# On the theorem of M Golomb

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## 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## Preview

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## References

- [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]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]Braess D and Pinkus A, Interpolation by ridge functions,
*J. Approx. Theory***73**(1993) 218–236zbMATHCrossRefMathSciNetGoogle Scholar - [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]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]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]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]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]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]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]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]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]Klopotowski A and Nadkarni M G, Shift invariant measures and simple spectrum,
*Colloq. Math.***84/85**(2000) 385–394MathSciNetGoogle Scholar - [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]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]Marshall D E and O’Farrell A G, Uniform approximation by real functions,
*Fund. Math.***104**(1979) 203–211zbMATHMathSciNetGoogle Scholar - [17]Navada K G, Some remarks on good sets,
*Proc. Indian Acad. Sci. (Math. Sci.)***114(4)**(2003) 389–397CrossRefMathSciNetGoogle Scholar - [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

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