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A moment problem for pseudo-positive definite functionals

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Arkiv för Matematik

Abstract

A moment problem is presented for a class of signed measures which are termed pseudo-positive. Our main result says that for every pseudo-positive definite functional (subject to some reasonable restrictions) there exists a representing pseudo-positive measure.

The second main result is a characterization of determinacy in the class of equivalent pseudo-positive representation measures. Finally the corresponding truncated moment problem is discussed.

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References

  1. Akhiezer, N. I., The Problem of Moments and Some Related Questions in Analysis, Fiz.-Mat. Lit., Moscow, 1961 (Russian). English transl.: Oliver & Boyd, Edinburgh, 1965.

    Google Scholar 

  2. Andrews, G. E., Askey, R. and Roy, R., Special Functions, Cambridge University Press, Cambridge, 1999.

    MATH  Google Scholar 

  3. Axler, S., Bourdon, P. and Ramey, W., Harmonic Function Theory, Springer, New York, 1992.

    MATH  Google Scholar 

  4. Baouendi, M., Goulaouic, C. and Lipkin, L., On the operator Δr2+μ(∂/∂r)r+λ, J. Differential Equations15 (1974), 499–509.

    Article  MATH  MathSciNet  Google Scholar 

  5. Berg, C., The multivariate moment problem and semigroups, in Moments in Mathematics (San Antonio, TX, 1987 ), Proc. Symp. Appl. Math. 37, pp. 110–124, Amer. Math. Soc., Providence, RI, 1987.

    Google Scholar 

  6. Berg, C., Christensen, J. P. R. and Ressel, P., Harmonic Analysis on Semigroups, Springer, New York, 1984.

    MATH  Google Scholar 

  7. Berg, C. and Thill, M., Rotation invariant moment problems, Acta Math.167 (1991), 207–227.

    Article  MATH  MathSciNet  Google Scholar 

  8. Butkovskii, A. G., Distributed Control Systems, Nauka, Moscow, 1965 (Russian). English transl.: American Elsevier, New York, 1969.

    Google Scholar 

  9. Butkovskii, A. G. and Pustylnikov, L. M., Characteristics of Distributed-Parameter Systems: Handbook of Equations of Mathematical Physics and Distributed-Parameter Systems, Kluwer, Dordrecht, 1993.

    Google Scholar 

  10. Chihara, T. S., Indeterminate symmetric moment problems, J. Math. Anal. Appl.85 (1982), 331–346.

    Article  MATH  MathSciNet  Google Scholar 

  11. Cohn, D. L., Measure Theory, Birkhäuser, Boston, 1980.

    MATH  Google Scholar 

  12. Curto, R. E. and Fialkow, L. A., The truncated complex K-moment problem, Trans. Amer. Math. Soc.352 (2000), 2825–2855.

    Article  MATH  MathSciNet  Google Scholar 

  13. Fuglede, B., The multivariate moment problem, Exposition Math.1 (1983), 47–65.

    MATH  MathSciNet  Google Scholar 

  14. Gelfand, I. and Vilenkin, N. Ya., Applications of Harmonic Analysis, Academic Press, New York, 1964.

    Google Scholar 

  15. Kounchev, O., Distributed moment problem and some related questions on approximation of functions of many variables, in Mathematics and Education in Mathematics, pp. 454–458, Publishing House of the Bulgarian Academy of Sciences, Sofia, 1985.

    Google Scholar 

  16. Kounchev, O., Duality properties for the extreme values of integrals in distributed moments, in Differential Equations and Applications, pp. 759–762, Publ. House of Tech. Univ. of Russe, Russe, 1985.

    Google Scholar 

  17. Kounchev, O., Extremal problems for the distributed moment problem, in Potential Theory (Prague, 1987 ), pp. 187–195, Plenum, New York, 1988.

    Google Scholar 

  18. Kounchev, O., Multivariate Polysplines. Applications to Numerical and Wavelet Analysis, Academic Press, San Diego, CA, 2001.

    MATH  Google Scholar 

  19. Kounchev, O. and Render, H., Pseudopositive multivariate moment problem, C. R. Acad. Bulgare Sci.58 (2005), 1243–1246.

    MATH  MathSciNet  Google Scholar 

  20. Kounchev, O. and Render, H., New PDE method for approximating multivariate integrals, C. R. Acad. Bulgare Sci.58 (2005), 1373–1378.

    MATH  MathSciNet  Google Scholar 

  21. Kounchev, O. and Render, H., A new method for approximating multivariate integrals, Preprint.

  22. McGregor, J. L., Solvability criteria for certain N-dimensional moment problems, J. Approx. Theory30 (1980), 315–333.

    Article  MATH  MathSciNet  Google Scholar 

  23. Pedersen, H. L., Stieltjes moment problems and the Friedrichs extension of a positive definite operator, J. Approx. Theory83 (1995), 289–307.

    Article  MATH  MathSciNet  Google Scholar 

  24. Putinar, M. and Vasilescu, F., Solving moment problems by dimensional extension, Ann. of Math.149 (1999), 1087–1107.

    Article  MATH  MathSciNet  Google Scholar 

  25. Schmuedgen, K., The K-moment problem for compact semi-algebraic sets, Math. Ann.289 (1991), 203–206.

    Article  MATH  MathSciNet  Google Scholar 

  26. Schulze, B.-W. and Wildenhain, G., Methoden der Potentialtheorie für elliptische Differentialgleichungen beliebiger Ordnung, Akademie-Verlag, Berlin, 1977; Birkhäuser, Basel, 1977.

    Google Scholar 

  27. Sobolev, S. L., Cubature Formulas and Modern Analysis: An Introduction, Gordon and Breach, Montreux, 1992.

    MATH  Google Scholar 

  28. Stein, E. M. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ, 1971.

    MATH  Google Scholar 

  29. Stochel, J., Solving the truncated moment problem solves the full moment problem, Glasgow Math. J.43 (2001), 335–341.

    Article  MATH  MathSciNet  Google Scholar 

  30. Stochel, J. and Szafraniec, F. H., The complex moment problem and subnormality: a polar decomposition approach, J. Funct. Anal.159 (1998), 432–491.

    Article  MATH  MathSciNet  Google Scholar 

  31. Zidarov, D., Inverse Gravimetric Problem in Geoprospecting and Geodesy, Elsevier, Amsterdam, 1990.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 18, BG-1113, Sofia, Bulgaria

    Ognyan Kounchev

  2. School of Mathematical Sciences, University College Dublin, Belfield, Ireland

    Hermann Render

Authors
  1. Ognyan Kounchev
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  2. Hermann Render
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Corresponding author

Correspondence to Ognyan Kounchev.

Additional information

Both authors acknowledge the support of the Institutes Partnership Project with the Alexander von Humboldt Foundation, Bonn. The first author was partially supported by a project DO-02-275, 2008 with the National Science Foundation of Bulgaria, and a bilateral research project B-Gr17 within the Greek-Bulgarian S&T Cooperation.

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Kounchev, O., Render, H. A moment problem for pseudo-positive definite functionals. Ark Mat 48, 97–120 (2010). https://doi.org/10.1007/s11512-009-0095-3

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  • DOI: https://doi.org/10.1007/s11512-009-0095-3

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