Abstract
The main goal of this paper is to study the truncated matricial moment problem on a finite closed interval in the case of an odd number of prescribed moments by using of the FMI method of V.P. Potapov. The solvability of this problem is characterized by the fact that two block Hankel matrices built from the data of the problem are nonnegative Hermitian (Theorem 1.3). An essential step to solve the problem under consideration is to derive an effective coupling identity between both block Hankel matrices (Proposition 2.5). In the case that these Hankel matrices are both positive Hermitian we parametrize the set of solutions via a linear fractional transformation the generating matrix-valued function of which is a matrix polynomial whereas the set of parameters consists of distinguished pairs of meromorphic matrix-valued functions.
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References
Alpay, D/ Bolotnikov, V.: On a general moment problem and certain matrix equations, Linear Alg. Appl. 203–204 (1994), 3–43.
Bolotnikov, V.A.; Sakhnovich, L.A.: On an operator approach to interpolation problems for Stieltjes functions, Integral Equations and Operator Theory 35 (1999), 423–470.
Choque Rivero, A.E/ Dyukarev, Y.M./ Fritzsche, B./ Kirstein, B.: A Truncated Matricial Moment Problem on a Finite Interval to appear in: Interpolation, Schur Functions and Moment Problems (Eds.: D. Alpay and I. Gohberg), Operator Theory: Advances and Applications, 165, Birkhäuser, Basel-Boston-Berlin 2006
Dubovoj, V. K./ Fritzsche, B./ Kirstein, B.: Matricial Version of the Classical Schur Problem. Teubner-Texte zur Mathematik Bd. 129, Teubner, Stuttgart Leipzig 1992
Dyukarev, Yu.M.; Katsnelson, V.E.: Multiplicative and additive classes of analytic matrix functions of Stieltjes type and associated interpolation problems (Russian). Teor. Funkcii, Funkcional. Anal. i Prilozen., Part I: 36 (1981), 13–27; Part III: 41 (1984), 64–70. 36: 13–27, 1981. Engl. transl. in: Amer. Math. Soc. Transl., Series 2, 131 (1986), 55–70.
Dubovoj, V.K.: Indefinite metric in the interpolation problem of Schur for analytic matrix functions (Russian), Teor. Funkcii, Funkcional. Anal. i. Prilozen., Part I: 37 (1982), 14–26; Part II: 38 (1982), 32–39; Part III: 41 (1984), 55–64; Part IV: 42 (1984), 46–57; Part V: 45 (1986), 16–21; Part VI: 47 (1987), 112–119.
Dyukarev, Yu.M.: Multiplicative and additive Stieltjes classes of analytic matrixvalued functions and interpolation problems connected with them. II. (Russian), Teor. Funkcii, Funkcional. Anal. i Prilozen. 38 (1982), 40–48.
Fritzsche, B.; Kirstein, B.: Schwache Konvergenz nichtnegativer hermitescher Borelmaße, Wiss. Z. Karl-Marx-Univ. Leipzig, Math. Naturwiss. R. 37 (1988) 4, 375–398.
Fritzsche, B.; Kirstein, B.; Krug, V.: On several types of resolvent matrices of nondegenerate matricial Carathéodory problems Linear Algebra Appl. 281 (1998), 137–170.
Golinskii, L.B.: On the Nevanlinna-Pick problem in the generalized Schur class of analytic matrix functions (Russian), in: Analysis in Indefinite Dimensional Spaces and Operator Theory (Ed.: V.A. Marčenko), Naukova Dumka, Kiev 1983, pp. 23–33.
Golinskii, L.B.: On some generalization of the matricial Nevanlinna-Pick problem (Russian), Izv. Akad. Nauk Armjan. SSR, Ser. Mat. 18 (1983), No. 3, 187–205.
Ivanchenko, T.S.; Sakhnovich, L.A.: An operator approach to the Potapov scheme for the solution of interpolation problems, in: Matrix and Operator Valued Functions-The Vladimir Petrovich Potapov Memorial Volume (Eds.: I. Gohberg, L.A. Sakhnovich), OT Series, Vol. 72, Birkhäuser, Basel-Boston-Berlin 1994, pp. 48–86.
Krein, M.G.: The ideas of P.L. Chebyshev and A.A. Markov in the theory of limit values of integrals and their further development (Russian), Usp. Mat. Nauk 5 (1951), no. 4, 3–66, Engl. transl.: Amer. Math. Soc. Transl., Series 2, 12 (1959), 1–121.
Katsnelson, V.E.: Continual analogues of the Hamburger-Nevanlinna theorem and fundamental matrix inequalities of classical problems (Russian), Teor. Funkcii, Funkcional. Anal. i Prilozen., Part I: 36 (1981), 31–48; Part II: 37 (1982), 31–48; Part III: 39 (1983), 61–73; Part IV: 40 (1983), 79–90. Engl. transl. in: Amer. Math. Soc. Transl., Series 2, 136 (1987), 49–108.
Katsnelson, V.E.: Methods of J-theory in continuous interpolation problems of analysis (Russian), deposited in VINITI 1983. Engl. transl. by T. Ando, Sapporo 1985.
Katsnelson, V.E.: On transformations of Potapov’s fundamental matrix inequality, in: Topics in Interpolation Theory (Eds.: H. Dym, B. Fritzsche, V.E. Katsnelson), OT Series, Birkhäuser, Basel-Boston-Berlin, 1997, pp. 253–281.
Krein, M.G.; Nudelman, A.A.: The Markov Moment Problem and Extremal Problems, Amer. Math. Soc. Transl., Vol. 50, Amer. Math. Soc., Providence, R.I. 1977.
Kovalishina, I.V.: Analytic theory of a class of interpolation problems (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 47 (1983), 455–497; Engl. transl. in: Math. USSR Izvestiya 22 (1984), 419–463.
Kovalishina, I.V.: The multiple boundary interpolation problem for contractive matrix functions in the unit disc (Russian), Teor. Funkcii, Funkcional. Anal. i Prilozen, 51 (1988), 38–55.
Kats, I.S.: On Hilbert spaces generated by Hermitian monotone matrix functions (Russian), Zapiski Nauc.-issled. Inst. Mat. i Mekh. i Kharkov. Mat. Obsh. 22 (1950), 95–113.
Rosenberg, M.: The square integrability of matrix-valued functions with respect to a non-negative Hermitian measure. Duke Math. J. 31, (1964), 291–298.
Rosenberg, M.: Operators as spectral integrals of operator-valued functions from the study of multivariate stationary stochastic processes. J. Multivariate Anal. 4, (1974), 166–209.
Rosenberg, M.: Spectral integrals of operator-valued functions — II. From the study of stationary processes. J. Multivariate Anal. 6, (1976), 538–571.
Sakhnovich, L.A.: Similarity of operators (Russian), Sibirskii Matematicheskii Zhurnal 13 (1972), 868–883. English transl. in: Sibirian Mathematical Journal 13 (1972), 604–615.
Sakhnovich, L.A.: Interpolation Theory and Its Applications, Kluwer, Dordrecht 1997.
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Choque Rivero, A.E., Dyukarev, Y.M., Fritzsche, B., Kirstein, B. (2007). A Truncated Matricial Moment Problem on a Finite Interval. The Case of an Odd Number of Prescribed Moments. In: Alpay, D., Vinnikov, V. (eds) System Theory, the Schur Algorithm and Multidimensional Analysis. Operator Theory: Advances and Applications, vol 176. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8137-0_2
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