Bibliography

[AA66] Vadim M. Adamjan and Damir Z. Arov, Unitary couplings of semi- unitary operators, (Russian) Mat. Issled. 1 (1966), no. 2, 3–64. [AA68] Vadim M. Adamjan and Damir Z. Arov, A general solution of a certain problem in the linear prediction of stationary processes, (Rus- sian) Teor. Verojatnost. i Primenen. 13 (1968) 419–431; English translation: Theor. Probability Appl. 13 (1968), 394–407 [AAK71] Vadim M. Adamjan, Damir Z. Arov and Mark G. Krein, Analytic properties of Schmidt pairs of a Hankel operator and the generalized Schur–Takagi problem, (Russian) Mat. Sb. (N.S.) 86(128) (1971), 34– 75; English translation: Math USSR Sbornik 15 (1971), no. 1, 31–73. [AR68] Naum I. Ahiezer, and Alexander M. Rybalko, Continual analogues of polynomials orthogonal on a circle, (Russian) Ukrain.Mat.Z.20 (1968), no. 1, 3–24. [AlD84] Daniel Alpay and Harry Dym, Hilbert spaces of analytic functions, inverse scattering and operator models, I., Integral Equations Operator Theory 7 (1984), no. 5, 589–641. [AlD85] Daniel Alpay and Harry Dym, Hilbert spaces of analytic functions, inverse scattering and operator models, II., Integral Equations Operator Theory 8 (1985), no. 2, 145–180. [AlD93] Daniel Alpay and Harry Dym, On a new class of structured reproducing kernel spaces, J. Functional Analysis 111 (1993), no. 1, 1–28. [AlG95] Daniel Alpay and Israel Gohberg, Inverse spectral problem for differential operators with rational scattering matrix functions, J. Differ- ential Equations 118 (1995), no. 1, 1–19. [AlG97] Daniel Alpay and Israel Gohberg, State space method for inverse spectral problems, in: Systems and Control in the Twenty-first Century (St. Louis, MO, 1996), Progr. Systems Control Theory 22, Birkhäuser Boston, Boston, MA, 1997, pp. 1–16. [AlG01] Daniel Alpay and Israel Gohberg, Inverse problems associated to a canonical differential system, in: Recent Advances in Operator The- ory and Related Topics (Szeged, 1999), Operator Theory: Advances Applications 127, Birkhäuser, Basel, 2001, pp. 1–27.

© Springer International Publishing AG, part of Springer Nature 2018 385 D. Z. Arov, H. Dym, Multivariate Prediction, de Branges Spaces, and Related Extension and Inverse Problems, Operator Theory: Advances and Applications 266, https://doi.org/10.1007/978-3-319-70262-9 386 Bibliography

[AGKS00] Daniel Alpay, Israel Gohberg, Marinus A. Kaashoek, Leonid Lerer and Alexander Sakhnovich, Direct and inverse scattering problem for canonical systems with a strictly pseudo-exponential potential, Math. Nachr. 215 (2000), 5–31. (Reviewer: Hendrik S. V. de Snoo) [AGKLS09] Daniel Alpay, Israel Gohberg, Marinus A. Kaashoek, Leonid Lerer and Alexander Sakhnovich, Krein systems, in: Modern Analysis and Applications. The Mark Krein Centenary Conference. Vol. 2: Differ- ential operators and mechanics, Operator Theory: Advances Applica- tions 191, Birkhäuser, Basel, 2009, pp. 19–36. [AGKLS10] Daniel Alpay, Israel Gohberg, Marinus A. Kaashoek, Leonid Lerer and Alexander L. Sakhnovich, Krein systems and canonical systems on a finite interval: accelerants with a jump discontinuity at the origin and continuous potentials, Integral Equations Operator Theory 68 (2010), no. 1, 115–150. [Arn50] Nachman Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. [Ar74] Damir Z. Arov, Scattering theory with dissipation of energy, (Rus- sian) Dokl. Akad. Nauk SSSR 216 (1974), 713–716; English translation: Soviet Math. Dokl. 15 (1974), 848–854. [Ar79a] Damir Z. Arov, Optimal and stable passive systems, (Russian) Dokl. Akad Nauk SSSR 247 (1979 ), no. 2, 265–268; English translation: Soviet Math. Dokl. 20 (1979), no. 4, 676–680. [Ar79b] Damir Z. Arov, Passive linear steady-state dynamical systems, (Rus- sian) Sibirsk. Mat. Zh. 20 (1979), no. 2, 211–228; English translation: Siberian Math. J. 20 (1979), no. 2, 149–162. [Ar90] Damir Z. Arov, Regular J-inner matrix-functions and related continuation problems, in: Linear Operators in Function Spaces (Timi¸soara, 1988), Operator Theory: Advances Applications 43, Birkhäuser, Basel, 1990, pp. 63–87. [Ar93] Damir Z. Arov, The generalized bitangent Carathéodory–Nevanlinna- Pick problem and (j, J0)-inner matrix functions, (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993), no. 1, 3–32; English translation: Russian Acad. Sci. Izv. Math. 42 (1994), no. 1, 1–26. [ArD98] Damir Z. Arov and Harry Dym, On three Krein extension problems and some generalizations, Integral Equations Operator Theory 31 (1998), no. 1, 1–91. [ArD02] Damir Z. Arov and Harry Dym, J-inner matrix functions, interpolation and inverse problems for canonical systems, V. The inverse input scattering problem for Wiener class and rational p × q input scattering matrices, Integral Equations Operator Theory 43 (2002), no. 1, 68–129. Bibliography 387

[ArD05b] Damir Z. Arov and Harry Dym, Strongly regular J-inner matrix- valued functions and inverse problems for canonical systems, in: Recent Advances in Operator Theory and its Applications (M.A. Kaashoek, S. Seatzu and C. van der Mee, eds.), Operator Theory: Advances Applications 160, Birkhäuser, Basel, 2005, pp. 101–160. [ArD05c] Damir Z. Arov and Harry Dym, Direct and inverse problems for differential systems connected with Dirac systems and related factorization problems, Indiana Univ. Math. J. 54 (2005), no. 6, 1769–1815. [ArD07] Damir Z. Arov and Harry Dym, Direct and inverse asymptotic scattering problems for Dirac–Krein systems, (Russian) Funktsional. Anal. i Prilozhen. 41 (2007), no. 3, 17–33; English translation: Func- tional Anal. Appl. 41 (2007), no. 3, 181–195. [ArD07b] Damir Z. Arov and Harry Dym, Bitangential direct and inverse problems for systems of differential equations, in: Probability, geometry and integrable systems (M. Pinsky and B. Birnir, eds.) Math. Sci. Res. Inst. Publ. 55, Cambridge University Press, Cambridge, 2008, pp. 1–28. [ArD08] Damir Z. Arov and Harry Dym, J-contractive Matrix Valued Func- tions and Related Topics, Encyclopedia of Mathematics and its Ap- plications 116, Cambridge University Press, Cambridge, 2008. [ArD12] Damir Z. Arov and Harry Dym, Bitangential Direct and Inverse Prob- lems for Systems of Integral and Differential Equations, Encyclope- dia of Mathematics and its Applications 145, Cambridge University Press, Cambridge, 2012. [ArK81] Damir Z. Arov and Mark G. Krein, The problem of finding the min- imum entropy in indeterminate problems of continuation, (Russian) Funktsional. Anal. i Prilozhen. 15 (1981), no. 2, 61–64; English translation: Functional Anal. Appl. 1 15 (1981), no. 2, 123–126. [ArK83] Damir Z. Arov and Mark G. Krein, Calculation of entropy functionals and their minima in indeterminate continuation problems, (Russian) Acta Sci. Math. (Szeged) 45 (1983), no. 1–4, 33–50. [ArN96] Damir Z. Arov and Mark A. Nudelman, Passive linear stationary dynamical scattering systems with continuous time, Integral Equations Operator Theory 24 (1996), no. 1, 1–45. [ArN02] Damir Z. Arov and Mark A. Nudelman, Conditions for the similarity of all minimal passive realizations of a given transfer function (scattering and resistance matrices), (Russian) Mat. Sb. 193 (2002), no. 6, 3–24; English translation Sb. Math. 193 (2002), no. 5–6, 791–810. [AS17] DamirZ.ArovandOlofJ.Staffans,Linear Stationary In- put/State/Output and State/Signal Systems, preprint available at http://users.abo.fi/staffans 388 Bibliography

[ArR12] Damir Z. Arov and Natalia A. Rozhenko, Realizations of stationary stochastic processes: applications of passive system theory, Methods Functional Anal. Topology 18 (2012), no. 4, 305–331. [Art84] A.P. Artemenko, Hermitian positive functions and positive functionals, I, II, (Russian) Teor.FunktsiiFunktsional.Anal.iPrilhozen.41 (1984), 3–16; 42 (1984), 3–21. [BKSZ15] Joseph A. Ball, Mikael Kurula, Olof J. Staffans and Hans Zwart, De Branges–Rovnyak realizations of operator-valued Schur functions on the complex right half-plane, Complex Anal. Oper. Theory 9 (2015), no. 4, 723–792. [BG002] Laurent Baratchart, Andrea Gombani and Martine Olivi, Parameter determination for surface acoustic wave filters, in: Proceedings of the 39th IEEE Conference on Decision and Control (Sydney, Australia, 2000), 2002, pp. 2011–2016. [BEGO07] Laurent Baratchart, Per Enqvist, Andrea Gombani and Martine Olivi, Minimal symmetric Darlington synthesis, Math. Control Sig- nals Systems 19 (2007), no. 4, 283–311. [BGK82] Harm Bart, Israel Gohberg and Marinus A. Kaashoek, Convolution equations and linear systems, Integral Equations Operator Theory 5 (1982), no. 3, 283–340. [B12] Nicholas H. Bingham, Multivariate prediction and matrix Szeg˝othe- ory, Probab. Surv. 9 (2012), 325–339. [Br59] Louis de Branges, Some Hilbert spaces of entire functions, Proc. Amer. Math. Soc. 10 (1959), 840–846. [Br60] Louis de Branges, Some Hilbert spaces of entire functions, Trans. Amer. Math. Soc. 96 (1960), 259–295. [Br61a] Louis de Branges, Some Hilbert spaces of entire functions. II, Trans. Amer. Math. Soc. 99 (1961), 118–152. [Br61b] Louis de Branges, Some Hilbert spaces of entire functions. III, Trans. Amer. Math. Soc. 100 (1961), 73–115. [Br61c] Louis de Branges, Some Hilbert spaces of entire functions, Bull. Amer. Math. Soc. 67 (1961) 129–134. [Br62] Louis de Branges, Some Hilbert spaces of entire functions. IV, Trans. Amer. Math. Soc. 105 (1962), 43–83. [Br63] Louis de Branges, Some Hilbert spaces of analytic functions I, Trans. Amer. Math. Soc. 106 (1963), 445–468. [Br65] Louis de Branges, Some Hilbert spaces of analytic functions II, J. Math. Anal. Appl. 11 (1965), 44–72. [Br68a] Louis de Branges, Hilbert Spaces of Entire Functions. Prentice-Hall, Englewood Cliffs, 1968. Bibliography 389

[Br68b] Louis de Branges, The expansion theorem for Hilbert spaces of entire functions, in: Entire Functions and Related Parts of Analysis, Proc. Sympos. Pure Math. (La Jolla, Calif., 1966), Amer. Math. Soc., Providence, 1968, pp. 79–148. [Br83] Louis de Branges, The comparison theorem for Hilbert spaces of entire functions, Integral Equations Operator Theory 6 (1983), no. 5, 603– 646. [CP93] Raymond Cheng and Mohsen Pourahmadi, Baxter’s inequality and convergence of finite predictors of multivariate stochastic processes, Probab. Theory Related Fields 95 (1993), no. 1, 115–124. [Ch61] Joshua Chover, On normalized entropy and the extensions of a positive-definite function, J. Math. Mech. 10 (1961), 927–945. [CZ95] Ruth F. Curtain and Hans Zwart, An Introduction to Infinite- Dimensional Linear Systems Theory, Texts in Applied Mathematics 21, Springer-Verlag, New York, 1995. [Den02] Sergey A. Denisov, To the spectral theory of Krein systems, Integral Equations Operator Theory 42 (2002), no. 2, 166–173. [Den06] Sergey A. Denisov, Continuous analogs of polynomials orthogonal on the unit circle and Krein systems, IMRS Int. Math. Res. Surv. 2006, Art. ID 54517. [DD14] Vladimir Derkach and Harry Dym, A generalized Schur–Takagi interpolation problem, Integral Equations Operator Theory 80 (2014), no. 2, 165–227. [Do53] Joseph L. Doob, Stochastic Processes, John Wiley, New York, 1953. [Du70] Peter L. Duren, Theory of Hp Spaces, Pure and Applied Mathematics 38, Academic Press, New York 1970. [Dy70] Harry, Dym, An introduction to de Branges spaces of entire functions with applications to differential equations of the Sturm–Liouville type, Advances in Math. 5 (1970), 395–471. [Dy89a] Harry Dym, J-Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation, CBMS Regional Conference Series in Mathematics 71. Published for the Conference Board of the Math- ematical Sciences, Washington, DC by the American Mathematical Society, Providence, RI, 1989. [Dy89b] Harry Dym, On reproducing kernel spaces, J unitary matrix functions, interpolation and displacement rank, in: TheGohbergAn- niversary Collection, II, Operator Theory: Advances Applications 41, Birkhäuser, Basel, 1989, pp. 173–239. [Dy90] Harry Dym, On reproducing kernels and the continuous covariance extension problem, in: Analysis and Partial Differential Equations: A 390 Bibliography

Collection of Papers Dedicated to Mischa Cotlar (C. Sadosky, ed.), Marcel Dekker, New York, 1990, pp. 427–482. [Dy94a] Harry Dym, On the zeros of some continuous analogues of matrix orthogonal polynomials and a related extension problem with negative squares, Comm. Pure Appl. Math. 47 (1994), no. 2, 207–256. [Dy94b] Harry Dym, Shifts, realizations and interpolation, redux, in: Non- selfadjoint Operators and Related Topics (Beer Sheva, 1992), Oper- ator Theory: Advances Applications 73 Birkhäuser, Basel, 1994, pp. 182–243. [Dy98] Harry Dym, A basic interpolation problem, in: Holomorphic Spaces (Berkeley, CA, 1995), Math. Sci. Res. Inst. Publ. 33, Cambridge Uni- versity Press, Cambridge, 1998, pp. 381–423. [Dy00] Harry Dym, M.G. Krein’s contributions to prediction theory, in: Op- erator Theory and Related Topics, Vol. II (Odessa, 1997), Operator Theory: Advances Applications 118, Birkhä user, Basel, 2000, pp. 1–15. [Dy12] Harry Dym, Tutorial on a Nehari problem and related reproducing kernel spaces, in: Mathematical Methods in Systems, Optimization, and Control, Operator Theory: Advances Applications 222, Birkhä user, Basel, 2012, pp. 129–149. [Dy13] Harry Dym, Linear Algebra in Action, Second edition, Graduate Studies in Mathematics 78 American Mathematical Society, Provi- dence, RI, 2013. [DG80] Harry Dym and Israel Gohberg, On an extension problem, generalized Fourier analysis, and an entropy formula, Integral Equations Operator Theory 3 (1980), no. 2, 143–215. [DI84] Harry Dym and Andrei Iacob, Positive definite extensions, canonical equations and inverse problems, in: Topics in Operator Theory, Sys- tems and Networks (H.DymandI.Gohberg,eds.),OperatorTheory: Advances Applications 12, Birkhäuser, Basel, 1984, pp. 141–240. [DK16] Harry Dym and David P. Kimsey, CMV matrices, a matrix version of Baxter’s theorem, scattering and de Branges spaces. EMS Surv. Math. Sci. 3 (2016), no. 1, 1–105. [DMc70a] Harry Dym and Henry P. McKean, Application of de Branges spaces of integral functions to the prediction of stationary Gaussian processes, Illinois J. Math. 14 (1970), 299–343. [DMc70b] Harry Dym and Henry P. McKean, Extrapolation and interpolation of stationary Gaussian processes, Ann. Math. Statist. 41 (1970), 1817– 1844. [DMc72] Harry Dym and Henry P. McKean, it Fourier Series and Integrals, Probability and Mathematical Statistics 14, Academic Press, New York, 1972. Bibliography 391

[DMc76] Harry Dym and Henry P. McKean, Gaussian Processes, Function Theory, and the Inverse Spectral Problem, Probability and Mathe- matical Statistics 31, Academic Press, New York, 1976; reprinted by Dover, New York, 2008. [DS17] Harry Dym and Santanu Sarkar, Multiplication operators with defi- ciency indices (p, p) and sampling formulas in Hilbert spaces of vector valued entire functions, J. Functional Analysis 273 (2017), 3671–3718. [DT81] Harry Dym and Shlomo Ta’assan, An abstract version of a limit theorem of Szeg˝o, J. Functional Analysis 43 (1981), no. 3, 294–312. [F91] David F. Findley, Convergence of finite multistep predictors from incorrect models and its role in model selection, Dedicated to the memory of Professor Gottfried Köthe, Note Mat. 11 (1991), 145–155. [GRS64] Israel M. Gelfand, Dmitry Raikov and Georgi E. Shilov, Commutative Normed Rings, Chelsea Publ. Co., New York, 1964. [GM10] Mamikon S. Ginovyan and Levon V. Mikaelyan, Prediction error for continuous-time stationary processes with singular spectral densities, Acta Appl. Math. 110 (2010), no. 1, 327–351. [GKS98a] Israel Gohberg, Marinus A. Kaashoek and Alexander L. Sakhnovich, Canonical systems with rational spectral densities: explicit formulas and applications. Math. Nachr. 194 (1998), 93–125. [GKS98b] Israel Gohberg, Marinus A. Kaashoek and Alexander L. Sakhnovich, Pseudo-canonical systems with rational Weyl functions: explicit formulas and applications, J. Differential Equations 146 (1998), no. 2, 375–398. [GKS98c] Israel Gohberg, Marinus A. Kaashoek and Alexander L. Sakhnovich, Sturm–Liouville systems with rational Weyl functions: explicit formulas and applications. Dedicated to the memory of Mark Grigorievich Krein (1907–1989). Integral Equations Operator Theory 30 (1998), no. 3, 338–377. [GKS02] Israel Gohberg, Marinus A. Kaashoek and Alexander L. Sakhnovich, Scattering problems for a canonical system with a pseudo-exponential potential, Asymptot. Anal. 29 (2002), no. 1, 1–38. [GK85] Israel Gohberg and Israel Koltracht, Numerical solution of integral equations, fast algorithms and Krein–Sobolev equation, Numer. Math. 47 (1985), no. 2, 237–288. [GKW91] Israel Gohberg, Marinus A. Kaashoek and Hugo J. Woerdeman, A maximum entropy principle in the general framework of the band method, J. Functional Analysis 95 (1991), no. 2, 231–254. [GGK93] Israel Gohberg, Seymour Goldberg and Marinus A. Kaashoek, Classes of Linear Operators, Vol. II, Birkhäuser, Basel, 1993. 392 Bibliography

[GK58] Israel C. Gohberg and Mark G. Krein, Systems of integral equations on a half-line with kernels depending on the difference of arguments, (Russian) Uspehi Mat. Nauk. 13 (1958), no. 2, 3–72; English translation: Amer. Math. Soc. Transl. (2) 14 (1960), 217–287. [GoZ69] Israel Gohberg and Mircea K. Zambickii, On the theory of linear operators in spaces with two norms, (Russian), Ukrain. Mat. Z. 18 (1966), no. 1, 11–23; English translation: Amer. Math. Soc. Transl. (2) 85 (1969), 145–163. [GoGo97] Myroslav L. Gorbachuk and Valentina I. Gorbachuk, M.G. Krein’s Lectures on Entire Operators, Operator Theory: Advances Applica- tions 97, Birkhäuser Verlag, Basel, 1997. [GL06] Sergei V. Gusev and Andrey L. Likhtarnikov, Kalman–Popov– Yakubovich lemma and the S-procedure: A historical essay, (Russian) Avtomatika i Telemekhanika 1 (2006), no. 11, 77–121; English translation: Automation and Control 67 (2006), no. 11, 1768–1810. [He74] J. William Helton, Discrete time systems, operator models, and scattering theory, J. Functional Analysis 16 (1974), no. 1, 15–38. [HPh57] Einar Hille and Ralph S. Phillips, Functional Analysis and Semi- groups, Third printing of the revised edition of 1957, American Mathematical Society Colloquium Publications, Vol. XXXI. Amer- ican Mathematical Society, Providence, R. I., 1974. [JZ12] Birgit Jacob and Hans Zwart, Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces, Operator Theory: Advances Applica- tions 223, subseries Linear Operators and Linear Systems, Birkhä user, Basel, 2012. [JPT16] Palle Jorgensen, Steen Pedersen and Feng Tian, Extensions of Pos- itive Definite Functions. Applications and their Harmonic Analysis, Lecture Notes in Mathematics 2160, Springer, Cham, Switzerland, 2016. [KaKr74a] Israel S. Kac and Mark G. Krein, R-functions – analytic functions mapping the upper half-plane into itself, Amer. Math. Soc. Transl. (2) 103 (1974), 1–18. [KaKr74b] Israel S. Kac and Mark G. Krein, On the spectral functions of the string, Amer. Math. Soc. Transl. (2) 103 (1974), 19–102. [Kl74] Thomas Kailath, A view of three decades of linear filtering theory, IEEE Trans. Inf. Theory 20 (1974), no. 2, 146–181. [Ka63a] Rudolf E. Kalman, Contributions to the theory of optimal control, Bol. Soc. Math. Mexicana (2) 5 (1960), 102–119, [Ka63b] Rudolf E. Kalman, Lyapunov functions for the problem of Luréin automatic control, Proc. Nat. Acad. Sci. U.S.A. 49 (1963), 201–205. Bibliography 393

[Ka65] Rudolf E. Kalman, Irreducible realizations and the degree of a rational matrix, J. Soc. Indust. Appl. Math. 13 (1965), 520–544. [KFA69] Rudolf E. Kalman, Peter L. Falb and M. A. Arbib, Topics in Math- ematical System Theory, McGraw-Hill Book Co., New York, 1969. [Ko40a] Andrei N. Kolmogorov, Kurven in Hilbertschen Raum, die gegenüber einer einparametrigen Gruppe von Bewegungen invariant sind, (Ger- man) C.R. (Doklady) Acad. Sci. U.R.S.S. (N.S.) 26 (1940), 6–9. [Ko40b] Andrei N. Kolmogorov, Wienersche Spiralen und einige andere in- teressante Kurven im Hilbertschen Raum, (German) C.R. (Doklady) Acad. Sci. U.R.S.S. (N.S.) 26 (1940), 115–118. [Kr40] Mark G. Krein, Sur le problème du prolongement des fonctions her- mitiennes positives et continues, (French) C.R. (Dokl.) Acad. Sci. U.R.S.S. (N.S.) 26 (1940) 17–22. [Kr44a] Mark G. Krein, On the logarithm of an infinitely decomposable Hermite-positive function, C.R. (Doklady) Acad. Sci. U.R.S.S. (N.S.) 45 (1944), 91–94. [Kr44b] Mark G. Krein, On a remarkable class of Hermitian operators, C.R. (Doklady) Acad. Sci. U.R.S.S. (N.S.) 44 (1944), 175–179. [Kr44c] Mark G. Krein, On the problem of continuation of helical arcs in Hilbert space, C.R. (Doklady) Acad. Sci. U.R.S.S. (N.S.) 45 (1944), 139–142. [Kr47] Mark G. Krein, A contribution to the theory of entire functions of exponential type, (Russian) Izv. Akad. Nauk SSSR 11 (1947), 309– 326. [Kr54] Mark G. Krein, On a fundamental approximation problem in the theory of extrapolation and filtering of stationary random processes, (Russian) Dokl. Akad. Nauk SSSR 94 (1954), 13–16; English translation: Selected Transl. in Math. Stat. and Prob. 4 (1963), 127–132 [Kr55] Mark G. Krein, Continuous analogues of propositions on polynomials orthogonal on the unit circle, (Russian) Dokl. Akad. Nauk SSSR (N.S.) 105 (1955), 637–640. [Kr56] Mark G. Krein, On the theory of accelerants and S-matrices of canonical differential systems, (Russian) Dokl. Akad. Nauk SSSR (N.S.) 111 (1956), 1167–1170. [KrL85] Mark G. Krein and Heinz Langer, On some continuation problems which are closely related to the theory of operators in spaces Πκ.IV. Continuous analogues of orthogonal polynomials on the unit circle with respect to an indefinite weight and related continuation problems for some classes of functions, J. Operator Theory 13 (1985), no. 2, 299–417. 394 Bibliography

[KrL14] Mark G. Krein and Heinz Langer, Continuation of Hermitian positive definite functions and related questions, Integral Equations Operator Theory 78 (2014), no. 1, 1–69. [KrMA68] Mark G Krein and Felix E. Melik-Adamyan, On the theory of S- matrices of canonical differential systems with summable potential, (Russian) Dokl. Akad. Nauk Armyan SSR 46 (1968), no. 4, 150–155. [KrMA70] Mark G Krein and Felix E. Melik-Adamyan, Certain applications of theorems on the factorization of a unitary matrix, (Russian) Funkt- sional. Anal. i Prilozhen. 4 (1970), no. 4, 73–75; English translation: Functional Anal. Appl. 4 (1970), no. 4. 327–329. [KrMA84] Mark G Krein and Felix E. Melik-Adamyan, Hankel integral operators operators and related problems of continuation. I and II (Russian) Izv. Akad. Nauk Armyan SSR, Ser. Mat. 19 (1984), no. 4, 311–332; 19 (1984), no. 5, 339–360; English translation: Soviet J. Contemporary Math. Anal. 19 (1984), no. 4, 47–68; 19 (1984), no. 5, 1–22. [KrMA86] Mark G. Krein and Felix E. Melik-Adamyan, Matrix-continuous analogues of the Schur and the Carathéodory–Toeplitz problem, (Rus- sian) Izv. Akad. Nauk Armyan. SSR, Ser. Mat. 21 (1986), no. 2, 107–141; English translation: Soviet J. Contemporary Math. Anal. 21 (1986), no. 2, 1–37. [LMc64] Norman Levinson and Henry P. McKean, Weighted trigonometrical approximation on R1 with application to the germ field of a stationary Gaussian noise, Acta Math. 112 (1964), 99–143. [LP15] Anders Lindquist and Giorgio Picci, Linear Stochastic Systems. A Geometric Approach to Modeling, Estimation and Identification,Se- ries in Contemporary Mathematics 1, Springer, Heidelberg, 2015. [MSW06] Jarmo Malinen, Olof J. Staffans and George Weiss, When is a linear system conservative? Quart. Appl. Math. 64 (2006), no. 1, 61–91. [Ma60] Pesi Masani, The prediction theory of multivariate stochastic processes, III., Unbounded spectral densities, Acta Math. 104 (1960) 141– 162. [Ma62] Pesi Masani, Shift invariant spaces and prediction theory, Acta Math. 107 (1962), 275–290. [PW34] Raymond E.A.C. Paley and Norbert Wiener, Fourier Transforms in the Complex Domain, American Mathematical Society Colloquium Publications XIX, American Math. Soc., Providence, R.I. 1934. [LP67] Peter Lax and Ralph Phillips, Scattering Theory, Pure and Applied Mathematics 26, Academic Press, New York, 1967. [NF70] Béla Sz.-Nagy and Ciprian Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970. Bibliography 395

[PK82] Vladimir V. Peller and Sergei V. Khrushchev, Hankel operators, best approximations and stationary Gaussian processes, (Russian) Uspekhi Mat. Nauk 37 (1982), no. 1, (223) 53–124; English translation: Rus- sian Math. Surveys 37 (1982), no. 1, 61–144. [Pe03] Vladimir V. Peller, Hankel Operators and their Applications, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. [P72] Loren D. Pitt, On problems of trigonometrical approximation from the theory of stationary Gaussian processes, J. Multivariate Anal. 2 (1972), 145–161. [Po73] Vasilie M. Popov, Hyperstability of Control Systems, Die Grundlehren der mathematischen Wissenschaften 204, Springer-Verlag, Berlin, 1973. [Ra96] Anders Rantzer, On the Kalman–Yakubovich–Popov lemma, Systems Control Lett. 28 (1996), no. 1, 7–10. [RoRo71] Marvin Rosenblum and James Rovnyak, Factorization of operator valued entire functions, Indiana Univ. Math. J. 20 (1971), no. 2, 157– 173. [RoRo94] Marvin Rosenblum and James Rovnyak, Topics in Hardy Classes and Univalent Functions, Birkhäuser, Basel, 1994. [RoRo97] Marvin Rosenblum and James Rovnyak, Hardy Classes and Operator Theory, Corrected reprint of the 1985 original, Dover Publications, Inc., Mineola, NY, 1997. [Rov68] James Rovnyak, Characterization of spaces H(M), 1968; available at: http://www.people.virginia.edu/ jlr5m/Papers/HM.pdf. [Roz67] Yuri A. Rozanov, Stationary Random Processes (translated from the 1963 Russian edition by A. Feinstein), Holden-Day, Inc., San Fran- cisco, 1967. [Ry66] Alexander M. Rybalko, On the theory of continual analogues of orthogonal polynomials, (Russian) Teor. Funkcii. 29 Funkcional. Anal. i Prilozen. 3 (1966), 42–60. [vNS41] John von Neumann, and Isaac J. Schoenberg, Fourier integrals and metric geometry, Trans. Amer. Math. Soc. 50 (1941), 226–251. [Re62] Raymond Redheffer, On the relation of transmission-line theory to scattering and transfer, J. Math. and Phys. 41 (1962), 1–41. [Sa98a] Lev A. Sakhnovich, Spectral analysis of a class of canonical differential systems, (Russian) Algebra i Analiz 10 (1998), no. 1, 187–201; English translation: St. Petersburg Math. J. 10 (1998), no. 1, 147–158. [Sa98b] Lev A. Sakhnovich, On a class of canonical systems on half-axis, In- tegral Equations Operator Theory 31 (1998), no. 1, 92–112. 396 Bibliography

[Sa99] Lev A. Sakhnovich, Spectral Theory of Canonical Differential systems. Method of Operator Identities, Operator Theory: Advances Applica- tions 107, Birkhäuser Verlag, Basel, 1999. [Sa00a] Lev A. Sakhnovich, On the spectral theory of a class of canonical differential systems, (Russian) Funktsional. Anal. i Prilozhen. 34 (2000), no. 2, 50–62; English translations: Functional Analysis. Appl. 34 (2000), no. 2, 119–128. [Sa00b] Lev A. Sakhnovich, On the spectral theory of the generalized differential system of M.G. Krein, (Russian) Ukrain. Mat. Zh. 52 (2000), no. 5, 717–721; English translation: Ukrain. Math. J. 52 (2001), no. 5, 821–826. [S58] Frank Smithies, Integral Equations, Cambridge Tracts in Mathemat- ics and Mathematical Physics 49, Cambridge University Press, New York, 1958. [St02] Olof J. Staffans, Passive and conservative continuous-time impedance and scattering systems, I. Well-posed systems, Math. Control Signals Systems 15 (2002), no. 4, 291–315. [St05] Olof J. Staffans, Well-Posed Linear Systems, Encyclopedia of Math- ematics and its Applications 103, Cambridge University Press, Cam- bridge, 2005. [T05] Alexander Teplyaev, A note on the theorems of M.G. Krein and L.A. Sakhnovich on continuous analogs of orthogonal polynomials on the circle, J. Functional Analysis 226 (2005), no. 2, 257–280. [TrV97] Sergei Treil and Alexander Volberg, Wavelets and the angle between past and future, J. Functional Analysis 143 (1997), no. 2, 269–308. [WM57] Norbert Wiener and Pesi Masani, The prediction theory of multivariate stochastic processes, I., The regularity condition, Acta Math. 98 (1957), 111–150. [WM58] Norbert Wiener and Pesi Masani, The prediction theory of multivariate stochastic processes, II., The linear predictor, Acta Math. 99 (1958), 93–137. [Wi71] Jan C. Willems, Least squares stationary optimal control and the algebraic Riccati equation, IEEE Trans. Automatic Control AC-16 (1971), 321–351. [Wi72a] Jan C. Willems, Dissipative dynamical systems, I. General theory, Arch. Rational Mech. Anal. 45 (1972), 321–351. [Wi72b] Jan C. Willems, Dissipative dynamical systems, II. Linear systems with quadratic supply rates, Arch. Rational Mech. Anal. 45 (1972), 352–393. Bibliography 397

[Za41] Victor N. Zasuhin, On the theory of multidimensional stationary random processes, C.R. (Doklady) Acad. Sci. U.R.S.S. (N.S.) 33 (1941), 435–437. [ZDG95] Kemin Zhou with John C. Doyle and Keith Glover, Robust and Op- timal Control, Prentice Hall, 1995. Index

p×p Sample references to frequently used Qa ,57 symbols are listed here; see also R0,93 Sections 1.2 and 2.1. R0-invariant spaces of entire vvf’s, 246 (B(E))sf , 72, 154 subspaces, 104, 226 ∗-optimal, 321 Rα, 7, 32, 67, 310 ∗-optimal minimal passive Rα-invariant, 79 impedance system, 309, 332 R•, 331 + AT , 165 R♦, 353, 354, 356 Aβ(λ), 90 TU [X], 38 Am(λ), 48 TU [x], 38 [∞] Aa , 137 TV , 261 [0] U2,94 Aa , 154 Vα,8 C0-group, 302 − C0-semigroups, 289 X ,24 • E+(λ), 5 X , 302 t X◦, 302 E+(λ), 362 Xo , 302 E−(λ), 5 Σ+ o t X , 302 E−(λ), 362 Σ− T Fc(λ), 338 Z (Δ), 157 +/− Fo(λ), 338 Z (Δ), 273, 278 p −/+ H2 ,68 Z (Δ), 273 + J-inner mvf’s, 37 ZT (Δ), 162 A ZT −(Δ), 157, 162 Kω (λ), 92 c Z[0,a](Δ), 190 Kω(λ), 338 o Z[a,b](Δ), 1 Kω(λ), 338 E [a,b] Kω (λ), 6 Ze (Δ), 2 Nγ,δ, 195 [0,d, 181 ν P•, 330, 332 C+, 289 P♦, 330, 332, 350, 354, 356, 363, Γ, 337 365, 367, 368 Γ♦, 354–356 (1) Pc, 338 Λ (t), 358 (2) Po, 338 Λ (t), 358 p×p Q∞ ,57 Ωsf , 153

© Springer International Publishing AG, part of Springer Nature 2018 399 D. Z. Arov, H. Dym, Multivariate Prediction, de Branges Spaces, and Related Extension and Inverse Problems, Operator Theory: Advances and Applications 266, https://doi.org/10.1007/978-3-319-70262-9 400 INDEX

M ΠZ+/− , 276 c, 338, 339 M ΠZ−/+ , 276 o, 338, 339 p×p [ ] N ΠZ a,b (Δ+), 172 + , 112 Real, 50 def Nα =, 67 Real ∩ Symm, 269 p×p Nout , 112 ∩ ◦ Real Symp, 268 P(f ; a), 17, 114 p×q Real ∩X , 255 Pn×n, 330 ∗ Σ , 292, 331 p×p P∞ , 16, 55 d × Σ (V; Nγ,δ), 196 Pp p sf a , 17, 55 d p×p Σsf (dM), 183 S ,6 p×p Symm, 51 Sin ,6 Symm ∩X p×p, 255 T ,20 Symp, 261, 264 U(J), 37 fW , 207 U H (J), 41 A(h◦; a), 18 U ◦(J), 40 p×p A∞ ,58 US(J), 40 Ap×p U a , 18, 58 S(Jp), 44, 99 B(E), 6 UAR(Jp), 228 C(A), 39 Uconst(J), 40 C0(m), 48 UrR(J), 40 H E∩U (Jp), 14 UrsR(J), 41 ◦ E∩U (Jp), 14 UrsR(Jp), 41, 100 E∩UrR(Jp), 14 V(t), 195 E∩UH Y rR(Jp), 14 , 200 b F, 183 ω(λ), 69 F+, 304 D+, 304 F−, 304 D−, 304 F2, 183, 190 E(λ), 5, 6 ◦ G(g ; a), 15, 112 EA(λ), 88 p×p E G∞ (0), 14, 52, 54 a(λ), 172 Gp×p v a (0), 14, 52, 62 (t), 195, 362, 363, 368–370 H(A), 92, 93, 241 f,gB(E),6 H(b), 69 C, 329 O∗ H∗(b), 69 , 329 A˚p×p Hω,84 ∞ , 18, 58, 59 A˚p×p I(jp), 6, 74, 89 a ,58 ◦ H ˚ I (jp), 7 AEP(h ; a), 18 ◦ ˚ ◦ I (jp), 7, 74 AEP(h ; a), 116 ρ(A), 290 IR(jp), 7, 105, 234 IH ρ(Δ), 277 R (jp), 7, 105 I◦ Σ=(−iA,B,C,D; X, Cp), 291 R(jp), 106 τ(f), 33 IsR(jp), 235 I◦ ± def sR(jp), 245 τf =, 33 INDEX 401

τ±(f), 33 inverse spectral problem, 242 τf ,33 Blaschke–Potapov factor, 346 τL, 226 Blaschke–Potapov products, 351 ∗ C algebras, 224 Bochner, S., 55 × A , 346 bounded passive ap(E), 74 impedance system, 324 apII(A), 40 bounded real lemma, 332 b(λ), 355 bounded system, 291, 319 bc(λ), 339, 340 bo(λ), 339, 340 canonical integral equations, 11 br(λ), 355 canonical integral system, 182 cf (λ), 17, 111 Carath´eodory class, 28 cg(λ), 15, 111 Carath´eodory extension problem, ch(λ), 111 15, 111 −itA× e , 359 Cartwright condition, 34, 98, 157 × eitA , 358, 359 Cayley transform, 39 b ◦ kω(λ), 69 CEP (c ; a), 15, 111 ◦ C(c ; a), 111 chain, 165, 181, 243 ◦ C(c ; b3,b4), 225 maximal nondecreasing ∼, 250 p×p C0 , 45, 56 maximal strictly increasing ∼, p×p C0 (m), 377 250 D (TU ), 38 of subspaces, 226 p×p St ,50 characteristic matrix function, 325 I(Δ), 247 ci, 117, 226, 239 I R(Δ), 247 classical trajectory, 291 completely indeterminate (ci), 117 accelerant, 18, 58, 63 compression, 305–309 extension problem, 18, 116 computations, 375, 377, 380 adjoint systems, 292, 331 conservative impedance system, 293, AEP (h◦; a), 18 298, 300, 308, 309 Alpay, D., 384 conservative scattering system, 293 Aronszajn, N., 66, 70 continuous, 13 Arov, D.Z., 384 continuous chains, 181, 182 associated pairs, 7, 74, 229, 247 controllable, 333 of the second kind, 14, 40 controllable pair, 329 band extension, 216 controllable systems, 299 Baratchart, 288 correlation, 272 Baxter’s inequality, 207, 224 correlation function, 21, 23 Bellman, R., 384 Beurling–Lax theorem, 31 de Branges bitangential identity, 93, 94, 340 generalizations, 225 matrix, 69, 317 helical extension problem, 238 space, 6, 70, 317, 318 402 INDEX

◦ de Branges, L., viii, 2, 26, 84, 101, GHEP(g ; F, Fr), 238 ◦ 102, 180 GHEP(g ; F, {0}), 238 ◦ de Branges–Rovnyak space, 324 GHEP(g ; {0}, Fr), 238 description of H(A), 140 Gohberg, I., 384 determinate, 117 Gombani, A., 288 dilation, 305 Dirac system, 195 Hamiltonian, 187, 381 Dirac–Krein system, 195 Hankel operator, 273, 337, 354 direct spectral problem, 183 Hardy class, 28 DK system, 195, 198, 205 Hardy space, 68 Dym, H., 25 Hardy space facts, 30 helical extension problem, 15, 112 entire, 33 helical function, 15, 52 entire de Branges matrix, 6 Helton, J.W., 384 entropy, 215 HEP (g◦; a), 15, 112 even transform, 268 Hille–Phillips, 324 evolution semigroup, 291 Hille–Yosida theorem, 290 exact type, 33 holomorphic in a neighborhood of exponential type, 33 ∞, 327 extremal solutions, 330 homogeneous J-inner mvf’s, 41 homogeneous de Branges matrices, 7 factorization, 22, 35, 90, 133, 278, homogeneous regular de Brangers 348 matrices, 7 factorization problems, 172 Feller–Krein string equation, vii, 188, 223 Iacob, A., ix Findley, D.F., 224 indeterminate, 117 formulas for inner, 6, 28 computing a de Branges matrix, inverse spectral problem, 183, 198 172 inverse spectral problem for DK resolvent matrices, 140, 240 systems, 198 Fourier transform, 29 involution, 259 generalized ∼, 10, 20, 183, 190, isometric inclusion, 103 195, 245, 304 full-rank process, 22 Kac, I.S., viii functional model, 24, 304 Kalman, R.E., 327, 328, 384 Kalman–Bucy ﬁlters, 288 generalized Kalman–Popov–Yakubovich backward-shift, 32 inequality, 298, 330 Carath´eodory extension problem, Khrushchev, S.V., 288 226 Kolmogorov, A.N., 23, 25, 52, 288 trajectory, 291 Kolmogorov–Krein theorem, 55 generator of a C0-semigroup, 289, Krein system, 195, 201, 210, 361, 290 362 INDEX 403

Krein, M.G., vii, viii, 14, 15, 25, 34, right denominator, 274, 316, 340, 35, 41, 52, 56, 59, 63, 151, 180, 355 222, 223, 288 system, 305 Krein–Langer, 152 monodromy matrix, 182 Krein–Sobolev equation, 199, 223 Moore–Penrose inverse, 330 KYP lemma, 384 Muckenhoupt (A2) condition, 9, 278

Langer, H., viii, 63, 151 Nevanlinna class, 28 Lax, P., 304 nondecreasing, 13 Lax–Phillips, 324, 384 nondecreasing chains, 181, 182 Lax–Phillips scheme, 288, 304, 305 nondegenerate, 82 left denominator, 273 normalized, 13 left tangential helical extension chains, 181, 182 problem, 238 de Branges matrices, 7 left-continuity, 244 J-inner mvf’s, 40 Levinson, N., 180 Lindquist, A., 288 observable, 333 linear fractional transformation, 13, pair, 329 38, 137 systems, 299 Livsic–Brodskii node, 325 odd transform, 268 Lyapunov equation, 335–338 Olivi, M., 288 optimal minimal passive impedance Masani, P., 25, 288 system, 309, 320 mass function, 182 orthogonal projection, 172 matrix, entire de Branges, 6 outer, 28, 302 matrizant, 11, 182 outer factors, 350 of the DK system, 205 maximum entropy, 178 Paley–Wiener, 30, 34 McKean, H.P., 26, 180 theorem, 31 McMillan degree, 328 partial isometry, 94 meromorphic de Branges matrices, passive 284 impedance system, 294 meromorphic pseudocontinuation, scattering system, 294 34 passive impedance system, 292, 293, minimal, 306 295, 296, 299, 300, 309, 319, 321 ∗-optimal passive impedance passive scattering system, 293–295, system, 318 299 left denominator, 273, 317, 340, Peller, V.V., 288 355 PEP(f ◦; a), 17, 114, 115 optimal passive impedance perfect, 13, 42, 91, 94, 96, 104–106, system, 317, 332 108, 186, 246 passive impedance system, 308, Phillips, R., 304 332 Picci, G., 288 positive deﬁnite solution, 350, 353 Pitt, L.D., vii, 26, 180 404 INDEX

Plancherel formula, 29 Sakhnovich, L.A., 224 Poisson formula, 30 scattering matrix, 304 Popov, V.M., 384 scattering operator, 304 positive extension problem, 17, 114 scattering suboperator, 304 positive real lemma, 330 Schoenberg, I.J., 151 Potapov–Ginzburg transform, 37 Schur class, 6, 28 potential, 362 sci, 117, 226, 239 prediction, 20, 221 Siegert, A.J.F., 384 error, 377 signature matrix, 4 problem, 24 similar systems, 292 projection simple conservative impedance error, 206 system, 301, 302, 304 formulas, 176 simple systems, 299 onto Z[0,a](Δ), 190, 206 singular J-inner mvf’s, 40 singular value decomposition, 343 singular values, 355 rational, 332 Smirnov class, 28 rational spectral density, 344, 361, Smirnov maximum principle, 36 369 spectral density, 12, 22, 23, 45, 72, real, 255, 272 221 real mvf’s, 50 spectral function, 12, 21, 23, 45, 51, realizations, 289, 292, 327 52, 72, 153, 183, 195 regular de Branges matrices, 7 Staﬀans, O.J., 324 regular process, 22 standard inner product, 4 reproducing kernel, 65 Stieltjes Hilbert spaces, 65 class, 50 resolvent identity, 93 inversion formula, 45, 51 resolvent matrix, 16, 112, 118, 126, strictly completely indeterminate 130, 133, 136, 154, 164, 172, 189, (sci), 117 205, 226, 239, 243, 266, 380 strictly contractive, 103 Riccati equation, 330, 331, 347, 350, string equation, 188 353 strongly regular entire de Branges Riccati inequality, 330, 331 matrix, 235 right denominator, 274 summable spectral densities, 285 right strongly regular J-inner mvf’s, symbol, 273 41 symmetric mvf’s, 51 right strongly regular Jp-inner symplectic, 261, 266, 269 mvf’s, 41 system, controllable, 307, 309, right tangential helical extension 327–329 problem, 238 system, minimal, 307, 309, 327, 328 right-continuity, 245 system, observable, 307, 309, right-regular J-inner mvf’s, 40 327–329 Rosenblum–Rovnyak theorem, 35 system, similar, 328 Rozanov, Y.A., 25 system, simple, 309 INDEX 405

Sz. Nagy–Foias, 384 Szeg˝o condition, 118, 119, 287 Szeg˝o matrix polynomials, 222 transfer function, 291 Treil–Volberg condition, 9, 278 unitarily similar system, 292, 308

Volterra operator, 93 von Neumann, J., 151 weakly stationary, 21 Wiener algabra, 9 Wiener class, 27 Wiener, N., 25, 30, 288 Willems, J.C., 331

Zasuhin, V.N., 25