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Inequalities in Hilbert modules of matrix-valued functions

Author: Adhemar Bultheel
Journal: Proc. Amer. Math. Soc. 85 (1982), 369-372
MSC: Primary 46E20; Secondary 60G25
MathSciNet review: 656105
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Abstract: The classical Cauchy-Schwarz inequality and extremality properties of reproducing kernels are generalized for a module of matrix-valued functions on which a matrix-valued inner product is defined. Reference to an application in the field of linear prediction of multivariate stochastic processes is made.

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  • [1] N. Aronzajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337-404. MR 0051437 (14:479c)
  • [2] A. Ben-Israel and T. N. E. Greville, Generalized inverses, theory and applications, Wiley, New York, 1974. MR 0396607 (53:469)
  • [3] J. Bognar, Indefinite inner product spaces, Springer-Verlag, Berlin, 1974. MR 0467261 (57:7125)
  • [4] A. Bultheel, Orthogonal matrix functions related to the multivariable Nevanlinna-Pick problem, Bull. Soc. Math. Belg. Ser. B 31 (1980), 149-170. MR 682639 (84e:93047)
  • [5] P. Delsarte, Y. Genin and Y. Kamp, Orthogonal polynomial matrices on the unit circle, IEEE Trans. Circuits and Systems CAS-25 (1978), 1051-1059. MR 0481886 (58:1981)
  • [6] -, The Nevanlinna-Pick problem for matrix valued functions, SIAM J. Appl. Math. 36 (1978), 47-61. MR 519182 (80h:30035)
  • [7] P. Dewilde and H. Dym, Lossless chain scattering matrices and optimum linear prediction: the vector case, Circuit Theory 9 (1981), 135-175. MR 612268 (82d:94053)
  • [8] H. Meschkowski, Hilbertsche Räume mit Kernfunktionen, Springer-Verlag, Berlin, 1962. MR 0140912 (25:4326)
  • [9] M. Rosenberg, The square integrability of matrix valued functions with respect to a non-negative hermitian measure, Duke J. Math. 31 (1964), 291-298. MR 0163346 (29:649)
  • [10] N. Wiener and P. Masani, The prediction theory of multivariable stochastic processes. I: Regularity conditions, Acta Math. 98 (1957), pp. 111-150; II: The linear predictor, Acta Math. 99 (1958), 93-139. MR 0097856 (20:4323)

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Keywords: Cauchy-Schwarz, reproducing kernel, matrix optimization
Article copyright: © Copyright 1982 American Mathematical Society

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