Advertisement

Complex Analysis and Operator Theory

, Volume 11, Issue 3, pp 627–649 | Cite as

Complex Positive Definite Functions on Strips

  • Jorge Buescu
  • A. C. Paixão
  • A. Symeonides
Article
  • 143 Downloads

Abstract

We characterize a holomorphic positive definite function f defined on a horizontal strip of the complex plane as the Fourier–Laplace transform of a unique exponentially finite measure on \({\mathbb R}\). With this characterization, the classical theorems of Bochner on positive definite functions and of Widder on exponentially convex functions become respectively the real and imaginary sections of the corresponding complex integral representation. We provide minimal holomorphy assumptions for this characterization and derive conclusions for meromorphic functions under minimal positive definiteness conditions. Further characterizations are derived from conditions on the derivatives of f arising from the study of the usual concepts of moment, moment-generating function and characteristic function in this context.

Keywords

Positive definite functions Fourier–Laplace transform  Complex analysis Characteristic functions Moment-generating functions Exponentially convex functions Meromorphic functions Zeta function 

Mathematics Subject Classification

Primary 42A82 Secondary 30A10 30C40 60E10 

References

  1. 1.
    Akhiezer, N.: The Classical Moment Problem and Some Related Questions in Analysis. Oliver and Boyd, Edinburgh (1965)zbMATHGoogle Scholar
  2. 2.
    Berg, C., Christensen, J., Ressel, P.: Harmonic Analysis on Semigroups. Graduate Texts in Mathematics, 100. Springer, New York (1984)Google Scholar
  3. 3.
    Berg, C., Maserick, P.: Exponentially bounded positive definite functions. Ill. J. Math. 28(1), 162–179 (1984)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bernstein, S.: Sur les fonctions absolument monotones. Acta Math. 52, 1–66 (1929)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bisgaard, T., Sasvári, Z.: Characteristic Functions and Moment Sequences. Nova Science, New York (2000)zbMATHGoogle Scholar
  6. 6.
    Buescu, J., Paixão, A.: A linear algebraic approach to holomorphic reproducing kernels in \({\mathbb{C}}^n\). Linear Algebra Appl. 412(2–3), 270–290 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Buescu, J., Paixão, A.: Positive definite matrices and differentiable reproducing kernel inequalities. J. Math. Anal. Appl. 320(1), 279–292 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Buescu, J., Paixão, A.: On differentiability and analyticity of positive definite functions. J. Math. Anal. Appl. 375(1), 336–341 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Buescu, J., Paixão, A.: Real and complex variable positive definite functions. São Paulo J. Math. Sci. 6(2), 155–169 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Buescu, J., Paixão, A.: Complex variable positive definite functions. Complex Anal. Oper. Theory 8(4), 937–954 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cramér, H.: Mathematical Methods of Statistics. Reprint of the 1946 original. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1999)Google Scholar
  12. 12.
    Devinatz, A.: Integral representations of positive definite functions. Trans. Am. Math. Soc. 74, 56–77 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Devinatz, A.: Integral representations of positive definite functions II. Trans. Am. Math. Soc. 77, 455–480 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ehm, W., Genton, M., Gneiting, T.: Stationary covariances associated with exponentially convex functions. Bernoulli 9(4), 607–615 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Graczyk, P., Loeb, J.: Bochner and Schoenberg theorems on symmetric spaces in the complex case. Bull. Soc. Math. France 122(4), 571–590 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hamburger, H.: Bemerkungen zu einer Fragestellung des Herrn Pólya. Math. Z. 7, 302–322 (1920)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Loeb, J., Youssfi, E.: Fonctions holomorphes définies positives sur les domaines tubes. C. R. Math. Acad. Sci. Paris 343(2), 87–90 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lukacs, E.: Characteristic Functions, 2nd edn. Griffin, London (1970)zbMATHGoogle Scholar
  19. 19.
    Mathias, M.: Über positive Fourier-integrale. Math. Z. 16(1), 103–125 (1923)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Stewart, J.: Positive definite functions and generalizations, an historical survey. Rocky Mt. J. Math. 6(3), 409–434 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Youssfi, E.: Harmonic analysis on conelike bodies and holomorphic functions on tube domains. J. Funct. Anal. 155(2), 381–435 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Whittaker, E., Watson, G.: A Course in Modern Analysis, 4th edn. Cambridge University Press, Cambridge (1996)CrossRefzbMATHGoogle Scholar
  23. 23.
    Widder, D.: Necessary and sufficient conditions for the representation of a function by a doubly infinite Laplace integral. Bull. Am. Math. Soc. 40(4), 321–326 (1934)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2015

Authors and Affiliations

  1. 1.Dep. MatemáticaFCUL and CMAFLisbonPortugal
  2. 2.Área Departamental de MatemàticaISELLisbonPortugal
  3. 3.Dep. MatemáticaFCULLisbonPortugal

Personalised recommendations