Skip to main content
SpringerLink
Log in

Superresolution of principal semi-algebraic sets

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

The Hölder continuity of the truncated moment map of a shade function in Euclidean space is established in the vicinity of a principal semi-algebraic set. The proof combines volume bounds of semi-algebraic sets and convex optimization methods. The main estimate is applied to a potential type transform specific to two real variables, for perturbations of quadrature domains.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aharonov, D., Shapiro, H.S.: Domains on which analytic functions satisfy quadrature identities. J. d’Analyse Math. 30, 39–73 (1976)

    Article  MathSciNet  Google Scholar 

  2. Ahiezer, N.I., Krein, M.: Some questions in the theory of moments. translated by W. Fleming and D. Prill. Translations of Mathematical Monographs, Vol. 2. American Mathematical Society, Providence, RI (1962)

  3. Ammari, H., Putinar, M., Steenkamp, A., Triki, F.: Identification of an algebraic domain in two dimensions from a finite number of its generalized polarization tensors. Math. Ann. 375(3–4), 1337–1354 (2019)

    Article  MathSciNet  Google Scholar 

  4. Cuyt, A., Golub, G., Milanfar, P., Verdonk, B.: Multidimensional integral inversion, with applications in shape reconstruction. SIAM J. Sci. Comput. 27(3), 1058–1070 (2005)

    Article  MathSciNet  Google Scholar 

  5. Diaconis, P.: The Markov moment problem and de Finetti’s theorem. I. Math. Z. 247(1), 183–199 (2004)

    Article  MathSciNet  Google Scholar 

  6. Dieu, N.Q., Hung, D.H., Pham, T.S., Anh, H.T.: Volume estimates of sublevel sets of real polynomials. Ann. Polon. Math. 121(2), 157–174 (2018)

    Article  MathSciNet  Google Scholar 

  7. Ebenfelt, P., Gustafsson, B., Khavinson, D., Putinar, M. (eds.) Quadrature domains and their applications, volume 156 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel (2005). The Harold S. Shapiro anniversary volume, Papers from the conference held at the University of California at Santa Barbara, Santa Barbara, CA, March 2003

  8. Fatemi, M., Amini, A., Vetterli, M.: Sampling and reconstruction of shapes with algebraic boundaries. IEEE Trans. Signal Process. 64(22), 5807–5818 (2016)

    Article  MathSciNet  Google Scholar 

  9. Gamboa, F., Gassiat, E.: Sets of superresolution and the maximum entropy method on the mean. SIAM J. Math. Anal. 27(4), 1129–1152 (1996)

    Article  MathSciNet  Google Scholar 

  10. Golub, G.H., Milanfar, P., Varah, J.: A stable numerical method for inverting shape from moments. SIAM J. Sci. Comput. 21(4):1222–1243 (1999/00)

  11. Greenblatt, M.: Oscillatory integral decay, sublevel set growth, and the Newton polyhedron. Math. Ann. 346(4), 857–895 (2010)

    Article  MathSciNet  Google Scholar 

  12. Gustafsson, B.: Singular and special points on quadrature domains from an algebraic geometric point of view. J. Anal. Math. 51, 91–117 (1988)

    Article  MathSciNet  Google Scholar 

  13. Gustafsson, B., He, C., Milanfar, P., Putinar, M.: Reconstructing planar domains from their moments. Inverse Probl. 16(4), 1053–1070 (2000)

    Article  MathSciNet  Google Scholar 

  14. Gustafsson, B., Putinar, M.: Hyponormal quantization of planar domains, volume 2199 of Lecture Notes in Mathematics. Springer, Cham. Exponential transform in dimension two (2017)

  15. Kreĭn, M.G., Nudel’ man, A.A.: The Markov moment problem and extremal problems. American Mathematical Society, Providence, RI (1977). Ideas and problems of P. L. Čebyšev and A. A. Markov and their further development, Translated from the Russian by D. Louvish, Translations of Mathematical Monographs, Vol. 50

  16. Lasserre, J.B., Putinar, M.: Algebraic-exponential data recovery from moments. Discrete Comput. Geom. 54(4), 993–1012 (2015)

    Article  MathSciNet  Google Scholar 

  17. Lewis, A.S.: Superresolution in the Markov moment problem. J. Math. Anal. Appl. 197(3), 774–780 (1996)

    Article  MathSciNet  Google Scholar 

  18. Phong, D.H., Stein, E.M., Sturm, J.: Multilinear level set operators, oscillatory integral operators, and Newton polyhedra. Math. Ann. 319(3), 573–596 (2001)

    Article  MathSciNet  Google Scholar 

  19. Putinar, M.: Extremal solutions of the two-dimensional \(L\)-problem of moments. J. Funct. Anal. 136(2), 331–364 (1996)

    Article  MathSciNet  Google Scholar 

  20. Putinar, M.: On a diagonal Padé approximation in two complex variables. Numer. Math. 93(1), 131–152 (2002)

    MathSciNet  MATH  Google Scholar 

  21. Varčenko, A.N.: Newton polyhedra and estimates of oscillatory integrals. Funkcional. Anal. i Priložen. 10(3), 13–38 (1976)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

Partially supported by a Simons Foundation collaboration grant for mathematicians.

Author information

Authors and Affiliations

  1. University of California at Santa Barbara, Santa Barbara, CA, USA

    Mihai Putinar

  2. Newcastle University, Newcastle upon Tyne, UK

    Mihai Putinar

Authors
  1. Mihai Putinar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mihai Putinar.

Ethics declarations

Conflict of interest

The author declare that he has no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Harold S. Shapiro, in memoriam.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Putinar, M. Superresolution of principal semi-algebraic sets. Anal.Math.Phys. 11, 107 (2021). https://doi.org/10.1007/s13324-021-00540-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13324-021-00540-1

Keywords

Mathematics Subject Classification

Search

Navigation