Title: Integral Representations of Semi-Inner Products in Function Spaces
Author(s): Florian-Horia Vasilescu
Pages: 107-133
Cite as:
Florian-Horia Vasilescu, Integral Representations of Semi-Inner Products in Function Spaces, Int. J. Anal. Appl., 14 (2) (2017), 107-133.

Abstract


Various spaces of measurable functions are usually endowed with semi-inner products expressed in terms of positive measures. Trying to give answers to the inverse problem, we present integral representations for some semi-inner products on function spaces of measurable functions, obtained either directly or by adapting and extending techniques from the theory of moment problems.

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References


  1. J. Agler and J. E. McCarthy, Pick Interpolaton and Hilbert Function Spaces, AMS Graduate Studies in Mathematics, Vol 44, Providence, Rhode Island, 2002. Google Scholar

  2. D. Alpay, The Schur Algorithm, Reproducing Kernel Spaces and System Theory, SMF/AMS Texts and Monographs, Vol. 5, 2001. Google Scholar

  3. S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, Springer-Verlag, New York/Berlin/Heidelberg, 2001. Google Scholar

  4. C. Bayer and J. Teichmann, The proof of Tchakaloff’s theorem, Proc. Amer. Math. Soc., 134 (10) (2006), 3035-3040. Google Scholar

  5. C. Berg, J. P. R. Christensen and P. Ressel, Harmonic analysis on semigroups. Theory of positive definite and related functions, Graduate Texts in Mathematics, 100. Springer-Verlag, New York, 1984. Google Scholar

  6. M. S. Birman and M.Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, D. Reidel Publishing Company, Dordrecht, 1987. Google Scholar

  7. J.B. Conway, A Course in Abstract Analysis, Graduate Studies in Mathematics Vol. 141, AMS, Providence, Rhode Island, 2012. Google Scholar

  8. R. E. Curto and L. A. Fialkow, Recursiveness, positivity, and truncated moment problems, Huston J. Math. 17 (4) (1991), 603-635. Google Scholar

  9. R. E. Curto and L. A. Fialkow, Solution of the truncated complex moment problem for flat data, Memoirs of the AMS, Number 568, 1996. Google Scholar

  10. R. E. Curto and L. A. Fialkow, Flat extensions of positive moment matrices: Recursively generated relations, Memoirs of the AMS, Number 648, 1998. Google Scholar

  11. R. E. Curto and L. A. Fialkow, A duality proof of Tchakaloff’s theorem, J. Math. Anal. Appl., 269 (2002), 519-532. Google Scholar

  12. R. E. Curto and L. A. Fialkow, Truncated K-moment problems in several variables, J. Operator Theory, 54 (1) (2005), 189-226. Google Scholar

  13. R. E. Curto and L. A. Fialkow, An analogue of the Riesz-Haviland theorem for the truncated moment problem, J. Funct. Anal. 255 (2008), 2709-2731. Google Scholar

  14. R. E. Curto, L. A. Fialkow and H. M. M¨ oller, The extremal truncated moment problem, Integral Equations Oper. Theory, 60 (2008), 177-200. Google Scholar

  15. N. Dunford and J.T. Schwartz, Linear Operators, Part I: General Theory, Interscience Publishers, New York/London, 1958. Google Scholar

  16. L. Fialkow, Solution of the truncated moment problem with variety y = x 3 , Trans. Amer. Math. Soc. 363 (2011), 3133-3165. Google Scholar

  17. L. Fialkow and J. Nie, Positivity of Riesz functionals and solutions of quadratic and quartic moment problems, J. Funct. Anal. 258 (2010), 328-356. Google Scholar

  18. E. Hille, Introduction to general theory of reproducing kernels, Rocky Mountain J. Math. 2 (1972), 321-368. Google Scholar

  19. J. H. B. Kemperman, The general moment problem, a geometric approach, Ann. Math. Statist. 39 (1968), 93-122. Google Scholar

  20. M. Laurent, Sums of squares, moment matrices and optimization over polynomials, Emerging applications of algebraic geometry, IMA Vol. Math. Appl., 149, 157-270, Springer, New York, 2009. Google Scholar

  21. H. M. Möller, On square positive extensions and cubature formulas, J. Comput. Appl. Math. 199 (2006), 80-88. Google Scholar

  22. M. Putinar, On Tchakaloff’s theorem, Proc. Amer. Math. Soc. 125 (1997), 2409-2414. Google Scholar

  23. J. Stochel, Solving the truncated moment problem solves the full moment problem, Glasg. Math. J. 43(2001), 335-341. Google Scholar

  24. K. Schmüdgen, Unbounded self-adjoint operators on Hilbert space, Graduate Texts in Mathematics, 265. Springer, Dordrecht, 2012. Google Scholar

  25. V. Tchakaloff, Formule de cubatures mecaniques à coefficients non negatifs, Bull. Sci. Math. 81 (2), 1957, 123-134. Google Scholar

  26. F.-H. Vasilescu, Analytic Functional Calculus and Spectral Decompositions, D. Reidel Publishing Company, Dordrecht, 1982. Google Scholar

  27. F.-H. Vasilescu, Operator theoretic characterizations of moment functions, 17th OT Conference Proceedings, Theta, 2000, 405-415. Google Scholar

  28. F.-H. Vasilescu, Spaces of fractions and positive functionals, Math. Scand. 96 (2005), 257-279. Google Scholar

  29. F.-H. Vasilescu, Dimensional stability in truncated moment problems, J. Math. Anal. Appl. 388 (2012), 219-230. Google Scholar

  30. F.-H. Vasilescu, An Idempotent Approach to Truncated Moment Problems, Integral Equations Oper. Theory 79 (3) (2014), 301-335. Google Scholar

  31. F.-H. Vasilescu, Square Positive Functionals in an Abstract Setting, Operator Theory: the State of the Art, 145-167, Theta, 2016. Google Scholar