A Chebyshev-type theorem characterizing best approximation of a continuous function by elements of the sum of two algebras (Q2037621)

Let \(X\) be a compact metric space, let \(A_1,A_2\) be two closed subalgebras of continuous functions on \(X\), containg constants. The authors assume an additional condition in lieu of point separation. For \(u \in C(X)\), the paper addresses the question of describing a best approximation to \(u\) in \(A_1+A_2\).The description involves the notion of a `lightning bolt' due to \textit{D. E. Marshall} and \textit{A. G. O'Farrell} [Fundam. Math. 104, 203--211 (1979; Zbl 0439.41030); J. Funct. Anal. 52, 353--368 (1983; Zbl 0536.46035)].