Notes on Generalized Lemniscates

  • Mihai Putinar
Part of the Operator Theory: Advances and Applications book series (OT, volume 157)


A series of analytic and geometric features of generalized lemniscates are presented from an elementary and unifying point of view. A novel interplay between matrix theory and elementary geometry of planar algebraic curves is derived, with a variety of applicationvalue problem and Hardy space estimates to a root separation algorithm.


lemniscate rational embedding determinantal variety Schwarz reflection quadrature domain Hardy space 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Agler, J., McCarthy, J., Pick Interpolation and Hilbert Function Spaces, Amer. Math. Soc., Providence, R.I., 2002.Google Scholar
  2. [2]
    Aharonov, D., Shapiro, H.S., Domains on which analytic functions satisfy quadrature identities, J. Analyse Math. 30(1976), 39–73.Google Scholar
  3. [3]
    Akhiezer, N.I., On a minimum problem in function theory and the number of roots of an algebraic equation inside the unit disc (in Russian), Izv. Akad. Nauk SSSR. 9(1930), 1169–1189.Google Scholar
  4. [4]
    Alpay, D., Dym, H., On a new class of realization formulas and their applications, Linear Alg. Appl. 241–243(1996), 3–84.CrossRefGoogle Scholar
  5. [5]
    Aplay, D., Putinar, M., Vinnikov, V., A Hilbert space approach to bounded analytic extension in the ball, Comm. Pure Appl. Analysis 2(2003), 139–145.Google Scholar
  6. [6]
    Carey, R.W. and Pincus, J.D., An exponential formula for determining functions, Indiana Univ. Math.J. 23 (1974), 1031–1042.CrossRefGoogle Scholar
  7. [7]
    Crowdy, D., Constructing multiply-connected quadrature domains I: algebraic curves, preprint 2002.Google Scholar
  8. [8]
    Ph.J. Davis, The Schwarz function and its applications, Carus Math. Mono. vol. 17, Math. Assoc. Amer., 1974.Google Scholar
  9. [9]
    Ebenfelt, P., Khavinson, D., Shapiro, H.S., An inverse problem for the double layer potential, Comput. Methods. Funct. Theory 1 (2001), 387–401.Google Scholar
  10. [10]
    Eremenko, A., Hayman, W., On the length of lemniscates, Paul Erdös and his mathematics, I (Budapest, 1999), Bolyai Soc. Math. Stud. 11, János Bolyai Math. Soc., Budapest, 2002, pp. 241–242.Google Scholar
  11. [11]
    Foias, C. and Frazho, A.E., The commutant lifting approach to interpolation problems, Birkhäuser Verlag, Basel, 1990.Google Scholar
  12. [12]
    Golub, G., Gustafsson, B., Milanfar, P., Putinar, M. and Varah, J., Shape reconstruction from moments: theory, algorithms, and applications, Signal Processing and Image Engineering, SPIE Proceedings vol. 4116(2000), Advanced Signal Processing, Algorithms, Architecture, and Implementations X (Franklin T. Luk, ed.), pp. 406–416.Google Scholar
  13. [13]
    Griffiths, P., Harris, J., Principles of Algebraic Geometry, J. Wiley Sons, New York, 1994.Google Scholar
  14. [14]
    Gustafsson, B., Quadrature identities and the Schottky double, Acta Appl. Math. 1 (1983), 209–240.CrossRefGoogle Scholar
  15. [15]
    Gustafsson, B., Singular and special points on quadrature domains from an algebraic point of view, J. d’Analyse Math. 51(1988), 91–117.Google Scholar
  16. [16]
    Gustafsson, B. and Putinar, M., An exponential transform and regularity of free boundaries in two dimensions, Ann. Sc. Norm. Sup. Pisa, 26 (1998), 507–543.Google Scholar
  17. [17]
    Gustafsson, B. and Putinar, M., Linear analysis of quadrature domains. II, Israel J. Math. 119(2000), 187–216.Google Scholar
  18. [18]
    Gustafsson, B. and Putinar, M., Linear analysis of quadrature domains. IV, Quadrature Domains and Applications, The Harold S. Shapiro Anniversary Volume, (P. Ebenfeldt et al. eds.), Operator Theory: Advances Appl. vol. 156, Birkhäuser, Basel, 2004, 147–168.Google Scholar
  19. [19]
    Horn, R.A. and Johnson, C.R., Matrix Analysis, Cambridge Univ. Press, Cambridge, 1985.Google Scholar
  20. [20]
    Král, J., Integral Operators in Potential Theory, Lect. Notes Math. vol. 823, Springer, Berlin, 1980.Google Scholar
  21. [21]
    Kravitsky, N., Rational operator functions and Bezoutian operator vessels, Integral Eq. Operator Theory 26(1996), 60–80.CrossRefGoogle Scholar
  22. [22]
    Kuznetsova, O.S., Tkachev, V.G., Length functions of lemniscates, Manuscripta Math. 112 (2003), 519–538.CrossRefGoogle Scholar
  23. [23]
    Livsic, M.S., Kravitsky, N., Markus, A.S., Vinnikov, V., Theory of commruting non-selfadjoint operators, Kluwer Acad. Publ. Group, Dordrecht, 1995.Google Scholar
  24. [24]
    Martin, M. and Putinar, M., Lectures on Hyponormal Operators, Birkhäuser, Basel, 1989.Google Scholar
  25. [25]
    Pincus, J.D. and R.ovnyak, J., A representation for determining functions, Proc. Amer. Math. Soc. 22(1969), 498–502.Google Scholar
  26. [26]
    Putinar, G., Putinar, M., Root separation on generalized lemniscates, Hokkaido Math. J. 30(2001), 705–716.Google Scholar
  27. [27]
    Putinar, M., Linear analysis of quadrature domains, Ark. Mat. 33 (1995), 357–376.Google Scholar
  28. [28]
    Putinar, M., A renormalized Riesz potential and applications, in vol. Advances in Constructive Approximation: Vanderbilt 2003, (M. Neamtu and E. Saff, eds.), Nash-boro Press, Brentwood, TN, pp. 433–466.Google Scholar
  29. [29]
    Putinar, M., Sandberg, S., A skew normal dilation on the numerical range, Math. Ann., to appear.Google Scholar
  30. [30]
    Ransford, T., Potential Theory in the Complex Domain, Cambridge Univ. Press, Cambridge, 1995.Google Scholar
  31. [31]
    Riesz, F. and Sz.-Nagy, B., Functional analysis, Dover Publ., New York, 1990.Google Scholar
  32. [32]
    Sakai, M., Quadrature Domains, Lect. Notes Math. 934, Springer-Verlag, Berlin-Heidelberg 1982.Google Scholar
  33. [33]
    Shapiro, Alex., personal communication.Google Scholar
  34. [34]
    Shapiro, H.S., The Schwarz function and its generalization to higher dimensions, Univ. of Arkansas Lect. Notes Math. Vol. 9, Wiley, New York, 1992.Google Scholar
  35. [35]
    Vinnikov, V., Complete description of determinantal representations of smooth irreducible curves, Linear Alg. Appl. 125 (1989), 103–140.CrossRefGoogle Scholar
  36. [36]
    Vinnikov, V., Elementary transformations of determinantal representations of algebraic curves, Linear Alg. Appl. 135 (1990), 1–18.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • Mihai Putinar
    • 1
  1. 1.Mathematics DepartmentUniversity of CaliforniaSanta BarbaraUSA

Personalised recommendations