Proceedings - Mathematical Sciences

, Volume 127, Issue 2, pp 361–374 | Cite as

Diliberto–Straus algorithm for the uniform approximation by a sum of two algebras



In 1951, Diliberto and Straus [5] proposed a levelling algorithm for the uniform approximation of a bivariate function, defined on a rectangle with sides parallel to the coordinate axes, by sums of univariate functions. In the current paper, we consider the problem of approximation of a continuous function defined on a compact Hausdorff space by a sum of two closed algebras containing constants. Under reasonable assumptions, we show the convergence of the Diliberto–Straus algorithm. For the approximation by sums of univariate functions, it follows that Diliberto–Straus’s original result holds for a large class of compact convex sets.


Uniform approximation levelling algorithm proximity map bolt. 

Mathematics Subject Classifications:

41A30 41A65 46B28 65D15. 



The authors are grateful to the referee for numerous comments and suggestions that improved the original manuscript.


  1. [1]
    Arnold V I, On functions of three variables (Russian), Dokl. Akad. Nauk SSSR 114 (1957) 679–681 ; English transl: Amer. Math. Soc. Trans. 28 (1963) 51–54MathSciNetGoogle Scholar
  2. [2]
    Aumann G, Uber approximative nomographie, II, (German), Bayer. Akad. Wiss. Math.-Nat. Kl. S.-B. (1959) 103–109Google Scholar
  3. [3]
    Cowsik R C, Klopotowski A and Nadkarni M G, When is f(x,y) = u(x) + v(y)? Proc. Indian Acad. Sci. (Math. Sci.) 109 (1999) 57–64MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Deutsch F, The alternating method of von Neumann, Multivariate approximation theory (Proc. Conf., Math. Res. Inst., Oberwolfach, 1979), in: Internat. Ser. Numer. Math., 51 (1979) (Birkhauser: Basel-Boston) pp. 83–96Google Scholar
  5. [5]
    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) 195–210MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Engelking R, General topology, Sigma Series in Pure Mathematics, 6 (1989) (Berlin: Heldermann Verlag) 529 pp.Google Scholar
  7. [7]
    Franklin S P, Spaces in which sequences suffice, Fund. Math. 57 (1965) 107–115MathSciNetzbMATHGoogle Scholar
  8. [8]
    Franklin S P, Spaces in which sequences suffice. II, Fund. Math. 61 (1967) 51–56MathSciNetzbMATHGoogle Scholar
  9. [9]
    Golomb M, Approximation by functions of fewer variables, in: On numerical approximation. Proceedings of a Symposium (ed.) R E Langer (1959) (Madison: The University of Wisconsin Press) pp. 275–327Google Scholar
  10. [10]
    Halperin I, The product of projection operators, Acta Sci. Math. (Szeged) 23 (1962) 96–99MathSciNetzbMATHGoogle Scholar
  11. [11]
    Ismailov V E, On the theorem of M. Golomb, Proc. Indian Acad. Sci. (Math. Sci.) 119 (2009) 45–52MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Ismailov V E, On error formulas for approximation by sums of univariate functions, Int. J. Math. and Math. Sci. 2006 (2006) Article ID 65620, 11 pp.Google Scholar
  13. [13]
    Ismailov V E, Methods for computing the least deviation from the sums of functions of one variable, (Russian), Sibirskii Mat. Zhurnal. 47 (2006) 1076–1082 ; translation in Siberian Math. J. 47 (2006) 883–888zbMATHGoogle Scholar
  14. [14]
    Khavinson S Ya, Best approximation by linear superpositions (approximate nomography), Translated from the Russian manuscript by D Khavinson, Translations of Mathematical Monographs, 159 (1997) (Providence, RI: American Mathematical Society) 175 pp.Google Scholar
  15. [15]
    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) 77–86MathSciNetCrossRefGoogle Scholar
  16. [16]
    Klopotowski A and Nadkarni M G, Shift invariant measures and simple spectrum, Colloq. Math. 84/85 (2000) 385–394MathSciNetzbMATHGoogle Scholar
  17. [17]
    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) 305–322MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Light W A and Cheney E W, Approximation theory in tensor product spaces, Lecture Notes in Mathematics, 1169 (1985) (Berlin: Springer-Verlag) 157 pp.Google Scholar
  19. [19]
    Marshall D E and O’Farrell A G, Approximation by a sum of two algebras, The lightning bolt principle, J. Funct. Anal. 52 (1983) 353–368MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Marshall D E and O’Farrell A G, Uniform approximation by real functions, Fund. Math. 104 (1979) 203–211MathSciNetzbMATHGoogle Scholar
  21. [21]
    Medvedev V A, On the sum of two closed algebras of continuous functions on a compact space, (Russian), Funktsional. Anal. i Prilozhen. 27 (1) (1993) 33–36 ; translation in Funct. Anal. Appl. 27(1) (1993) 28–30MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Medvedev V A, Refutation of a theorem of Diliberto and Straus, Mat. Zametki 51 (1992) 78–80 ; English transl: Math. Notes 51 (1992) 380–381MathSciNetzbMATHGoogle Scholar
  23. [23]
    von Neumann J, Functional Operators, II, The Geometry of Orthogonal Spaces (1950) (Princeton University Press) 107 pp. (This is a reprint of mimeographed lecture notes first distributed in 1933)Google Scholar
  24. [24]
    Pinkus A, The alternating algorithm in a uniformly convex and uniformly smooth Banach space, J. Math. Anal. Appl. 421 (2015) 747–753MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Indian Academy of Sciences 2017

Authors and Affiliations

  1. 1.Institute of Mathematics and MechanicsNational Academy of Sciences of AzerbaijanBakuAzerbaijan

Personalised recommendations