Flat extensions of positive moment matrices: Recursively generated relations

Memoirs of the American Mathematical Society

Volumn 136, Issue 648, 1998, Pages

  Curto, Raúl E ^a   Fialkow, Lawrence A ^b  

^a UNIVERSITY OF IOWA   (United States)

^b STATE UNIVERSITY OF NEW YORK   (United States)

Author keywords

Algebraic variety of a moment sequence; Flat extensions of positive matrices; Moment matrix extension block; Recursively generated relations; Truncated complex moment problem

Indexed keywords

EID: 21944445317     PISSN: 00659266     EISSN: None     Source Type: Journal    
DOI: 10.1090/memo/0648     Document Type: Article

References (67)

1. N.I. Ahiezer and M.G. Krein, Some Questions in the Theory of Moments, Transl. Math. Monographs vol. 2, Amer. Math. Soc., Providence, 1962.
2. N.I. Akhiezer, The Classical Moment Problem, Hafner Publ. Co., New York, 1965.
3. T. Ando, Truncated moment problems for operators, Acta Scient. Math. 31 (1970), 319-333.
4. A. Atzmon, A moment problem for positive measures on the unit disc, Pacific J. Math. 59 (1975), 317-325.
5. C. Berg, The multidimensional moment problem and semigroups, Moments in Mathematics, Proc. Symposia Appl. Math. vol. 37, Amer. Math. Soc., Providence, 1987, pp. 110-124.
6. C. Berg, J.P.R. Christensen and C.U. Jensen, A remark on the multidimensional moment problem, Math. Ann. 223 (1979), 163-169.
7. C. Berg and P.H. Maserick, Polynomially positive definite sequences, Math. Ann. 259 (1982), 487-495.
8. n et décomposition des polynômes à plusieurs variables, J. Funct. Anal. 58 (1984), 254-266.
9. J. Conway, Subnormal Operators, Pitman Publ. Co., London, 1981.
10. R. Cools and P. Rabinowitz, Monomial cubature rules since "Stroud": a compilation, J. Comp. Appl. Math. 48 (1993), 309-326.
11. R. Curto, Quadratically hyponormal weighted shifts, Integral Eq. Operator Th. 13 (1990), 49-66.
12. R. Curto, Joint hyponormality: A bridge between hyponormality and subnormality, Proc. Symp. Pure Math., vol. 51, Part 2, Amer. Math. Soc., Providence, 1990, pp. 69-91.
13. R. Curto, Polynomially hyponormal operators on Hilbert space, Rev. Union Mat. Arg. 37 (1991), 29-56.
14. R. Curto and L. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, Integral Equations and Operator Theory 17 (1993), 202-246.
15. R. Curto and L. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, II, Integral Equations and Operator Theory 18 (1994), 369-426.
16. R. Curto and L. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston J. Math. 17 (1991), 603-635.
17. R. Curto and L. Fialkow, Solution of the truncated complex moment problem for flat data, Memoirs Amer. Math. Soc. no. 568, Amer. Math. Soc., Providence, 1996.
18. R. Curto and L. Fialkow, Flat extensions of positive moment matrices: Relations in analytic or conjugate terms, Operator Th.: Adv. Appl., to appear.
19. R. Curto and L. Fialkow, The truncated complex K-moment problem, preprint, 1998.
20. R. Curto and L. Fialkow, The quadratic moment problem for the unit disk and unit circle, preprint, 1998.
21. R. Curto and M. Putinar, Existence of non-subnormal polynomially hyponormal operators, Bull. Amer. Math. Soc. 25 (1991), 373-378.
22. R. Curto and M. Putinar, Nearly subnormal operators and moment problems, J. Funct. Anal. 115 (1993), 480-497.
23. P.J. Davis and P. Rabinowitz, Methods of Numerical Integration, Academic Press, Inc., Orlando, 1984.
24. R.G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert spaces, Proc. Amer. Math. Soc. 17 (1966), 413-415.
25. L. Fialkow, Positivity, extensions and the truncated complex moment problem, Contemporary Math. 185 (1995), 133-150.
26. L. Fialkow, Multivariable quadrature and extensions of moment matrices, preprint, 1996.
27. L. Fialkow, Minimal representing measures arising from, rank-increasing moment matrix extensions, preprint, 1997.
28. B. Fuglede, The multidimensional moment problem, Expo. Math. 1 (1983), 47-65.
29. F. Hausdorff, Momentprobleme für ein endliches Intervall, Math. Zeit. 16 (1923), 220-248.
30. E.K. Haviland, On the momentum problem for distributions in more than one dimension, Amer. J. Math. 57 (1935), 562-568.
31. E.K. Haviland, On the momentum problem for distributions in more than one dimension, Part II, Amer. J. Math. 58 (1936), 164-168.
32. D. Hilbert, Über die Darstellung definiter Formen als Summen von Formenquadraten, Math. Ann. 32 (1888), 342-350.
33. N. Jewell and A. Lubin, Commuting weighted shifts and analytic function theory in several variables, J. Operator Th. 1 (1979), 207-223.
34. M.G. Krein and A. Nudel'man, The Markov Moment Problem and Extremal Problems, Transl. Math. Monographs, vol. 50, Amer. Math. Soc., Providence, 1977.
35. H. Landau, Classical background of the moment problem, Moments in Mathematics, Proc. Symposia Appl. Math., vol. 37, Amer. Math. Soc., Providence, 1987, pp. 1-15.
36. X. Li, Moment Sequences and Their Applications, Ph.D. dissertation, Virginia Tech. Univ., 1994.
37. J.E. McCarthy, private communication.
38. J.E. McCarthy and L. Yang, Subnormal operators and quadrature domains, Advances in Mathematics, to appear.
39. J.L. McGregor, Solvability criteria for certain N-dimensional moment problems, J. Approx. Theory 30 (1980), 315-333.
40. I.P. Mysovskikh, On Chakalov's Theorem, USSR Comp. Math. 15 (1975), 221-227.
41. F. J. Narcowich, R-operators II: On the approximation of certain operator-valued analytic functions and the Hermitian moment problem, Indiana Univ. Math. J. 26 (1977), 483-513.
42. M. Putinar, A two-dimensional moment problem, J. Funct. Anal. 80 (1988), 1-8.
43. M. Putinar, The L problem of moments in two dimensions, J. Funct. Anal. 94 (1990), 288-307.
44. M. Putinar, Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J. 42 (1993), 969-984.
45. M. Putinar, Extremal solutions of the two-dimensional L-problem of moments, J. Funct. Anal. 136 (1996), 331-364.
46. M. Putinar, Extremal solutions of the two-dimensional L-problem of moments, II, MSRI Preprint 105-95.
47. M. Putinar, Linear analysis of quadrature domains, Ark. for Mat. 33 (1995), 357-376.
48. M. Putinar, On Tchakaloff's Theorem, preprint 1995.
49. A. Ralston, A first Course in Numerical Analysis, McGraw-Hill, New York, 1965.
50. B. Reznick, Sums of Even Powers of Real Linear Forms, Memoirs Amer. Math. Soc., vol. 96, no. 463, Amer. Math. Soc., Providence, 1992.
51. B. Reznick, private communication.
52. D. Sarason, Moment problems and operators on Hilbert space, Moments in Mathematics, Proc. Symposia Appl. Math., vol. 37, Amer. Math. Soc., Providence, 1987, pp. 54-70.
53. K. Schmüdgen, 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.
54. K. Schmüdgen, The K-moment problem for semi-algebraic sets. Math. Ann. 289 (1991), 203-206.
55. Z. Sebestyén and J. Stochel, Restrictions of positive self-adjoint operators, Acta Scient. Math. 55 (1991), 149-154.
56. J. Shohat and J. Tamarkin, The Problem of Moments, Math. Surveys I, Amer. Math. Soc., Providence, 1943.
57. J.L. Smul'jan, An operator Hellinger integral (Russian), Mat. Sb. 91 (1959), 381-430.
58. J. Stochel, private communication.
59. J. Stochel and F. Szafraniec, A characterization of subnormal operators, Spectral theory of linear operators and related topics, Birkhäuser (1984), pp. 261-263.
60. J. Stochel and F. Szafraniec, Unbounded weighted shifts and subnormality, Integral Eq. Operator Th. 12 (1989), 146-153.
61. J. Stochel and F. Szafraniec, On normal extensions of unbounded operators, III: Spectral properties, Publ. RIMS 25 (1989), 105-139.
62. J. Stochel and F. Szafraniec, Algebraic operators and moments on algebraic sets, Portugaliae Math. 51 (1994), 1-21.
63. A.H. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall, 1971.
64. F. Szafraniec, Boundedness of the shift operator related to positive definite forms: An application to moment problems, Ark. Mat. 19 (1981), 251-259.
65. F. Szafraniec, Moments on compact sets, Prediction Theory and Harmonic Analysis (V. Mandrekar and H. Salehi, eds.), North-Holland, Amsterdam, 1983, pp. 379-385.
66. V. Tchakaloff, Formules de cubatures mécaniques à coefficients non négatifs, Bull. Sc. Math. 81 (1957), 123-134.
67. Wolfram Research, Inc., Mathematica, Version 2.1, Wolfram Research, Inc., Champaign, Illinois, 1992.