Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 109, 48 pages      arXiv:1801.06013      https://doi.org/10.3842/SIGMA.2018.109
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)

Inverse of Infinite Hankel Moment Matrices

Christian Berg a and Ryszard Szwarc b
a) Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark
b) Institute of Mathematics, University of Wroclaw, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Received January 19, 2018, in final form October 02, 2018; Published online October 06, 2018

Abstract
Let $(s_n)_{n\ge 0}$ denote an indeterminate Hamburger moment sequence and let $\mathcal H=\{s_{m+n}\}$ be the corresponding positive definite Hankel matrix. We consider the question if there exists an infinite symmetric matrix $\mathcal A=\{a_{j,k}\}$, which is an inverse of $\mathcal H$ in the sense that the matrix product $\mathcal A\mathcal H$ is defined by absolutely convergent series and $\mathcal A\mathcal H$ equals the identity matrix $\mathcal I$, a property called (aci). A candidate for $\mathcal A$ is the coefficient matrix of the reproducing kernel of the moment problem, considered as an entire function of two complex variables. We say that the moment problem has property (aci), if (aci) holds for this matrix $\mathcal A$. We show that this is true for many classical indeterminate moment problems but not for the symmetrized version of a cubic birth-and-death process studied by Valent and co-authors. We consider mainly symmetric indeterminate moment problems and give a number of sufficient conditions for (aci) to hold in terms of the recurrence coefficients for the orthonormal polynomials. A sufficient condition is a rapid increase of the recurrence coefficients in the sense that the quotient between consecutive terms is uniformly bounded by a constant strictly smaller than one. We also give a simple example, where (aci) holds, but an inverse matrix of $\mathcal H$ is highly non-unique.

Key words: indeterminate moment problems; Jacobi matrices; Hankel matrices; orthogonal polynomials.

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References

  1. Akhiezer N.I., The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965.
  2. Berg C., Indeterminate moment problems and the theory of entire functions, J. Comput. Appl. Math. 65 (1995), 27-55.
  3. Berg C., Chen Y., Ismail M.E.H., Small eigenvalues of large Hankel matrices: the indeterminate case, Math. Scand. 91 (2002), 67-81, math.CA/9907110.
  4. Berg C., Christensen J.P.R., Density questions in the classical theory of moments, Ann. Inst. Fourier (Grenoble) 31 (1981), 99-114.
  5. Berg C., Pedersen H.L., On the order and type of the entire functions associated with an indeterminate Hamburger moment problem, Ark. Mat. 32 (1994), 1-11.
  6. Berg C., Pedersen H.L., Logarithmic order and type of indeterminate moment problems (with an appendix by Walter Hayman), in Difference Equations, Special Functions and Orthogonal Polynomials, World Sci. Publ., Hackensack, NJ, 2007, 51-79.
  7. Berg C., Szwarc R., The smallest eigenvalue of Hankel matrices, Constr. Approx. 34 (2011), 107-133, arXiv:0906.4506.
  8. Berg C., Szwarc R., On the order of indeterminate moment problems, Adv. Math. 250 (2014), 105-143, arXiv:1310.0247.
  9. Berg C., Szwarc R., Symmetric moment problems and a conjecture of Valent, Sb. Math. 208 (2017), 335-359, arXiv:1509.06540.
  10. Berg C., Valent G., The Nevanlinna parametrization for some indeterminate Stieltjes moment problems associated with birth and death processes, Methods Appl. Anal. 1 (1994), 169-209.
  11. Bochkov I., Polynomial birth-death processes and the second conjecture of Valent, arXiv:1712.03571.
  12. Chen Y., Ismail M.E.H., Some indeterminate moment problems and Freud-like weights, Constr. Approx. 14 (1998), 439-458.
  13. Chihara T.S., Indeterminate symmetric moment problems, J. Math. Anal. Appl. 85 (1982), 331-346.
  14. Christiansen J.S., Ismail M.E.H., A moment problem and a family of integral evaluations, Trans. Amer. Math. Soc. 358 (2006), 4071-4097.
  15. Escribano C., Gonzalo R., Torrano E., On the inversion of infinite moment matrices, Linear Algebra Appl. 475 (2015), 292-305, arXiv:1311.3473.
  16. Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Vol. 96, 2nd ed., Cambridge University Press, Cambridge, 2004.
  17. Gilewicz J., Leopold E., Valent G., New Nevanlinna matrices for orthogonal polynomials related to cubic birth and death processes, J. Comput. Appl. Math. 178 (2005), 235-245.
  18. Gradshteyn I.S., Ryzhik I.M., Table of integrals, series, and products, 6th ed., Academic Press, Inc., San Diego, CA, 2000.
  19. Ismail M.E.H., Masson D.R., $q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals, Trans. Amer. Math. Soc. 346 (1994), 63-116.
  20. Koekoek R., Swarttouw R.F., The Askey-scheme of hypergeometric orthogonal polynomials and its $q$-analogue, Report 98-17, 1998, available at http://aw.twi.tudelft.nl/~koekoek/askey/.
  21. Levin B.Ya., Lectures on entire functions, Translations of Mathematical Monographs, Vol. 150, Amer. Math. Soc., Providence, RI, 1996.
  22. Romanov R., Order problem for canonical systems and a conjecture of Valent, Trans. Amer. Math. Soc. 369 (2017), 1061-1078, arXiv:1502.04402.
  23. Shohat J.A., Tamarkin J.D., The Problem of Moments, American Mathematical Society Mathematical Surveys, Vol. 1, Amer. Math. Soc., New York, 1943.
  24. Valent G., Indeterminate moment problems and a conjecture on the growth of the entire functions in the Nevanlinna parametrization, in Applications and Computation of Orthogonal Polynomials (Oberwolfach, 1998), Internat. Ser. Numer. Math., Vol. 131, Birkhäuser, Basel, 1999, 227-237.
  25. Yafaev D.R., Unbounded Hankel operators and moment problems, Integral Equations Operator Theory 85 (2016), 289-300, arXiv:1601.08042.

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