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    • On the Theorem of 𝑀 Golomb

      Vugar E Ismailov

      Volume 119 Issue 1 February 2009 pp 45-52

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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/119/01/0045-0052

    • Keywords

       

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

    • Abstract

       

      Let $X_1,\ldots,X_n$ be compact spaces and $X=X_1\times\cdots\times X_n$. Consider the approximation of a function $f\in C(X)$ by sums $g_1(x_1)+\cdots+g_n(x_n)$, where $g_i\in C(X_i),i=1,\ldots,n$. 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.

    • Author Affiliations

       

      Vugar E Ismailov1

      1. Mathematics and Mechanics Institute, Azerbaijan National Academy of Sciences, Az-1141, Baku, Azerbaijan

    • Dates

       
      Published
      February 2009
      Manuscript received
      25 August 2007
      Manuscript revised
      19 August 2008

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