 2, 370–392. MR 2001i:42010

    Article  MathSciNet  Google Scholar 

  42. S.V. Hruščev, The problem of simultaneous approximation and of removal of the singularities of Cauchy type integrals, Trudy Mat. Inst. Steklov. 130 (1978), 124–195. 223, Spectral theory of functions and operators. MR 80j:30055

    MathSciNet  Google Scholar 

  43. S.V. Hruščev and S.A. Vinogradov, Free interpolation in the space of uniformly convergent Taylor series, Complex analysis and spectral theory (Leningrad, 1979/1980), Springer, Berlin, 1981, pp. 171–213. MR 83b:30032

    Google Scholar 

  44. _____, Inner functions and multipliers of Cauchy type integrals, Ark. Mat. 19 (1981), no. 1, 23–42. MR 83c:30027

    MathSciNet  Google Scholar 

  45. R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. MR 47 #701

    MathSciNet  Google Scholar 

  46. J.-P. Kahane, Best approximation in L1(T), Bull. Amer. Math. Soc. 80 (1974), 788–804. MR 52 #3845

    MATH  MathSciNet  Google Scholar 

  47. S. Kakutani, Concrete representation of abstract (L)-spaces and the mean ergodic theorem, Ann. of Math. (2) 42 (1941), 523–537. MR 2,318d

    MATH  MathSciNet  Google Scholar 

  48. Y. Katznelson, An introduction to harmonic analysis, corrected ed., Dover Publications Inc., New York, 1976. MR 54 #10976

    Google Scholar 

  49. S.V. Kisljakov, The Dunford-Pettis, Petczyński and Grothendieck conditions, Dokl. Akad. Nauk SSSR 225 (1975), no. 6, 1252–1255. MR 53 #1241

    MATH  MathSciNet  Google Scholar 

  50. A.N. Kolmogorov, Sur les fonctions harmoniques conjuquées et les séries de Fourier, Fund. Math. 7 (1925), 24–29.

    Google Scholar 

  51. P. Koosis, Introduction to H p spaces, second ed., Cambridge University Press, Cambridge, 1998. MR 2000b:30052

    Google Scholar 

  52. B.I. Korenblum, Closed ideals of the ring An, Funkcional. Anal. i Priložen. 6 (1972), no. 3, 38–52. MR 48 #2776

    MathSciNet  Google Scholar 

  53. P.I. Kuznetsov and E.D. Solomentsev, Ivan Ivanovich Privalov (on the ninetieth anniversary of his birth), Uspekhi Mat. Nauk 37 (1982), no. 4(226), 193–211 (1 plate). MR 84b:01049

    MathSciNet  Google Scholar 

  54. A.I. Markushevich, Theory of functions of a complex variable. Vol. I, II, III, English ed., Chelsea Publishing Co., New York, 1977. MR 56 #3258

    Google Scholar 

  55. L.A. Markushevich and G.Ts. Tumarkin, On a class of functions that can be represented in a domain by an integral of Cauchy-Stieltjes type, Uspekhi Mat. Nauk 52 (1997), no. 3(315), 169–170. MR 98j:30045

    MathSciNet  Google Scholar 

  56. V.G. Maz’ya and T.O. Shaposhnikova, Theory of multipliers in spaces of differentiable functions, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 87j:46074 [57]_M. Mooney, A theorem on bounded analytic functions, Pacific J. Math. 43 (1972), 457–463. MR 47 #2374

    Google Scholar 

  57. N.I. Muskhelishvili, Singular integral equations, Dover Publications Inc., New York, 1992, Boundary problems of function theory and their application to mathematical physics, Translated from the second (1946) Russian edition and with a preface by J. R. M. Radok, Corrected reprint of the 1953 English translation. MR 94a:45001

    Google Scholar 

  58. D.J. Newman, The nonexistence of projections from L1to H1, Proc. Amer. Math. Soc. 12 (1961), 98–99. MR 22 #11276

    MATH  MathSciNet  Google Scholar 

  59. N.K. Nikol’skiĭ, Treatise on the shift operator, Springer-Verlag, Berlin, 1986. MR 87i:47042

    Google Scholar 

  60. H. Pajot, Analytic capacity, rectifiability, Menger curvature and the Cauchy integral, Lecture Notes in Mathematics, vol. 1799, Springer-Verlag, Berlin, 2002. MR 1 952 175

    Google Scholar 

  61. A. Pełczyński, Banach spaces of analytic functions and absolutely summing operators, American Mathematical Society, Providence, R.I., 1977, Expository lectures from the CBMS Regional Conference held at Kent State University, Kent, Ohio, July 11–16, 1976, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 30. MR 58 #23526

    Google Scholar 

  62. S.K. Pichorides, On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov, Studia Math. 44 (1972), 165–179. (errata insert), Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, II. MR 47 #702

    MATH  MathSciNet  Google Scholar 

  63. J. Plemelj, Ein Ergänzungssatz zur Cauchyschen Integraldarstellung analytischer Functionen, Randwerte betreffend, Monatshefte für Math. u. Phys. XIX (1908), 205–210.

    Google Scholar 

  64. A. Poltoratski, Boundary behavior of pseudocontinuable functions, Algebra i Analiz 5 (1993), no. 2, 189–210. MR 94k:30090

    Google Scholar 

  65. _____, On the distributions of boundary values of Cauchy integrals, Proc. Amer. Math. Soc. 124 (1996), no. 8, 2455–2463. MR 96j:30057

    Article  MATH  MathSciNet  Google Scholar 

  66. _____, Kreĭn’s spectral shift and perturbations of spectra of rank one, Algebra i Analiz 10 (1998), no. 5, 143–183. MR 2000d:47028

    Google Scholar 

  67. _____, Maximal properties of the normalized Cauchy transform, J. Amer. Math. Soc. 16 (2003), no. 1, 1–17 (electronic). MR 2003j:30056

    Article  MATH  MathSciNet  Google Scholar 

  68. I.I. Privalov, Sur les fonctions conjuguées, Bull. Soc. Math. France 44 (1916), 100–103.

    MathSciNet  Google Scholar 

  69. _____, Intégral de Cauchy, Ph.D. thesis, Saratov, 1919.

    Google Scholar 

  70. _____, Randeigenschaften analytischer Funktionen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1956. MR 18,727f

    Google Scholar 

  71. F. Riesz and M. Riesz, Über die Randwerte einer analytischen Function, IV Skand. Math. Kongr., Mittag-Leffer, Uppsala (1920), 27–44.

    Google Scholar 

  72. M. Riesz, Sur les fonctions conjugées, Math. Z. 27 (1927), 218–244.

    MATH  MathSciNet  Google Scholar 

  73. W.W. Rogosinski and H.S. Shapiro, On certain extremum problems for analytic functions, Acta Math. 90 (1953), 287–318. MR 15,516a

    MathSciNet  Google Scholar 

  74. W.T. Ross and H.S. Shapiro, Generalized analytic continuation, University Lecture Series, vol. 25, American Mathematical Society, Providence, RI, 2002. MR 2003h:30003

    Google Scholar 

  75. W. Rudin, The closed ideals in an algebra of analytic functions, Canad. J. Math. 9 (1957), 426–434. MR 19,641c

    MATH  MathSciNet  Google Scholar 

  76. _____, Real and complex analysis, third ed., McGraw-Hill Book Co., New York, 1987. MR 88k:00002

    Google Scholar 

  77. _____, Functional analysis, second ed., McGraw-Hill Inc., New York, 1991. MR 92k:46001

    Google Scholar 

  78. A. Rybkin, On an analogue of Cauchy’s formula for Hp, 1/2 ≤ p < 1, and the Cauchy type integral of a singular measure, Complex Variables Theory Appl. 43 (2000), no. 2, 139–149. MR 2001i:30042

    MATH  MathSciNet  Google Scholar 

  79. D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391–405. MR 51 #13690

    Google Scholar 

  80. _____, Composition operators as integral operators, Analysis and partial differential equations, Dekker, New York, 1990, pp. 545–565. MR 92a:47040

    Google Scholar 

  81. _____, Sub-Hardy Hilbert spaces in the unit disk, University of Arkansas Lecture Notes in the Mathematical Sciences, 10, John Wiley & Sons Inc., New York, 1994, A Wiley-Interscience Publication. MR 96k:46039

    Google Scholar 

  82. J. Shapiro and C. Sundberg, Compact composition operators on L1, Proc. Amer. Math. Soc. 108 (1990), no. 2, 443–449. MR 90d:47035

    MathSciNet  Google Scholar 

  83. N.A. Shirokov, Analytic functions smooth up to the boundary, Springer-Verlag, Berlin, 1988. MR 90h:3008

    Google Scholar 

  84. V.I. Smirnov, Sur les valeurs limites des fonctions, régulières à l’intérieur d’un cercle, Journal de la Société Phys.-Math. de Léningrade 2 (1929), 22–37.

    Google Scholar 

  85. F. Smithies, Cauchy and the creation of complex function theory, Cambridge University Press, Cambridge, 1997. MR 99b:01013

    Google Scholar 

  86. Yu.B. Sokhotskii, On definite integrals and functions using series expansions, Ph.D. thesis, St. Petersburg, 1873.

    Google Scholar 

  87. S. Spanne, Sur l’interpolation entre les espaces \(\mathcal{L}_k ^{p\Phi }\), Ann. Scuola Norm. Sup. Pisa (3) 20 (1966), 625–648. MR 35 #728

    MATH  MathSciNet  Google Scholar 

  88. D. Stegenga, Multipliers of the Dirichlet space, Illinois J. Math. 24 (1980), no. 1, 113–139. MR 81a:30027

    MATH  MathSciNet  Google Scholar 

  89. E.M. Stein, Singular integrals, harmonic functions, and differentiability properties of functions of several variables, Singular integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Amer. Math. Soc., Providence, R.I., 1967, pp. 316–335. MR 58 #2467

    Google Scholar 

  90. _____, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 44 #7280

    Google Scholar 

  91. E.M. Stein and G. Weiss, An extension of a theorem of Marcinkiewicz and some of its applications, J. Math. Mech. 8 (1959), 263–284. MR 21 #5888

    MathSciNet  Google Scholar 

  92. O.D. Tsereteli, Remarks on the theorems of Kolmogorov and of F. and M. Riesz, Proceedings of the Symposium on Continuum Mechanics and Related Problems of Analysis (Tbilisi, 1971), Vol. 1 (Russian), Izdat. “Mecniereba”, Tbilisi, 1973, pp. 241–254. MR 52 #6306

    Google Scholar 

  93. _____, Conjugate functions, Mat. Zametki 22 (1977), no. 5, 771-783. MR 58 #12166

    Google Scholar 

  94. G.C. Tumarkin, On integrals of Cauchy-Stieltjes type, Uspehi Mat. Nauk (N.S.) 11 (1956), no. 4(70), 163–166. MR 18,725f

    MATH  MathSciNet  Google Scholar 

  95. P.L. Ul’yanov, On the A-Cauchy integral. I, Uspehi Mat. Nauk (N.S.) 11 (1956), no. 5(71), 223–229. MR 18,726a

    MathSciNet  Google Scholar 

  96. S.A. Vinogradov, Properties of multipliers of integrals of Cauchy-Stieltjes type, and some problems of factorization of analytic functions, Mathematical programming and related questions (Proc. Seventh Winter School, Drogobych, 1974), Theory of functions and functional analysis (Russian), Central Èkonom.-Mat. Inst. Akad. Nauk SSSR, Moscow, 1976, pp. 5–39. MR 58 #28518

    Google Scholar 

  97. S.A. Vinogradov, M.G. Goluzina, and V.P. Havin, Multipliers and divisors of Cauchy-Stieltjes type integrals, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 19 (1970), 55–78. MR 45 #562

    MathSciNet  Google Scholar 

  98. P. Wojtaszczyk, Banach spaces for analysts, Cambridge University Press, Cambridge, 1991. MR 93d:46001

    Google Scholar 

  99. A. Zygmund, Sur les fonctions conjugées, Fund. Math. 13 (1929), 284–303.

    MATH  Google Scholar 

  100. _____, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 21 #6498

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Birkhäuser Verlag Basel/Switzerland

About this paper

Cite this paper

Cima, J.A., Matheson, A., Ross, W.T. (2005). The Cauchy Transform. In: Ebenfelt, P., Gustafsson, B., Khavinson, D., Putinar, M. (eds) Quadrature Domains and Their Applications. Operator Theory: Advances and Applications, vol 156. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7316-4_4

Download citation

Publish with us

Policies and ethics