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Über ein Turánsches Problem für ℓ-1 radiale, positiv definite Funktionen

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Abstract

Turán’s problem is to determine the greatest possible value of the integral ∫ℝ df(x)dx/ f (0) for positive definite functions f (x), x ∈ ℝd, supported in a given convex centrally symmetric body D ⊂ ℝd. In this note we consider the 2-dimensional Turán problem for positive definite functions of the form f(x) = φ (∥x∥1), x ∈ ℝ2, with φ supported in [0,π].

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Literatur

  1. Abramowitz, M. and I. Stegun, Handbook of Mathematical Functions, Dover Publ. Inc., New York, 1970.

  2. Andreev, N.N., Extremal problems for periodic functions with small supports, Vestn. Mosk. Univ., Ser. I, No. 1 (1997) 29-32 (in Russian); English transi.: Mosc. Univ. Math. Bull. 52, No.1 (1997) 29–32.

    MATH  Google Scholar 

  3. Arestov, V.V. and E.E. Berdysheva, The Turán problem for a class of polytopes, East J. Approx., 8, (2002), 381–388.

    MathSciNet  MATH  Google Scholar 

  4. Arestov, V.V., E.E. Berdysheva and H. Berens, On pointwise Turán’s problem for positive definite functions, East J. Approx., 9, (2003), 31–42.

    MathSciNet  MATH  Google Scholar 

  5. Berens, H. & W. zu Castell, Hypergeometric functions as a tool for summability of the Fourier integral, Result. Math. 34 (1998), 69–84.

    Article  MATH  Google Scholar 

  6. Berens, H. and Y. Xu, ℓ-1 summability of multiple Fourier integrals and positivity, Math. Proc. Camb. Phil. Soc. 122 (1997), 149–172.

    Article  MathSciNet  MATH  Google Scholar 

  7. Boas, Jr., R.P., and M. Kac, Inequalities for Fourier transforms of positive functions, Duke Math. J. 12 (1945), 189–206. Errata: Boas, Jr., R.P., More inequalities for Fourier transforms, Duke Math. J. 15 (1948), 105-109.

    Article  MathSciNet  MATH  Google Scholar 

  8. Boas, Jr., R.P., Fourier series with positive coefficients, J. Math. Anal Appl. 17 (1967), 463–483.

    Article  MathSciNet  MATH  Google Scholar 

  9. Ehm, W., T. Gneiting and D. Richards, Convolution roots of radial positive definite functions with compact support, Trans. Amer. Math. Soc. 356 (2004), 4655–4685.

    Article  MathSciNet  MATH  Google Scholar 

  10. Gneiting, T., Kuttner’s problem and a Pólya type criterion for characteristic functions, Proc. Amer. Math. Soc. 128 (2000), 1721–1728.

    Article  MathSciNet  MATH  Google Scholar 

  11. Goldfarb, D. and M.J. Todd, Linear Programming in: G.L. Nemhauser et al., Eds., Handbook in OR & MS, Vol. 1, Chap. II, Elsevier Sci. Publ. B.V., 1989.

  12. Gorbachev, D.V. and A.S. Manoshina, Turán extremal problem for periodic function with small support and its applications, Mat. Zametki 76(5) (2004), 688-700 (in Russian); English transi.: Math. Notes 76 (5–6) (2004), 640-652.

    Google Scholar 

  13. Kolountzakis, M.N. and S.G. Révész, Turán’s extremal problem for positive definite functions on groups, Preprint of the Alfréd Rényi Mathematical Institute, Hung. Acad. Sci., 5/2003.

  14. Stechkin, S.B., An extremal problem for trigonometric series with nonnegative coefficients, Acta Math. Acad. Sci. Hung. 23 (1972), 289–291 (in Russian).

    Article  MATH  Google Scholar 

  15. Wendland, H., Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math. 4 (1995), 389–396.

    Article  MathSciNet  MATH  Google Scholar 

  16. Williamson, R.E., Multiply monotone functions and their Laplace transforms, Duke Math. J. 23 (1956), 189–207.

    Article  MathSciNet  MATH  Google Scholar 

  17. Zastavnyi V.P., On positive definiteness of some functions, J. Multivariate Anal. 73 (2000), 55–81.

    Article  MathSciNet  MATH  Google Scholar 

  18. Zygmund, A., Trigonometric Series, Vol. I & II, 2nd printing, Cambridge University Press, London 1959.

    Google Scholar 

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Authors and Affiliations

  1. Institut für Angewandte Mathematik und Statistik, Universität Hohenheim, 70593, Stuttgart, Deutschland

    Elena E. Berdysheva

  2. Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstr. 1 1/2, 91054, Erlangen, Deutschland

    Hubert Berens

Authors
  1. Elena E. Berdysheva
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  2. Hubert Berens
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Correspondence to Elena E. Berdysheva.

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Berdysheva, E.E., Berens, H. Über ein Turánsches Problem für ℓ-1 radiale, positiv definite Funktionen. Results. Math. 47, 17–32 (2005). https://doi.org/10.1007/BF03323009

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  • DOI: https://doi.org/10.1007/BF03323009

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