Advertisement

When isf(x1, x2, ..., xn) =u 1(x1) +u 2(x2) + ... +u n(xn)?

  • A. Kłopotowski
  • M G. Nadkarni
  • K. P. S. Bhaskara Rao
Article

Abstract

We discuss subsetsS of ℝn such that every real valued functionf onS is of the formf(x1, x2, ..., xn) =u 1(x1) +u 2(x2) +...+u n(xn), and the related concepts and situations in analysis.

Keywords

Good set sequentially good set linked component sequentially good measure simplicial measure 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Abhyankar S S, Hilbert’s 13th problem (preprint)Google Scholar
  2. [2]
    Arnold VI and Shimura G, Superposition of algebraic functions, mathematical developements arising from Hilbert problems,Proc. Symposia in Pure Mathematics, Am. Math. Soc.XXVIII(1) (1976) 45–46Google Scholar
  3. [3]
    Beneš V and Štěpán J, The support of extremal probability measures with given marginals, in: Mathematical Statistics and Probability Theory (Eds) M L Puriet al (D. Reidel Publishing Company) (1987) vol. A, pp. 33–41Google Scholar
  4. [4]
    Beneš V and Štěpán J, Extremal solutions in the marginal problem, in: Advances in probability distributions with given marginals (Dordrecht: Kluwer Academic Publishers) (1991) pp. 189–207Google Scholar
  5. [5]
    Cowsik R C, Klopotowski A and Nadkarni M G, When isf(x, y) =u (x) +v (y) ?Proc. Indian Acad. Sci. (Math. Sci.) 109 (1999) 57–64MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    Douglas R G, On extremal measures and subspace density,Michigan Math. J. 11 (1964) 243–246MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Hestir K and Williams S, Supports of doubly stochastic measures,Bernoulli 1(3) (1995) 217–243MATHMathSciNetGoogle Scholar
  8. [8]
    Klopotowski A and Nadkarni M G, On transformations with simple Lebesgue spectrum,Proc. Indian Acad. Sci. (Math. Sci.) 109 (1999) 47–55MATHMathSciNetGoogle Scholar
  9. [9]
    Klopotowski A and Nadkarni M G, Shift invariant measure and simple spectrum,Colloquium Mathematicum 84/85 (2000) 385–394MathSciNetGoogle Scholar
  10. [10]
    Klopotowski A and Nadkarni M G, Sets with doubleton sections, good sets and ergodic theory,Fund. Math. 173 (2002) 133–158MATHMathSciNetGoogle Scholar
  11. [11]
    Kahane J-P, Sur le Théorème de Superposition de Kolmogorov,J. Approximation Theory 13 (1975) 229–234MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Kolmogorov A N, On the representation of continuous functions of several variables as superposition of continuous function of one variable and addition,Dokl. Acad. Nauk SSSR 114 (1957) 679–681; Am. Math. Soc. Transl.28 (1963) 55–59MathSciNetGoogle Scholar
  13. [13]
    Lindenstrauss J, A remark on doubly stochastic measures,Am. Math. Monthly 72 (1965) 379–382MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Lorentz G G, The 13th problem of Hilbert, mathematical developements arising from Hilbert problems,Proc. Symposia in Pure Mathematics, Am. Math. Soc.XXVIII(2) (1976) 419–430MathSciNetGoogle Scholar
  15. [15]
    Marshall D E and O’Farrell A G, Uniform approximation by real functions,Fund. Math. 104(3) (1979) 203–211MATHMathSciNetGoogle Scholar
  16. [16]
    Marshall D E and O’Farrell A G, Approximation by sums of two algebras: The lightening bolt principle,J. Funct. Anal. 52 (1983) 353–368MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Mehta R D and Vasavada M H, Algebra direct sum decomposition of Cr(X),Proc. Am. Math. Soc. 98 (1986) 71–74MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Mehta R D and Vasavada M H, Algebra direct sum decomposition ofC r(X), II,Proc. Am. Math. Soc. 100 (1987) 123–126MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    Sproston J P and Strauss D, Sums subalgebras ofC(X), J. Lond. Math. Soc. 45(2) (1992) 265–278MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    Štěpán J, Simplicial measures and sets of uniqueness in the marginal problem, in: Statistics and decision (München: R. Oldenbourg Verlag) (1993) vol. 11, pp. 289–299Google Scholar
  21. [21]
    Sternfeld Y, Uniform separation of points and measures and representation of sums of algebras,Israel J. Math. 55 (1986) 350–363MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Indian Academy of Sciences 2003

Authors and Affiliations

  • A. Kłopotowski
    • 1
  • M G. Nadkarni
    • 2
  • K. P. S. Bhaskara Rao
    • 3
  1. 1.Institut GaliléeUniversité Paris XIIIVilletaneuse CedexFrance
  2. 2.Department of MathematicsUniversity of MumbaiKalina, MumbaiIndia
  3. 3.Stat-Math UnitIndian Statistical InstituteBangloreIndia

Personalised recommendations