Abstract
The aim of the present paper is to give a short historical survey on the extension problem for positive definite functions.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
N.I. Akhiezer, The classical moment problem. Edinburgh: Oliver and Boyd 1965.
E.J. Akutowicz, On extrapolating a positive definite function from a finite interval. Math. Scand. 7 (1959), 157–169.
A.P. Artjomenko, Hermitian positive functions and positive functionals. (Russian), Dissertation, Odessa State University (1941). Published in Teor. Funkcii, Funkcional. Anal. i Priložen. 41, (1983) 1–16; 42 (1984), 1–21.
C. Berg, J.P.R. Christensen, P. Ressel, Harmonic Analysis on Semigroups. Theory of Positive Definite and Related Functions. Berlin-Heidelberg-New York-Tokyo: Springer-Verlag 1984.
T.M. Bisgaard, Z. Sasvári, Characteristic Functions and Moment Sequences. Positive Definiteness in Probability. Huntigton, NY: Nova Science Publishers 2000.
S. Bochner, Vorlesungen über Fouriersche Integrale. Leipzig: Akademische Verlagsgesellschaft 1932.
S. Bochner, Spektralzerlegung linearer Scharen unitärer Operatoren. Sitzungsber. Preuss. Akad. Wiss. phys.-math. (1933), 371–376.
S. Bochner, Monotone Funktionen, Stieltjessche Integrale und harmonische Analyse. Math. Ann. 108 (1933), 378–410.
A. Calderón, R. Pepinsky, On the phases of Fourier coefficients for positive real periodic functions. Computing Methods and the Phase Problem in X-Ray Crystal Analysis, published by The X-Ray Crystal Analysis Laboratory, Department of Physics, The Pennsylvania State College (1952), 339–348.
C. Carathéodory, Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen. Math. Ann. 64 (1907), 95–115.
C. Carathéodory, Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen. Rend. Circ. Mat. Palermo, 32 (1911), 193–217.
J. Chover, On normalized entropy and the extensions of a positive-definite function. J. Math. Mech. 10 (1961), 927–945.
A. Devinatz, On the extensions of positive definite functions. Acta Math. 102 (1959), 109–134.
T. Gneiting, Z. Sasvári, The Characterization Problem for Isotropic Covariance Functions Math. Geology 31(1) (1999), 105–111.
T. Gneiting, Z. Sasvári, M. Schlather, Analogies and correspondences between variograms and covariance functions. Adv. Appl. Probab. 33 (2001), 617–630.
I. Gohberg, M.A. Kaashoek, H.J. Woerdemann, A Maximum Entropy Principle in the General Framework of the Band Method. J. Funct. Anal. 95 (1991), 231–254.
G. Herglotz, Über Potenzreihen mit positivem, reellen Teil im Einheitskreis. Leipziger Berichte, math.-phys. Kl. 63 (1911), 501–511.
D. Hilbert, Über die Darstellung definiter Formen als Summe von Formenquadraten. Math. Ann. 32 (1988), 342–350.
I.S. Iohvidov, Unitary and selfadjoint operators in spaces with an indefinite metric. (Russian) Dissertation, Odessa (1950).
I.S. Iohvidov, On the theory of indefinite Toeplitz forms. (Russian) Dokl. Akad. Nauk SSSR 101(2) (1955), 213–216.
I.S. Iohvidov, M.G. Krein, Spectral theory of operators in spaces with an indefinite metric II. (Russian) Trudy Moskov. Mat. Obšč. 8 (1959), 413–496. English translation: Amer. Math. Soc. Translations 2(3), (1963), 283–373.
M. Kaltenbäck, H. Woracek, On extensions of hermitian functions with a finite number of negative squares. J. Operator Theory 40 (1998), 147–183.
A. Khintchin, Korrelationstheorie der stationären stochastischen Prozesse. Math. Ann. 109 (1934), 604–615.
M.G. Krein, Sur le problème du prolongement des fonctions hermitiennes positives et continues. Dokl. Akad. Nauk SSSR 26 (1940), 17–22.
M.G. Krein, On the representation of functions by Fourier-Stieltjes integrals. (Russian) Učenije Zapiski Kuibishevskogo Gosud. Pedag. i Učitelskogo Inst. 7 (1943), 123–148.
M.G. Krein, Screw lines in infinite-dimensional Lobachevski space and the Lorentz transformation. (Russian) Usp. Mat. Nauk 3(3) (1948), 158–160.
M.G. Krein, On the integral representation of a continuous Hermitian-indefinite function with a finite number of negative squares. (Russian) Dokl. Akad. Nauk SSSR, 125(1) (1959), 31–34.
M.G. Krein, On measurable Hermitian-positive functions. (Russian) Mat. Zametki 23 (1978), 79–89. English translation: Math. Notes 23 (1978), 45–50.
M.G. Krein, Solution of the inverse Sturm-Liouville problem. Dokl. Akad. Nauk. SSSR 76:1 (1951), 21–24.
M.G. Krein, On the transition function of the one-dimensional second order boundary value problem. Dokl. Akad. Nauk. SSSR 88:3 (1953), 405–408.
M.G. Krein, On the determination of the potential of a particle by its S-function. Dokl. Akad. Nauk. SSSR 105:3 (1955), 433–436.
M.G. Krein, H. Langer, On the indefinite power moment problem. Soviet Math. Dokl. 17 (1976), 90–93.
M.G. Krein, H. Langer, Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Π κ zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen. Math. Nachr. 77 (1977), 187–236.
M.G. Krein, H. Langer, Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Π κ zusammenhängen. II. Verallgemeinerte Resolventen, u-Resolventen und ganze Operatoren. J. Funct. Anal. 30 (1978), 390–447.
M.G. Krein, H. Langer, On some extension problems which are closely related with the theory of hermitian operators in a space Π κ . III. Indefinite analogues of the Hamburger and Stieltjes problems. Part (I). Beiträge zur Analysis 14 (1979), 25–40.
M.G. Krein, H. Langer, On some continuation problems which are closely related to the theory of operators in spaces Π κ . IV: Continuous analogues of orthogonal polynomials on the unit circle with respect to an indefinite weight and related coninuation problems for some classes of functions. J. Operator Theory 13 (1985), 299–417.
M.G. Krein, H. Langer, Continuation of Hermitian positive definite functions and related questions. (Manuscript)
H.J. Landau, Maximum Entropy and the Moment Problem, Bull. Amer. Math. Soc. 16(1), (1987), 47–77.
H. Langer, M. Langer, Z. Sasvári, Continuations of Hermitian Indefinite Functions and Corresponding Canonical Systems: An Example. Methods. of Funct. Anal. and Topology, 10(1) 2004, 39–53.
P. Lévy, Théorie de l’addition des variables aléatoires. Paris: Gauthier-Villars, 1937.
M. Mathias, Über positive Fourier-Integrale. Math. Zeitschrift 16 (1923), 103–125.
S. Mitra, T. Gneiting, Z. Sasvári, Polynomial covariance functions on intervals. Bernoulli 9(2) 2003, 229–241.
T.S. Motzkin, The arithmetic-geometric inequality. Published in: Inequalities (Ed. by O. Shisha). New York-London: Academic Press 1967.
G. Pólya, Remarks on characteristic functions. Proc. First Berkeley Conf. on Math. Stat. and Prob. Berkeley: Univ. of Calif. Press (1949), 115–123.
D.A. Raikov, On the decomposition of Gauss and Poisson laws. (Russian) Izv. Akad. Nauk. SSSR, ser. math. 2 (1937), 91–124.
D.A. Raikov, Sur les fonctions positivement définies. Dokl. Akad. Nauk SSSR, 26 (1940), 860–865.
F. Riesz, Über Sätze von Stone und Bochner. Acta Sci. Math. 6 (1932–1934), 184–198.
W. Rudin, The extension problem for positive-definite functions. Illinois J. Math. 7 (1963), 532–539.
W. Rudin, An extension theorem for positive-definite functions. Duke Math. J. 37 (1970), 49–53.
Z. Sasvári, Positive Definite and Definitizable Functions. Berlin: Akademie Verlag 1994.
Z. Sasvári, The extension problem for measurable positive definite functions. Math. Zeitschrift 191 (1986), 475–478.
J. Shohat, J. Tamarkin, The problem of moments. Math. Surveys No. 1, Providence, R. I.: Amer. Math. Soc. 1943.
I.J. Schoenberg, Metric spaces and competely monotone functions. Ann. of Math. 39(4) (1938), 811–841.
M.H. Stone, Linear transformations in Hilbert space. III. Operational methods and group theory. Proc. Nat. Acad. Sci. U.S.A. 16 (1930), 172–175.
M.H. Stone, On one-parameter unitary groups in Hilbert space. Ann. of Math. 33(2) (1932) 643–648.
O. Toeplitz, Über die Fourier’sche Entwickelung positiver Funktionen. Rend. Circ. Mat. Palermo 32 (1911), 191–192.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to Heinz Langer on the occasion of his retirement
Rights and permissions
Copyright information
© 2005 Birkhäuser Verlag Basel/Switzerland
About this chapter
Cite this chapter
Sasvári, Z. (2005). The Extension Problem for Positive Definite Functions. A Short Historical Survey. In: Langer, M., Luger, A., Woracek, H. (eds) Operator Theory and Indefinite Inner Product Spaces. Operator Theory: Advances and Applications, vol 163. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7516-7_16
Download citation
DOI: https://doi.org/10.1007/3-7643-7516-7_16
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7515-7
Online ISBN: 978-3-7643-7516-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)