Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Maximum entropy and the moment problem

Author: H. J. Landau
Journal: Bull. Amer. Math. Soc. 16 (1987), 47-77
MSC (1985): Primary 42A70; Secondary 42A05, 62M15, 94A17, 60G25
MathSciNet review: 866018
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • 1. N. I. Akhiezer, The classical moment problem, Hafner, New York, 1965.
  • 2. L. Breiman, Probability and stochastic processes, Houghton-Mifflin, Boston, 1969.
  • 3. A. M. Bruckstein and T. Kailath, Inverse scattering for discrete transmission-line models, SIAM Review 7 (1986), 1332-1349. MR 902563
  • 4. A. M. Bruckstein, B. C. Levy, and T. Kailath, Differential methods in inverse scattering, SIAM J. Appl. Math. 45 (1985), 312-335. MR 781110
  • 5. A. Bultheel, Error analysis of incoming and outgoing schemes for the trigonometric moment problem, Padé Approximation and Applications, Lecture Notes in Math., vol. 888, Springer-Verlag, Berlin and New York, 1981, pp. 100-109. MR 649088
  • 6. J. P. Burg, Maximum entropy spectral analysis, Proc. 37th Meet. Soc. Exploration Geophysicists, 1967; reprinted in Modern Spectrum Analysis (D. G. Childers, ed.), IEEE Press, New York, 1978, pp. 34-39.
  • 7. J. P. Burg, A new analysis technique for time series data, NATO Adv. Study Inst, on Signal Processing, Enschede, Netherlands, 1968; reprinted in Modern Spectrum Analysis (D. G. Childers, ed.), IEEE Press, New York, 1978, 42-48.
  • 8. J. P. Burg, The relationship between maximum entropy spectra and maximum likelihood spectra, Geophysics, 1972; reprinted in Modern Spectrum Analysis (D. G. Childers, ed.), IEEE Press, New, York, 1978, pp. 132-133.
  • 9. J. P. Burg, Maximum entropy spectral analysis, Ph.D. dissertation, Stanford University, Stanford, California, 1975.
  • 10. D. G. Childers, ed., Modern spectrum analysis, IEEE Press, New York, 1978.
  • 11. B. S. Choi and T. M. Cover, An information-theoretic proof of Burg's maximum entropy spectrum, Proc. IEEE 72 (1984), 1094-1095.
  • 12. I. Csiszàr and G. Tusnàdy, Information geometry and alternating minimization procedures, Statist. Decisions (1984), Suppl. 1, 205-237. MR 785210
  • 13. A. P. Dempster, N. M. Laird, and D. B. Rubin, Maximum likelihood from incomplete data via the EM algorithm, J. Royal Stat. Soc. Ser. B 39 (1977), 1-38. MR 501537
  • 14. H. Dym and A. Iacob, Applications of factorization and Toeplitz operators to inverse problems, Toeplitz Centennial Memorial Conference (I. Gohberg, ed.), Birkhäuser, Basel-Boston-Stuttgart, 1982, pp. 233-260. MR 669911
  • 15. I. C. Gohberg and I. A. Fel'dman, Convolution equations and projection methods for their solution, Transl. Math. Monographs, vol. 41, Amer. Math. Soc. Providence, RI, 1974. MR 355675
  • 16. B. Gopinath and M. M. Sondhi, Inversion of the telegraph equation and the synthesis of non-uniform lines, Proc. IEEE 59 (1971), 383-392. MR 339916
  • 17. U. Grenander and G. Szegö, Toeplitz forms and their applications, Univ. of Calif. Press, Berkeley, 1957. MR 94840
  • 18. S. Haykin, ed., Nonlinear methods of spectral analysis, Springer-Verlag, New York, 1979. MR 601216
  • 19. K. Hoffman, Banach spaces of analytic functions, Prentice-Hall, New York, 1962. MR 133008
  • 20. E. T. Jaynes, On the rationale of maximum entropy methods, Proc. IEEE 70 (1982), 939-952.
  • 21. T. Kailath, A theorem of I. Schur and its impact on modern signal processing, I. Schur Methods in Operator Theory and Signal Processing (I. Gohberg, ed.), Operator Theory: Advances and Applications, vol. 18, Birkhàuser, Basel-Boston-Stuttgart, 1986, pp. 9-30. MR 902601
  • 22. T. Kailath, A. Bruckstein, D. Morgan, Fast matrix factorizations via discrete transmission lines, Linear Algebra Appl. 75 (1986), 1-25. MR 825396
  • 23. T. Kailath and H. Lev-Ari, On mappings between covariance matrices and physical systems, Contemporary Mathematics (B. Datta, ed.), Amer. Math. Soc. vol. 47, Providence, R.I., 1985, pp. 241-252. MR 828304
  • 24. T. Kailath, A. Vieira, and M. Morf, Inverses of Toeplitz operators, innovations, and orthogonal polynomials, SIAM Review 20 (1978), 106-119. MR 512865
  • 25. S. J. Karlin and W. J. Studden, Tchebycheff systems: With applications in analysis and statistics, Interscience, New York, 1966.
  • 26. M. G. Krein, The ideas of P. L. Čebyšev and A. A. Markov in the theory of limiting values of integrals and their further development, Uspehi Mat. Nauk (NS) 6 (1951), 3-120; Amer. Math. Soc. Transl. Ser. 2, 12 (1959), 1-122. MR 44591
  • 27. M. G. Krein, Solution of the inverse Sturm-Liouville problem, Dokl. Akad. Nauk SSSR 76 (1951), 21-24. (Russian) MR 39895
  • 28. M. G. Krein, On integral equations which generate second-order differential equations, Dokl. Akad. Nauk SSSR 97 (1954), 21-24. (Russian) MR 65016
  • 29. M. G. Krein, Continuous analogues of propositions on polynomials orthogonal on the unit circle, Dokl. Akad. Nauk SSSR 105 (1955), 637-640. (Russian) MR 80735
  • 30. R. T. Lacoss, Autoregressive and maximum likelihood spectral analysis methods, Aspects of Signal Processing, Part 2, NATO Adv. Study Inst., La Spezia, Italy, 1976 (G. Tacconi, ed.), D. Reidel, Boston, pp. 591-615.
  • 31. H. J. Landau, The inverse problem for the vocal tract and the moment problem, SIAM J. Math. Anal. 14 (1983), 1019-1035. MR 711183
  • 32. H. Lev-Ari and T. Kailath, Lattice-filter parametrization and modeling of nonstationary processes, IEEE Trans. Inf. Theory IT-30 (1984), 2-16. MR 730996
  • 33. T. L. Marzetta and S. W. Lang, Power spectral density bounds, IEEE Trans. Inf. Theory IT-30 (1984), 117-122.
  • 34. A. Papoulis, Maximum entropy and spectral estimation: a review, IEEE Trans. Acoustics, Speech, and Signal Proc. ASSP-29 (1981), 1176-1186. MR 642904
  • 35. E. A. Robinson, Spectral approach to geophysical inversion by Lorentz, Fourier, and Radon transforms, Proc. IEEE 70 (1982), 1039-1054.
  • 36. J. E. Shore and R. W. Johnson, Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy, IEEE Trans. Inf. Theory IT-26 (1980), 26-37; Comments and corrections, IEEE Trans. Inf. Theory IT-29 (1983), 942-943. MR 733203
  • 37. J. E. Shore and R. W. Johnson, Properties of cross-entropy minimization, IEEE Trans. Inf. Theory IT-27 (1981) 472-482. MR 635526
  • 38. J. M. Van Campenhout and T. M. Cover, Maximum entropy and conditional probability, IEEE Trans. Inf. Theory IT-27 (1981), 483-489. MR 635527
  • 39. Y. Vardi, L. A. Shepp, and L. Kaufman, A statistical model for positron emission tomography, J. Amer. Statist. Assoc. 80 (1985), 8-37. MR 786595

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1985): 42A70, 42A05, 62M15, 94A17, 60G25

Retrieve articles in all journals with MSC (1985): 42A70, 42A05, 62M15, 94A17, 60G25

Additional Information


American Mathematical Society