Abstract
We detail the history and present complete proofs of the spectral properties of totally positive kernels and matrices.
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References
Ando, T., Totally positive matrices, Lin. Alg. and Appl. 90 (1987), 165– 219.
Anselone, P. M., and J. W. Lee, Spectral properties of integral operators with nonnegative kernels, Lin. Alg. and Appl. 9 (1974), 67–87.
Birkhoff, G. D., In Memoriam: The Mathematical work of Oliver Dimon Kellogg, Bull. Amer. Math. Soc. 39 (1933), 171–177.
Buslaev, A. P., Extremal problems of approximation theory, and nonlinear oscillations, Dokl. Akad. Nauk SSSR 305 (1989), 1289–1294;
Buslaev, A. P., English transl. in Soviet Math. Dokl. 39 (1989), 379–384.
Buslaev, A. P., A variational description of the spectra of totally positive matrices, and extremal problems of approximation theory, Mat. Zametki 47 (1990), 39–46;
Buslaev, A. P., English transl. in Math. Notes 47 (1990), 26–31.
Gantmacher, F., Sur les noyaux de Kellogg non symétriques, Comptes Rendus (Doklady) de l’Academie des Sciences de l’URSS 1 (10) (1936), 3–5.
Gantmacher, F. R., The Theory of Matrices, Gostekhizdat, MoscowLeningrad, 1953; English transl. as Matrix Theory, Chelsea, New York, 2 vols., 1959.
Gantmacher, F. R., Obituary, in Uspekhi Mat. Nauk 20 (1965), 149–158;
Gantmacher, F. R., English transl. as Russian Math. Surveys, 20 (1965), 143–151.
Gantmacher, F. R., and M. G. Krein, Sur une classe spéciale de déterminants ayant le rapport aux noyaux de Kellog, Recueil Mat. (Mat. Sbornik) 42 (1935), 501–508.
Gantmacher, F. R., and M. G. Krein, Sur les matrices oscillatoires, C. R. Acad. Sci. (Paris) 201 (1935), 577–579.
Gantmacher, F. R., and M. G. Krein, Sur les matrices complètement non négatives et oscillatoires, Composito Math. 4 (1937), 445–476.
Gantmacher, F. R., and M. G. Krein, Oscillation Matrices and Small Oscillations of Mechanical Systems (Russian), Gostekhizdat, MoscowLeningrad, 1941.
Gantmacher, F. R., and M. G. Krein, Ostsillyatsionye Matritsy i Yadra i Malye Kolebaniya Mekhanicheskikh Sistem, Gosudarstvenoe Izdatel’stvo, Moskva-Leningrad, 1950; German transl. as Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer Systeme, Akademie Verlag, Berlin, 1960; English transl. as Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, USAEC, 1961.
Gohberg, I., Mathematical Tales, in The Gohberg Anniversary Collection, Eds. H. Dym, S. Goldberg, M. A. Kaashoek, P. Lancaster, pp. 17–56, Operator Theory: Advances and Applications, Vol. 40, Birkhäuser Verlag, Basel, 1989.
Gohberg, I., Mark Grigorievich Krein 1907–1989, Notices Amer. Math. Soc. 37 (1990), 284–285.
Goursat, E., A Course in Mathematical Analysis: Integral Equations, Calculus of Variations, Vol. III, Part 2, Dover Publ. Inc., New York, 1964.
Jentzsch, R., Über Integralgleichungen mit positivem Kern, J. Reine und Angewandte Mathematik (Crelle), 141 (1912), 235–244.
Karlin, S., The existence of eigenvalues for integral operators, Trans. Amer. Math. Soc. 113 (1964), 1–17.
Karlin, S., Oscillation properties of eigenvectors of strictly totally positive matrices, J. D’Analyse Math. 14 (1965), 247–266.
Karlin, S., Some extremal problems for eigenvalues of certain matrix and integral operators, Adv. in Math. 9, (1972), 93–136.
Karlin, S., Some extremal problems for eigenvalues of certain matrix and integral operators, Adv. in Math. 9, (1972), 93–136.
Karlin, S., and A. Pinkus, Oscillation Properties of Generalized Characteristic Polynomials for Totally Positive and Positive Definite Matrices, Lin. Alg. and Appl. 8 (1974), 281–312.
Kellogg, O. D., The oscillation of functions of an orthogonal set, Amer. J. Math. 38 (1916), 1–5.
Kellogg, O. D., Orthogonal function sets arising from integral equations, Amer. J. Math. 40 (1918), 145–154.
Kellogg, O. D., Interpolation properties of orthogonal sets of solutions of differential equations, Amer. J. Math. 40 (1918), 225–234.
Lee, J. W., and A. Pinkus, Spectral Properties and Oscillation Theorems for Mixed Boundary-Value Problems of Sturm-Liouville Type, J. Differential Equations 27 (1978), 190–213.
Perron, O., Zur Theorie der Matrices, Math. Annalen 64 (1907), 248–263.
Pinkus, A., Some Extremal Problems for Strictly Totally Positive Matrices, Lin. Alg. and Appl. 64 (1985), 141–156.
Pinkus, A., n-Widths of Sobolev Spaces in L p , Constr. Approx. 1 (1985), 15–62.
Schoenberg, I. J., Über variationsvermindernde lineare Transformationen, Math. Z. 32 (1930), 321–328.
Schoenberg, I. J., I. J. Schoenberg: Selected Papers, Ed. C. de Boor, 2 Volumes, Birkhäuser, Basel, 1988.
Schur, I., Zur Theorie der linearen homogenen Integralgleichungen, Math. Ann. 67 (1909), 306–339.
Smithies, F., Integral Equations, Cambridge University Press, Cambridge, 1970.
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Pinkus, A. (1996). Spectral Properties of Totally Positive Kernels and Matrices. In: Gasca, M., Micchelli, C.A. (eds) Total Positivity and Its Applications. Mathematics and Its Applications, vol 359. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8674-0_23
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DOI: https://doi.org/10.1007/978-94-015-8674-0_23
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