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On Polya frequency functions

I. The totally positive functions and their Laplace transforms

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Bibliography

  1. M. Aissen, A. Edrei, I. J. Schoenberg and A. Whitney,On the generating functions of totally positive sequences, Proceedings of the National Academy of Sciences,37 (1951), 303–307.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. L. Cauchy, Exercices de Mathématiques, IIe année, Paris, 1827.

  3. G. Doetsch,Theorie und Anwendung der Laplace-Transformation, Berlin, 1937.

  4. J. Hadamard,Essai sur l’étude des fonctions données par leur développement de Taylor, Journal de Mathématiques, 4e série,8 (1892).

  5. H. Hamburger,Bemerkungen zu einer Fragestellung des Herrn Pólya, Mathematishe Zeitschrift,7 (1920), 302–322.

    Article  MathSciNet  Google Scholar 

  6. I. I. Hirschman and D. V. Widder,The inversion of a general class of convolution transforms, Transactions of the American Mathematical Society,66 (1949), 135–201.

    Article  MATH  MathSciNet  Google Scholar 

  7. I. I. Hirschman and D. V. Widder,Generalized inversion formulas for convolution transforms, Duke Mathematical Journal,15 (1948), 659–696.

    Article  MathSciNet  Google Scholar 

  8. E. Laguerre,Sur les fonctions du genre zéro et du genre un, Oeuvres de Laguerre I, Paris, 1898, 174–177.

  9. Th. Motzkin,Beitraege zur Theorie der linearen Ungleichungen, Jerusalem 1936.

  10. G. Pólya und I. Schur,Ueber zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, Journal für Mathematik,144 (1914), 89–113.

    Google Scholar 

  11. G. Pólya,Ueber Annaeherung durch Polynome mit lauter reelen Wurzeln, Rendiconti di Palermo,36 (1913), 1–17.

    Google Scholar 

  12. G. Pólya,Algebraische Untersuchungen über ganze Funktionen vom Geschlechte Null und Eins, Journal für Mathematik,145 (1915), 224–249.

    Google Scholar 

  13. G. Pólya und G. Szegö,Aufgaben und Lebrsaetze aus der Analysis, II, Berlin, 1925.

  14. I. J. Schoenberg,Ueber variationsvermindernde lineare Transformationen, Mathematische Zeitschrift,32 (1930), 321–328.

    Article  MathSciNet  Google Scholar 

  15. I. J. Schoenberg,On totally positive functions, Laplace integrals and entire functions of the Laguerre-Polya-Schur type, Proceedings of the National Academy of Sciences,33 (1947), 11–17.

    Article  MATH  MathSciNet  Google Scholar 

  16. I. J. Schoenberg et Anne Whitney,Sur la positivité des déterminants de translation des fonctions de fréquence de Polya avec une application au problème d’interpolation par des fonctions “spline”, Comptes Rendus des séances de l’Académie des Sciences (1949).

  17. I. J. Schoenberg,On Polya frequency functions II: Variation-diminishing integral operators of the convolution type, Acta Scientiarum Mathematicarum, Szeged,12 (1950), 97–106.

    MathSciNet  Google Scholar 

  18. I. J. Schoenberg and Anne Whitney,A theorem on polygons in ndimensions with applications to variation-diminishing transformations, Compositio Mathematica,9 (1951), 141–160.

    MATH  MathSciNet  Google Scholar 

  19. W. Sierpinski,Sus les fonctions convexes mesurables, Fundamenta Mathematicae,1 (1920), 125–129.

    Google Scholar 

  20. D. V. Widder,The Laplace transform, Princeton, 1946.

  21. D. V. Widder,Weierstrass transforms of positive functions, Proceedings of the National Academy of Sciences,37 (1951), 315–317.

    Article  MATH  MathSciNet  Google Scholar 

  22. A. Wintner,On a class of Fourier transforms, American Journal of Mathematics,58 (1936), 45–90.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Philadelphia, Pa., U.S.A.

    I. J. Schoenberg

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  1. I. J. Schoenberg
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To the memory of Dolli Schoenberg née Landau

The present work was partly sponsored by the Office of Naval Research at the University of Pennsylvania. A brief account of the main results appeared in 1947; see [15] in the bibliography at the end of the paper.

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Schoenberg, I.J. On Polya frequency functions. J. Anal. Math. 1, 331–374 (1951). https://doi.org/10.1007/BF02790092

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  • DOI: https://doi.org/10.1007/BF02790092

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