Abstract
In this chapter we present a geometric approach for studying positive definite sequences. First, we will obtain the Naimark dilation theorem which is the fundamental result of this chapter. Then we will use this theorem to show that there is a one to one correspondence between the set of all positive definite sequences on H, and the set of all choice sequences initiated on H. We also solve the positive definite Carathéodory interpolation problem in the operator setting, and present a geometric proof for many of the results in Chapter II. Then a recursive inverse scattering algorithm to compute the choice sequence from the positive definite sequence is presented. Further, we use a special matrix representation of the Naimark dilation to demonstrate how this choice sequence occurs in the Levinson algorithm for block Toeplitz matrices. This leads to another inverse scattering algorithm to compute the choice sequence. Also shown is how the Naimark dilation occurs in an operator version of the marine seismology and layered medium models discussed in Chapter III. Finally, we will show that there is a one to one correspondence through the Cayley transform, between the set of all positive definite sequences on H, and the set of all contractive analytic functions in H∞(H, H). From this it will follow that the positive definite operator valued Carathéodory interpalation problem is equivalent to the usual operator valued Carathéodory interpalation problem involving n by n contractive analytic Toeplitz matrices.
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Naimark, M.A., Self-adjoint extensions of the second kind of a symmetric operator, Bulletin (Szvestiya) Acad. Sci. URSS (ser. Math.) 4 (1940) pp. 53–104 (Russian, with English summary)
Sz.-Nagy, B. and C. Foias, Harmonic analysis of operators on Hilbert space, North Holland Publishing Co., Amsterdam-Budapest, 1970.
Halmos, P.R., A Hilbert space problem book, Springer-Verlag, New York, 1982.
Constantinescu, T., On the structure of the Naimark Dilation, J. Operator Theory,12 (1984) pp. 159175.
Frazho, A.E., Schur contractions and bilinear stochastic systems, Proceedings of the 1984 Conference on Information Sciences and Systems,Princeton University pp. 190196.
Arsene, Gr., Ceausescu, Z. and C. Foias, On intertwining dilations VIII, J. Operator Theory, 4 (1980) pp. 55–91.
Schur, I., On power series which are bounded in the interior of the unit circle I, J. fir die Reine und Angewandte Mathematik, 147 (1917) pp. 205–232, English translation in I. Schur Methods in Operator Theory and Signal Processing; Operator Theory: Advances and Applications, 18, Ed. I. Gohberg (1986) pp. 31–59.
Bareiss, E.H., Numerical solution of linear equations with Toeplitz and vector Toeplitz matrices, Numer. Math., 13 (1969) pp. 404–424.
Gohberg, I. and A.A. Semencul, On the inversion of finite Toeplitz matrices and their continuous analogs, Mat. Issled., 2 (1972) pp. 201–233.
Kennett, B.L.N., Seismic Wave Propagation in Stratified Media, Cambridge University Press, Cambridge, 1983.
Marple, S.L., Digital Spectral Analysis with Applications, Prentice Hall, Englewood Cliffs, New Jersey 1987.
Moran, M.D., On intertwining Dilations, J. Math. Anal. and Appt., 141 (1989) pp. 219–234.
Rissanen, J., Algorithms for triangular decomposition of block Hankel and Toeplitz matrices with application to factoring positive matrix polynomials, Math. Comput., 27 (1973) pp. 147–154.
Whittle, P., On the fitting of multivariable autoregressions and the approximate canonical factorization of a spectral density matrix, Biometrika, 50 (1963) pp. 129–134.
Wiggins, R.A. and E.A., Robinson, Recursive solution to the multichannel filtering problem, J. Geophys. Res., 70 (1965) pp. 1885–1891.
Dudgeon, D.E. and R.M., Mersereau, Multidimensional Digital Signal Processing, Prentice Hall, Englewood Cliffs, New Jersey (1984).
Akhiezer, N.I., The Classical Moment Problem, Olivier and Boyd, Edinburgh, Scotland, 1965.
Alpay, D. and H. Dym, On applications of reproducing kernel spaces to the Schur algorithm and rational J unitary factorizations, I. Schur Methods in Operator Theory and Signal Processing; Operator Theory: Advances and Applications, 18, Ed. I. Gohberg (1986) pp. 89–159.
Arocena, R., On the parameterization of Adamjan, Arov and Krein, Publ. Math. Orsay, 83 (1983) pp. 7–23.
Arocena, R., On generalized Toeplitz kernels and their relation with a paper of Adamjan, Arov and Krein, Functional Analysis Homomorphy and Approximation Theory Math Studies, 86 North-Holland Amsterdam (1984) pp. 1–22.
Arocena, R., A theorem of Naimark, linear systems and scattering operators, J. Functional Analysis, 69 (1986) pp. 281–288.
Burg, J.P., Maximum Entropy Spectral Analysis, Ph.D. thesis, Department of Geophysics, Stanford University (1975).
Kailath, T., A theorem of I. Schur and its impact on modern signal processing, I. Schur Methods in Operator Theory and Signal Processing; Operator Theory: Advances and Applications, 18, Ed. I. Gohberg (1986) pp. 9–30.
Kailath, T. and A.M. Bruckstein, Naimark dilations, state-space generators and transmission lines, Advances in Invariant Subspaces and other results of Operator theory; Operator theory: Advances and Applications (1984) pp. 173–186.
Kailath, T., Bruckstein, A.M. and D. Morgan, Fast matrix factorizations via discrete transmission lines, Linear Algebra and Its Applications, 75 (1986) pp. 1–25.
Koehler, F. and Taner, M.T. Direct and inverse problems relating refraction coefficients and reflection response for horizontally layered media, Geophysics, 42 (1977) pp. 1199–1206.
Kolmogorov, A., Sur l’interpolation et l’exprapolation des suites stationnaires, C. R. Acad. Sci (Paris) 208 (1939) pp. 2043–2045.
Kolmogorov, A., Stationary sequences in Hilbert space. (Russian) Bull. Math. Univ., Moscow 2 (1941) 40 pp. (English translation by Natasha Artin.)
Krein, M.G., Analytical problems and results in the theory of linear opeators on Hilbert spaces, Proc. Internat. Congress. Math., Moscow - August 1966, Izd. Mir. Moscow (1968) pp. 189–216.
Krein, M. and M. A. Krasnoselski, Fundamental theorems on the extensions of Hermitian operators and some of their applications to the theory of orthogonal polynomials and the moment problem, Uspekhi mat. Nauk, 2 (1947).
Krein, M.G. and A.A. Nudel’man, The Markov Moment Problem and External Problems, Transi. Math. Monographs 50, American Math. Society, Providence, Rhode Island, 1977.
Sz.-Nagy, B. and A. Koranyi, Relations d’un problème de Nevanlinna et Pick avec la theorie des opérateurs de l’espace Hilbertien, Acta Sci. Math., 7 (1956) pp. 295–302.
Wiener, N. and P. Masani, The prediction theory of multivariate stochastic processes I, Acta Math., 98 (1957) pp. 111–150.
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Foias, C., Frazho, A.E. (1990). A Geometric Approach to Positive Definite Sequences. In: The Commutant Lifting Approach to Interpolation Problems. OT 44 Operator Theory: Advances and Applications, vol 44. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7712-1_15
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DOI: https://doi.org/10.1007/978-3-0348-7712-1_15
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