Skip to main content

Advertisement

SpringerLink
Log in

Two-Dimensional Half-Strong Real Moment Problem and the Corresponding Block Matrices. Part I

  • Published:
Ukrainian Mathematical Journal Aims and scope

We generalize the relationship between the classical moment problem and the spectral theory of Jacobi matrices. We present the solution of the two-dimensional half-strong moment problem and suggest an analog of Jacobi-type matrices associated with the two-dimensional half-strong moment problem and the corresponding system of polynomials orthogonal with respect to a measure with compact support in the real plane.

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. N. I. Akhiezer, Classical Problem of Moments [in Russian], Fizmatgiz, Moscow (1961).

    Google Scholar 

  2. Yu. M. Berezanskii, Expansion in Eigenfunctions of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1965).

    Google Scholar 

  3. Yu. M. Berezanskii, Self-Adjoint Operators in Spaces of Functions of Infinitely Many Variables [in Russian], Naukova Dumka, Kiev (1978).

    Google Scholar 

  4. Yu. M. Berezanskii and Yu. G. Kondrat’ev, Spectral Methods in Infinite-Dimensional Analysis [in Russian], Naukova Dumka, Kiev (1988).

  5. Yu. M. Berezanskii, G. F. Us, and Z. G. Sheftel’, Functional Analysis. A Course of Lectures [in Russian], Vyshcha Shkola, Kiev (1990).

  6. Yu. M. Berezansky, “Some generalizations of the classical moment problem,” Integr. Equat. Oper. Theory, 44, 255–289 (2002).

    Article  MathSciNet  Google Scholar 

  7. Yu. M. Berezansky and M. E. Dudkin, “The complex moment problem in the exponential form,” Methods Funct. Anal. Topology, 10, No. 4, 1–10 (2004).

    MathSciNet  MATH  Google Scholar 

  8. Yu. M. Berezansky and M. E. Dudkin, “The direct and inverse spectral problems for the block Jacobi-type unitary matrices,” Methods Funct. Anal. Topol., 11, No. 4, 327–345 (2005).

    MathSciNet  MATH  Google Scholar 

  9. Yu. M. Berezansky and M. E. Dudkin, “On the complex moment problem,” Math. Nachr., No. 1-2, 60–73 (2007).

    Article  MathSciNet  Google Scholar 

  10. Yu. M. Berezansky and M. E. Dudkin, Jacobian Matrices and Moment Problems, Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences [in Ukrainian], Kyiv (2019).

  11. C. Berg, J. P. R. Christensen, and C. U. Jesse, “A remark on the multidimension moment problem,” Math. Ann., 243, 163–169 (1979).

    Article  MathSciNet  Google Scholar 

  12. T. M. Bisgaard, “On note on factoring of positive definite functions on semigroups,” Math. Nachr., 236, 31–46 (2002).

    Article  MathSciNet  Google Scholar 

  13. M. J. Cantero, L. Moral, and L. Velázquez, “Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle,” Linear Alg. Appl., 362, 29–56 (2003).

    Article  MathSciNet  Google Scholar 

  14. T. Carleman, Les Fonctions Quasi Analytiques, Paris (1926).

  15. A. Devinatz, “Integral representations of positive definite functions. II,” Trans. Amer. Math. Soc., 77, 455–480 (1954).

    Article  MathSciNet  Google Scholar 

  16. A. Devinatz, “Two parameter moment problems,” Duke Math. J., 24, 481–498 (1957).

    Article  MathSciNet  Google Scholar 

  17. M. E. Dudkin, “The exact inner structure of the block Jacobi-type unitary matrices connected with the corresponding direct and inverse spectral problems matrices,” Methods Funct. Anal. Topol., 14, No. 2, 168–176 (2008).

    MathSciNet  MATH  Google Scholar 

  18. M. E. Dudkin, “The complex moment problem in the exponential form with direct and inverse spectral problems for the block Jacobi-type correspondence matrices,” Methods Funct. Anal. Topol., 18, No. 2, 111–139 (2012).

    MathSciNet  MATH  Google Scholar 

  19. M. E. Dudkin, “The inner structure of the Jacobi–Laurent matrix related to the strong Hamburger moment problem,” Methods Funct. Anal. Topol., 19, No. 2, 97–107 (2013).

    MathSciNet  MATH  Google Scholar 

  20. M. E. Dudkin and V. I. Kozak, “Direct and inverse spectral problems for the block Jacobi-type bounded symmetric matrices related to the two-dimensional moment problem,” Methods Funct. Anal. Topol,, 20, No. 3, 219–251 (2014).

    MathSciNet  MATH  Google Scholar 

  21. M. E. Dudkin and V. I. Kozak, “Direct spectral problem with block Jacobi-type matrices corresponding to the strong two-dimensional moment problem,” Nauk. Zap. NaUKMA, Fiz.-Mat. Nauk., 178, 16–22 (2016).

  22. W. B. Jones, W. J. Thron, and O. Njåstad, “Orthogonal Laurent polynomials and strong Hamburger moment problem,” J. Math. Anal. Appl., 98, No. 2, 528–554 (1984).

    Article  MathSciNet  Google Scholar 

  23. W. B. Jones and O. Njåstad, “Orthogonal Laurent polynomials and strong moment theory: a survey,” Continued Fractions and Geometric Function Theory (CONFUN) (Trondheim, 1997), J. Comput. Appl. Math., 105, No. 1-2, 51–91 (1999).

    Article  MathSciNet  Google Scholar 

  24. R. B. Zarkhina, “On the two-dimensional moment problem,” Dokl. Akad. Nauk SSSR, 124, No. 4, 743–746 (1959).

    MathSciNet  Google Scholar 

  25. I. S. Kats and A. A. Nudel’man, “Strong Stieltjes moment problem,” Alg. Anal., 8, No. 6, 26–56 (1996).

    MATH  Google Scholar 

  26. V. I. Kozak, “Construction of block Jacobi-type matrices corresponding to the strong two-dimensional real moment problem,” Nauk. Zap. NaUKMA, Fiz.-Mat. Nauk., 165, 19–26 (2015).

    Google Scholar 

  27. A. G. Kostyuchenko and B. S. Mityagin, “Multidimensional moment problem,” Dokl. Akad. Nauk SSSR, 131, No. 6, 1249–1252 (1960).

    Google Scholar 

  28. A. G. Kostyuchenko and B. S. Mityagin, “Positive-definite functionals on kernel spaces,” Tr. Mosk. Mat. Obshch., 9, 283–316 (1960).

    MATH  Google Scholar 

  29. M. G. Krein, “One general method of factorization of positive-definite kernels in elementary products,” Dokl. Akad. Nauk SSSR, 53, No. 1, 3–6 (1946).

    MATH  Google Scholar 

  30. M. G. Krein, “On the Hermitian operators with directing functionals,” in: Proc. of the Institute of Mathematics, Academy of Sciences of Ukr. RSR [in Ukrainian], No. 10 (1948), pp. 83–106.

  31. S. Mandelbrojt, Séries Adhérentes. Régularisation des Suites, Applications, Gauthier-Villars, Paris (1952).

  32. E. Nelson, “Analytic vectors,” Ann. Math., 70, 572–614 (1959).

    Article  MathSciNet  Google Scholar 

  33. A. E. Nussbaum, “Quasi-analytic vectors,” Ark. Math., 6, No. 10, 179–191 (1965).

    Article  MathSciNet  Google Scholar 

  34. A. E. Nussbaum, “A note on quasi-analytic vectors,” Stud. Math., 33, 305–309 (1969).

    Article  MathSciNet  Google Scholar 

  35. L. C. Petersen, “On the relation between the multidimensional moment problem and the one-dimensional moment problem,” Math. Scand., 51, 361–366 (1982).

    Article  MathSciNet  Google Scholar 

  36. K. K. Simonov, “Strong matrix moment problem of Hamburger,” Methods Funct. Anal. Topol., No. 2, 183–196 (2006).

    MathSciNet  MATH  Google Scholar 

  37. K. K. Simonov, “Orthogonal matrix Laurent polynomials,” Mat. Zametki, 79, No. 2, 316–320 (2006).

    Article  MathSciNet  Google Scholar 

  38. B. Fuglede, “The multidimensional moment problem,” Expo. Math., No. 1, 47–65 (1983).

  39. E. K. Haviland, “On the moment problem for distribution functions in more than one dimension,” Amer. J. Math., 57, 562–572 (1995).

    Article  Google Scholar 

  40. E. K. Haviland, “On the moment problem for distribution functions in more than one dimension. II,” Amer. J. Math., 58, 164–168 (1996).

    Article  Google Scholar 

  41. Y. Xu, On orthogonal polynomials in several variables,” Amer. Math. Soc., 14, 247–270 (1997).

    MathSciNet  MATH  Google Scholar 

  42. Y. Xu, “Block Jacobi matrices and zeros of multivariate orthogonal polynomials,” Amer. Math. Soc., 342, No. 2, 855–866 (1994).

    Article  MathSciNet  Google Scholar 

  43. G. I. Eskin, “Sufficient condition for the solvability of the multidimensional moment problem,” Dokl. Akad. Nauk SSSR, 133, No. 3, 540–543 (1960).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ”Sikorsky Kyiv Polytechnic Institute,”, National Technical University of Ukraine, Kyiv, Ukraine

    M. E. Dudkin & O. Yu. Dyuzhenkova

Authors
  1. M. E. Dudkin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. Yu. Dyuzhenkova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. E. Dudkin.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 8, pp. 1047–1063, August, 2020. Ukrainian DOI: 10.37863/umzh.v72i8.6062.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dudkin, M.E., Dyuzhenkova, O.Y. Two-Dimensional Half-Strong Real Moment Problem and the Corresponding Block Matrices. Part I. Ukr Math J 72, 1212–1229 (2021). https://doi.org/10.1007/s11253-020-01856-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-020-01856-0

Search

Navigation