Abstract
In this article we introduce some matrix manipulations that allow us to obtain a version of the original Christoffel-Darboux formula, which is of interest in many applications of linear algebra. Using these developments matrix and Jensen’s inequality, we obtain the main result of this proposal, which is the generalization of the maximum entropy theorem of Burg for multivariate processes.
References
Arov, D.Z.; Krein, M.G. (1981) “Problem of search of the minimun of entropy in indeterminate extension problems”, Funt. Anal. Appl. 15(2): 123–126.
Burg, J.P. (1975) Maximum Entropy Spectral Analysis. Ph.D. Dissertation, Stanford University, Stanford, California.
Choi, B.S. (1986) “On the relation bemtween the maximum entropy probability density function and the autoregressive model”, IEEE Trans. Acoust. Speech Signal Process. ASSP-34.
Choi, B.S. (1993) “Multivariate maximum entropy spectrum”, J. Multivariate Anal. 46(1): 56–60.
Delsarte, P.; Genin, Y.; Kamp, Y. (1978) “Orthogonal polynomial matrices on the unit circles”, IEEE Trans. Circuits Syst. 25(3): 149–160.
Dym, H. (1989) “J. contractive matrix function, reproducing kernel Hilbert spaces and interpolation”, CBMS Regional Conference Series in Mathematics 71, A.M.S., Providence, RI.
Gabardo, J.P. (1993) “Extension of positive definite distributions and maximum entropy”, Mem. Am. Math. Soc., Providence RI, 102(489).
Landau, H.J. (1987) “Maximum entropy and the moment problem”, Bull. Am. Math. Soc. 16(1): 47–77.
Marcano, J.G.; Moran, M.D. (2003) “The Arov-Grossman model and Burg multivariate entropy” J. Fourier Anal. Appl. 9(6): 623–647.
Marcano, J.G.; Moran, M.D. (2005) “The Arov-Grossman model and Burg’s entropy”, in: R. Baez-Yates, J. Glaz, H. Gzyl, J. Hüsler & J.L. Palacios (Eds.) Recent Adv. Appl. Probab., Springer, Boston: 329–364.
Masani, P. (1966) Recent trends in multivariate prediction theory, in: P.R. Krishnaiah (Ed.) Multivariate Analysis, Academic Press, New York: 351–382.