The complex moment problem: determinacy and extendibility

  • Dariusz Cichoń
  • Jan Stochel
  • Franciszek Hugon Szafraniec

Abstract

Complex moment sequences are exactly those which admit positive definite extensions on the integer lattice points of the upper diagonal half-plane. Here we prove that the aforesaid extension is unique provided the complex moment sequence is determinate and its only representing measure has no atom at $0$. The question of converting the relation is posed as an open problem. A partial solution to this problem is established when at least one of representing measures is supported in a plane algebraic curve whose intersection with every straight line passing through $0$ is at most one point set. Further study concerns representing measures whose supports are Zariski dense in $\mathbb{C} $ as well as complex moment sequences which are constant on a family of parallel “Diophantine lines”. All this is supported by a bunch of illustrative examples.

References

Akhiezer, N. I., The classical moment problem and some related questions in analysis, Translated by N. Kemmer, Hafner Publishing Co., New York, 1965.

Benedetti, R. and Risler, J.-J., Real algebraic and semi-algebraic sets, Actualités Mathématiques, Hermann, Paris, 1990.

Berg, C., Christensen, J. P. R., and Ressel, P., Harmonic analysis on semigroups: theory of positive definite and related functions, Graduate Texts in Mathematics, vol. 100, Springer-Verlag, New York, 1984. https://doi.org/10.1007/978-1-4612-1128-0

Bisgaard, T. M. and Sasvári, Z., Characteristic functions and moment sequences: positive definiteness in probability, Nova Science Publishers, Inc., Huntington, NY, 2000.

Bochnak, J., Coste, M., and Roy, M.-F., Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 36, Springer-Verlag, Berlin, 1998. https://doi.org/10.1007/978-3-662-03718-8

Bronshtein, I. N., Semendyayev, K. A., Musiol, G., and Muehlig, H., Handbook of mathematics, fifth ed., Springer, Berlin, 2007.

Cichoń, D., Stochel, J., and Szafraniec, F. H., Three term recurrence relation modulo ideal and orthogonality of polynomials of several variables, J. Approx. Theory 134 (2005), no. 1, 11–64. https://doi.org/10.1016/j.jat.2004.12.011

Cichoń, D., Stochel, J., and Szafraniec, F. H., Naimark extensions for indeterminacy in the moment problem. An example, Indiana Univ. Math. J. 59 (2010), no. 6, 1947–1970. https://doi.org/10.1512/iumj.2010.59.4380

Cichoń, D., Stochel, J., and Szafraniec, F. H., Extending positive definiteness, Trans. Amer. Math. Soc. 363 (2011), no. 1, 545–577. https://doi.org/10.1090/S0002-9947-2010-05268-7

Cichoń, D., Stochel, J., and Szafraniec, F. H., Riesz-Haviland criterion for incomplete data, J. Math. Anal. Appl. 380 (2011), no. 1, 94–104. https://doi.org/10.1016/j.jmaa.2011.02.035

van den Essen, A., Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, vol. 190, Birkhäuser Verlag, Basel, 2000. https://doi.org/10.1007/978-3-0348-8440-2

Fuglede, B., The multidimensional moment problem, Exposition. Math. 1 (1983), no. 1, 47–65.

Haviland, E. K., On the Momentum Problem for Distribution Functions in More than One Dimension, Amer. J. Math. 57 (1935), no. 3, 562–568. https://doi.org/10.2307/2371187

Haviland, E. K., On the Momentum Problem for Distribution Functions in More Than One Dimension. II, Amer. J. Math. 58 (1936), no. 1, 164–168. https://doi.org/10.2307/2371063

Kilpi, Y., Über das komplexe Momentenproblem, Ann. Acad. Sci. Fenn. Ser. A. I. no. 236 (1957), 32.

Kunz, E., Introduction to plane algebraic curves, Birkhäuser Boston, Inc., Boston, MA, 2005.

Parthasarathy, K. R., Probability measures on metric spaces, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967.

Petersen, L. C., On the relation between the multidimensional moment problem and the one-dimensional moment problem, Math. Scand. 51 (1982), no. 2, 361–366. https://doi.org/10.7146/math.scand.a-11986

Rudin, W., Real and complex analysis, third ed., McGraw-Hill Book Co., New York, 1987.

Schmüdgen, K., An example of a positive polynomial which is not a sum of squares of polynomials. A positive, but not strongly positive functional, Math. Nachr. 88 (1979), 385–390. https://doi.org/10.1002/mana.19790880130

Schmüdgen, K., The moment problem, Graduate Texts in Mathematics, vol. 277, Springer, Cham, 2017.

Shohat, J. A. and Tamarkin, J. D., The Problem of Moments, American Mathematical Society Mathematical surveys, vol. I, American Mathematical Society, New York, 1943.

Simon, B., The classical moment problem as a self-adjoint finite difference operator, Adv. Math. 137 (1998), no. 1, 82–203. https://doi.org/10.1006/aima.1998.1728

Stieltjes, T.-J., Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. 8 (1894), no. 4, J1–J122.

Stochel, J., Moment functions on real algebraic sets, Ark. Mat. 30 (1992), no. 1, 133–148. https://doi.org/10.1007/BF02384866

Stochel, J. and Stochel, J. B., On the κth root of a Stieltjes moment sequence, J. Math. Anal. Appl. 396 (2012), no. 2, 786–800. https://doi.org/10.1016/j.jmaa.2012.07.012

Stochel, J. and Szafraniec, F. H., Algebraic operators and moments on algebraic sets, Portugal. Math. 51 (1994), no. 1, 25–45.

Stochel, J. and Szafraniec, F. H., The complex moment problem and subnormality: a polar decomposition approach, J. Funct. Anal. 159 (1998), no. 2, 432–491. https://doi.org/10.1006/jfan.1998.3284
Published
2019-06-17
How to Cite
Cichoń, D., Stochel, J., & Szafraniec, F. (2019). The complex moment problem: determinacy and extendibility. MATHEMATICA SCANDINAVICA, 124(2), 263-288. https://doi.org/10.7146/math.scand.a-112091
Section
Articles