Integral Representations of Semi-Inner Products in Function Spaces
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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
- J. Agler and J. E. McCarthy, Pick Interpolaton and Hilbert Function Spaces, AMS Graduate Studies in Mathematics, Vol 44, Providence, Rhode Island, 2002.
- D. Alpay, The Schur Algorithm, Reproducing Kernel Spaces and System Theory, SMF/AMS Texts and Monographs, Vol. 5, 2001.
- S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, Springer-Verlag, New York/Berlin/Heidelberg, 2001.
- C. Bayer and J. Teichmann, The proof of Tchakaloff's theorem, Proc. Amer. Math. Soc., 134 (10) (2006), 3035-3040.
- 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.
- M. S. Birman and M.Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, D. Reidel Publishing Company, Dordrecht, 1987.
- J.B. Conway, A Course in Abstract Analysis, Graduate Studies in Mathematics Vol. 141, AMS, Providence, Rhode Island, 2012.
- R. E. Curto and L. A. Fialkow, Recursiveness, positivity, and truncated moment problems, Huston J. Math. 17 (4) (1991), 603-635.
- R. E. Curto and L. A. Fialkow, Solution of the truncated complex moment problem for flat data, Memoirs of the AMS, Number 568, 1996.
- R. E. Curto and L. A. Fialkow, Flat extensions of positive moment matrices: Recursively generated relations, Memoirs of the AMS, Number 648, 1998.
- R. E. Curto and L. A. Fialkow, A duality proof of Tchakaloff's theorem, J. Math. Anal. Appl., 269 (2002), 519-532.
- R. E. Curto and L. A. Fialkow, Truncated K-moment problems in several variables, J. Operator Theory, 54 (1) (2005), 189-226.
- 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.
- R. E. Curto, L. A. Fialkow and H. M. M ¨ oller, The extremal truncated moment problem, Integral Equations Oper. Theory, 60 (2008), 177-200.
- N. Dunford and J.T. Schwartz, Linear Operators, Part I: General Theory, Interscience Publishers, New York/London, 1958.
- L. Fialkow, Solution of the truncated moment problem with variety y = x 3 , Trans. Amer. Math. Soc. 363 (2011), 3133-3165.
- L. Fialkow and J. Nie, Positivity of Riesz functionals and solutions of quadratic and quartic moment problems, J. Funct. Anal. 258 (2010), 328-356.
- E. Hille, Introduction to general theory of reproducing kernels, Rocky Mountain J. Math. 2 (1972), 321-368.
- J. H. B. Kemperman, The general moment problem, a geometric approach, Ann. Math. Statist. 39 (1968), 93-122.
- 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.
- H. M. Möller, On square positive extensions and cubature formulas, J. Comput. Appl. Math. 199 (2006), 80-88.
- M. Putinar, On Tchakaloff's theorem, Proc. Amer. Math. Soc. 125 (1997), 2409-2414.
- J. Stochel, Solving the truncated moment problem solves the full moment problem, Glasg. Math. J. 43(2001), 335-341.
- K. Schmüdgen, Unbounded self-adjoint operators on Hilbert space, Graduate Texts in Mathematics, 265. Springer, Dordrecht, 2012.
- V. Tchakaloff, Formule de cubatures mecaniques à coefficients non negatifs, Bull. Sci. Math. 81 (2), 1957, 123-134.
- F.-H. Vasilescu, Analytic Functional Calculus and Spectral Decompositions, D. Reidel Publishing Company, Dordrecht, 1982.
- F.-H. Vasilescu, Operator theoretic characterizations of moment functions, 17th OT Conference Proceedings, Theta, 2000, 405-415.
- F.-H. Vasilescu, Spaces of fractions and positive functionals, Math. Scand. 96 (2005), 257-279.
- F.-H. Vasilescu, Dimensional stability in truncated moment problems, J. Math. Anal. Appl. 388 (2012), 219-230.
- F.-H. Vasilescu, An Idempotent Approach to Truncated Moment Problems, Integral Equations Oper. Theory 79 (3) (2014), 301-335.
- F.-H. Vasilescu, Square Positive Functionals in an Abstract Setting, Operator Theory: the State of the Art, 145-167, Theta, 2016.