Abstract
We study the intimate relation of the continuation problem for hermitian positive definite functions with direct and inverse spectral problems for canonical systems and strings and with the extrapolation problem for second order stochastic processes.
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References
De Branges L.: Some Hilbert spaces of entire functions. Trans. Am. Math. Soc. 96, 259–295 (1960)
De Branges L.: Some Hilbert spaces of entire functions, II. Trans. Am. Math. Soc. 99, 118–152 (1961)
De Branges L.: Some Hilbert spaces of entire functions, III. Trans. Am. Math. Soc. 100, 73–115 (1961)
De Branges L.: Some Hilbert spaces of entire functions, IV. Trans. Am. Math. Soc. 105, 43–83 (1962)
De Branges L.: Hilbert Spaces of Entire Functions. Prentice-Hall, Inc., Englewood Cliffs (1968)
Dym, H., McKean, H.P.: Gaussian processes, function theory, and the inverse spectral problem. Probability and Mathematical Statistics, vol. 31. Academic Press [Harcourt Brace Jovanovich, Publishers], New York (1976)
Gesztesy F., Simon B.: A new approach to inverse spectral theory. II. General real potentials and the connection to the spectral measure. Ann. Math. (2) 152, 593–643 (2000)
Gnedenko B.W.: Lehrbuch der Wahrscheinlichkeitsrechnung. Akademie-Verlag, Berlin (1962)
Gohberg I.C., Kreǐn M.G.: Theory and applications of Volterra operators in Hilbert space. Translations of Mathematical Monographs, vol. 24. American Mathematical Society, Providence (1970)
Gorbachuk M.L., Gorbachuk V.I.: M.G. Krein’s Lectures on Entire Operators. Operator Theory: Advances and Applications, Vol. 97. Birkhäuser Verlag, Basel (1997)
Großmann M., Langer H.: Über indexerhaltende Erweiterungen eines hermiteschen Operators im Pontrjaginraum (German). Math. Nachr. 64, 289–317 (1974)
Hassi S., de Snoo H.S.V., Winkler H.: Boundary-value problems for two-dimensional canonical systems. Integr. Equ. Oper. Theory 36, 445–479 (2000)
Hayman W.K.: Meromorphic Functions. Oxford University Press, Oxford (1964)
Iohvidov I.S., Krein M.G., Langer H.: Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric. Mathematical Research, vol. 9. Akademie-Verlag, Berlin (1982)
Kac, I.S.: Linear relations, generated by a canonical differential equation on an interval with a regular endpoint, and expansibility in eigenfunctions. (Russian), Deposited in Ukr NIINTI, No. 1453 (1984) (VINITI Deponirovannye Naučnye Raboty, No. 1 (195), b.o. 720, 1985)
Kac, I.S.: Expansibility in eigenfunctions of a canonical differential equation on an interval with singular endpoints and associated linear relations. (Russian), Deposited in Ukr NIINTI, No. 2111 (1986) (VINITI Deponirovannye Naučnye Raboty, No. 12 (282), b.o. 1536, 1986)
Kac I.S., Kreǐn M.G.: R-functions. Analytic functions mapping the upper half plane into itself. Am. Math. Soc. Transl. 103(2), 1–18 (1974)
Kac I.S., Kreǐn M.G.: On the spectral functions of the string. Am. Math. Soc. Transl. 103(2), 19–101 (1974)
Koosis P.: The Logarithmic Integral, II. Cambridge University Press, Cambridge (1992)
Kreǐn M.G.: Sur le problème du prolongement des fonctions hermitiennes positives et continues. (French) C. R. (Doklady) Acad. Sci. URSS (N.S.) 26, 17–22 (1940)
Kreǐn M.G.: On the representation of functions by Fourier-Stieltjes integrals (Russian). Učenije Zapiski Kuibyševskovo Ped. Instituta 7, 123–147 (1943)
Kreǐn M.G.: On hermitian operators with deficiency indices equal to one. I. Dokl. Akad. Nauk SSSR 43, 323–326 (1944)
Kreǐn M.G.: On hermitian operators with deficiency indices equal to one. II. Dokl. Akad. Nauk SSSR 44, 191–195 (1944)
Kreǐn M.G.: On the logarithm of an infinitely decomposible Hermite-positive function. C. R. (Doklady) Acad. Sci. URSS (N.S.) 45, 91–94 (1944)
Kreǐn M.G.: On the problem of continuation of helical arcs in Hilbert space. C. R. (Doklady) Acad. Sci. URSS (N.S.) 45, 139–142 (1944)
Kreǐn M.: On a problem of extrapolation of A. N. Kolmogoroff. C. R. (Doklady) Acad. Sci. URSS (N. S.) 46, 306–309 (1945)
Kreǐn M.G.: A contribution to the theory of entire functions of exponential type. (Russian). Bull. Acad. Sci. URSS. Sr. Math. [Izvestia Akad. Nauk SSSR] 11, 309–326 (1947)
Kreǐn M.G.: On hermitian operators with directed functionals (Ukrainian). Akad. Nauk Ukrain. RSR. Zbirnik Prac Inst. Mat. 10, 83–106 (1948)
Kreǐn M.G.: The fundamental propositions of the theory of representations of hermitian operators with deficiency index (m,m) (Russian). Ukrain. Mat. Żurnal1, 3–66 (1949)
Kreǐn M.G.: On the theory of entire matrix functions of exponential type (Russian). Ukrain. Mat. Żurnal3, 164–173 (1951)
Kreǐn M.G.: Determination of the density of a nonhomogeneous symmetric cord by its frequency spectrum (Russian). Doklady Akad. Nauk SSSR (N.S.) 76, 345–348 (1951)
Kreǐn M.G.: On some new problems of the theory of oscillations of Sturmian systems (Russian). Akad. Nauk SSSR. Prikl. Mat. Meh. 16, 555–568 (1952)
Kreǐn M.G.: On inverse problems for a nonhomogeneous cord (Russian). Doklady Akad. Nauk SSSR (N.S.) 82, 669–672 (1952)
Kreǐn M.G.: On a generalization of investigations of Stieltjes (Russian). Doklady Akad. Nauk SSSR (N.S.) 87, 881–884 (1952)
Kreǐn M.G.: On the indeterminate case of the Sturm–Liouville boundary problem in the interval (0,∞) (Russian). Izvestiya Akad. Nauk SSSR. Ser. Mat. 16, 293–324 (1952)
Kreǐn M.G.: On the transfer function of a one-dimensional boundary problem of the second order (Russian). Doklady Akad. Nauk SSSR (N.S.) 88, 405–408 (1953)
Kreǐn M.G.: On a basic approximation problem of the theory of extrapolation and filtration of stationary random processes (Russian). Doklady Akad. Nauk SSSR (N.S.) 94, 13–16 (1954)
Kreǐn M.G.: Čebyšev-Markov inequalities in the theory of the spectral functions of a string (Russian). Mat. Issled. 5, 77–101 (1970)
Kreǐn M.G.: Measurable positive Hermitian functions (Russian). Mat. Zametki 23, 79–89 (1978)
Kreǐn, M.G., Langer, H.: The defect subspaces and generalized resolvents of a Hermitian operator in the space Πκ (Russian). Funkcional. Anal. i Priložen 5, no. 2, 59–71; no. 3, 54–69 (1971)
Kreǐn M.G., Langer H.: Über die Q-Funktion eines π-hermiteschen Operators im Raume Πκ (German). Acta Sci. Math. (Szeged) 34, 191–230 (1973)
Kreǐn M.G., Langer H.: Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Πκ zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen (German). Math. Nachr. 77, 187–236 (1977)
Kreǐn M.G., Langer H.: Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Πκ zusammenhängen. II. Verallgemeinerte Resolventen, u-Resolventen und ganze Operatoren (German). J. Funct. Anal. 30, 390–447 (1978)
Kreǐn, M.G., Langer, H.: On some extension problems which are closely connected with the theory of Hermitian operators in a space Πκ. III. Indefinite analogues of the Hamburger and Stieltjes moment problems. Part I. Beiträge Anal. 14, 25–40 (1979)
Kreǐn, M.G., Langer, H.: On some extension problems which are closely connected with the theory of Hermitian operators in a space Πκ. III. Indefinite analogues of the Hamburger and Stieltjes moment problems. Part II: Beiträge Anal. 15, 27–45 (1980)
Kreǐn M.G., Langer H.: 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. Oper. Theory 13, 299–417 (1985)
Kreǐn M.G., Nudelman A.A.: The Markov Moment Problem and Extremal Problems. Ideas and Problems of P. L. Čebyšev and A. A. Markov and Their Further Development. Translations of Mathematical Monographs, vol. 50. American Mathematical Society, Providence (1977)
Langer H.: Über die Methode der richtenden Funktionale von M. G. Krein. Acta Math. Acad. Sci. Hungar. 21, 207–224 (1970)
Langer H.: Spektralfunktionen einer Klasse von Differentialoperatoren zweiter Ordnung mit nichtlinearem Eigenwertparameter (German). Ann. Acad. Sci. Fenn. Ser. A I Math. 2, 269–301 (1976)
Langer H., Langer M., Sasvári Z.: Continuations of hermitian indefinite functions and corresponding canonical systems: an example. Methods Funct. Anal. Topology 10, 39–53 (2004)
Langer H., Textorius B.: L-resolvent matrices of symmetric linear relations with equal defect numbers; applications to canonical differential relations. Integr. Equ. Oper. Theory 5, 208–243 (1982)
Langer, M., Woracek, H.: A local inverse spectral theorem for Hamiltonian systems. Inverse Problems 27(5), 055002 (2011)
Levin B.Ja.: Distribution of Zeros of Entire Functions. Translations of Mathematical Monographs, vol. 5. American Mathematical Society, Providence, R.I. (1980)
Lukacs E.: Characteristic Functions. Hafner Publishing Co., New York (1970)
von Neumann J., Schoenberg I.J.: Fourier integrals and metric geometry. Trans. Am. Math. Soc. 50, 226–251 (1941)
Potapov V.P.: The multiplicative structure of J-contractive matrix functions. Am. Math. Soc. Transl. (2) 15, 131–243 (1960)
Ravaska T.: Differential geometry of C ∞ helices in Hilbert space. J. Math. Anal. Appl. 75, 385–401 (1980)
Sasvári Z.: Positive Definite and Definitizable Functions. Akademie Verlag, Berlin (1994)
Simon B.: A new approach to inverse spectral theory. I. Fundamental formalism. Ann. Math. (2) 150, 1029–1057 (1999)
Šmuljan Ju.L.: A certain class of holomorphic operator-valued functions (Russian). Mat. Zametki 5, 351–359 (1969)
Winkler H.: The inverse spectral problem for canonical systems. Integr. Equ. Oper. Theory 22, 360–374 (1995)
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Mark G. Krein: 3.4.1907–17.10.1989.
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Krein, M.G., Langer, H. Continuation of Hermitian Positive Definite Functions and Related Questions. Integr. Equ. Oper. Theory 78, 1–69 (2014). https://doi.org/10.1007/s00020-013-2091-z
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DOI: https://doi.org/10.1007/s00020-013-2091-z