On the theorem of M.Golomb

2007/08/26 by Vugar Ismailov, Ismailov, Vugar · 1 citation
Mathematics · #41A30 #41A50 #41A63 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:41A30 #msc:41A50 #msc:41A63

paper · pdf · doi:10.48550/arxiv.0708.3503

8 pages; the previous main result has substantially improved

arxiv created 2008/07/10 · arxiv updated 2009/12/01

Abstract

Let X₁,...,X_n be compact spaces and X=X₁× ... × X_n. Consider the approximation of a function f∈ C(X) by sums g₁(x₁)+... g_n(x_n), where g_i∈ C(X_i), i=1,...,n. In [8], M.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.

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