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Proceedings - Mathematical Sciences

, Volume 119, Issue 1, pp 45–52 | Cite as

On the theorem of M Golomb

  • Vugar E. Ismailov
Article

Abstract

Let X 1, …, X n be compact spaces and X = X 1 × … × X n . Consider the approximation of a function ƒ ∈ C(X) by sums g 1(x 1)+…+g n (x n ), where g i C(X i ), i = 1, …, n. In [8], Golomb obtained a formula for the error of this approximation in terms of measures constructed on special points of X, called ‘projection cycles’. However, his proof had a gap, which was pointed out by Marshall and O’Farrell [15]. But the question if the formula was correct, remained open. The purpose of the paper is to prove that Golomb’s formula holds in a stronger form.

Keywords

Approximation error duality relation projection cycle lightning bolt orthogonal measure extreme measure 

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References

  1. [1]
    Arnold V I, On functions of three variables, Dokl. Akad. Nauk SSSR 114 (1957) 679–681; English transl., Am. Math. Soc. Transl. 28 (1963) 51–54MathSciNetGoogle Scholar
  2. [2]
    Babaev M-B A, Estimates and ways for determining the exact value of the best approximation of functions of several variables by superpositions of functions of a smaller number of variables (Russian), Special questions in the theory of functions (Russian), Izdat. “Elm”, Baku (1977) pp. 3–23Google Scholar
  3. [3]
    Braess D and Pinkus A, Interpolation by ridge functions, J. Approx. Theory 73 (1993) 218–236zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Diliberto S P and Straus E G, On the approximation of a function of several variables by the sum of functions of fewer variables, Pacific J. Math. 1 (1951) 195–210zbMATHMathSciNetGoogle Scholar
  5. [5]
    Dyn N, Light W A and Cheney E W, Interpolation by piecewise-linear radial basis functions, J. Approx. Theory. 59 (1989) 202–223zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Garkavi A L, Medvedev V A and Khavinson S Ya, On the existence of a best uniform approximation of functions of two variables by sums of the type ϕ(x) + ψ(y), Sibirskii Mat. Zh. 36 (1995) 819–827; English transl. in Siberian Math. J. 36 (1995) 707–713MathSciNetGoogle Scholar
  7. [7]
    Golitschek M V and Light W A, Approximation by solutions of the planar wave equation, Siam J. Numer. Anal. 29 (1992) 816–830zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Golomb M, Approximation by functions of fewer variables, On numerical approximation, Proceedings of a Symposium, Madison 1959, edited by R E Langer (The University of Wisconsin Press) pp. 275–327Google Scholar
  9. [9]
    Havinson S Ja, A Chebyshev theorem for the approximation of a function of two variables by sums of the type ϕ(x) + ψ(y), Izv. Acad. Nauk. SSSR Ser. Mat. 33 (1969) 650–666; English transl. Math. USSR Izv. 3 (1969) 617–632MathSciNetGoogle Scholar
  10. [10]
    Ismailov V E, Methods for computing the least deviation from the sums of functions of one variable, Sibirski Mat. Zh. 47 (2006) 1076–1082; English transl. in Siberian Math. J. 47 (2006) 883–888zbMATHMathSciNetGoogle Scholar
  11. [11]
    Khavinson S Ya, Best approximation by linear superpositions (approximate nomography), translated from the Russian manuscript by D Khavinson, Translations of Mathematical Monographs 159 (Providence, RI: American Mathematical Society) (1997) 175 pp.zbMATHGoogle Scholar
  12. [12]
    Klopotowski A, Nadkarni M G and Bhaskara Rao K P S, When is f(x 1, x 2, …, x n) = u 1(x 1) + u 2(x 2) + … + u n(x n)? Proc. Indian Acad. Sci. (Math. Sci.) 113 (2003) 77–86zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Klopotowski A and Nadkarni M G, Shift invariant measures and simple spectrum, Colloq. Math. 84/85 (2000) 385–394MathSciNetGoogle Scholar
  14. [14]
    Light W A and Cheney E W, On the approximation of a bivariate function by the sum of univariate functions, J. Approx. Theory 29 (1980) 305–323zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Marshall D E and O’Farrell A G, Approximation by a sum of two algebras, The lightning bolt principle, J. Funct. Anal. 52 (1983) 353–368zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Marshall D E and O’Farrell A G, Uniform approximation by real functions, Fund. Math. 104 (1979) 203–211zbMATHMathSciNetGoogle Scholar
  17. [17]
    Navada K G, Some remarks on good sets, Proc. Indian Acad. Sci. (Math. Sci.) 114(4) (2003) 389–397CrossRefMathSciNetGoogle Scholar
  18. [18]
    Ofman Ju P, Best approximation of functions of two variables by functions of the form ϕ(x) + ψ(y), Izv. Akad. Nauk. SSSR Ser. Mat. 25 (1961) 239–252; English transl. Am. Math. Soc. Transl. 44 (1965) 12–28zbMATHMathSciNetGoogle Scholar

Copyright information

© Indian Academy of Sciences 2009

Authors and Affiliations

  1. 1.Mathematics and Mechanics InstituteAzerbaijan National Academy of SciencesBakuAzerbaijan

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