On the Theorem of 𝑀 Golomb
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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.
Vugar E Ismailov^{1} ^{}
Volume 130, 2020
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