Bounded point derivations on \(R(X)\) and approximation with simultaneous interpolation (Q1814551)

A criterion for the existence of bounded point derivations was found. It is a generalization of a theorem on peak points by E. Bishop [\textit{T. Gamelin} Uniform algebras. Englewood Cliffs, N. J.: Prentice-Hall (1969; Zbl 0213.40401)] and it is equivalent to the criteria by \textit{D. R. Wilken} [Proc. Am. Math. Soc. 24, 371--373 (1970; Zbl 0188.44903)] and \textit{A. P. Hallström} [J. Funct. Anal. 4, 153--165 (1969; Zbl 0176.44102)]. The criterion was derived on the basis of a theorem on uniform approximation with simultaneous interpolation of \(R(X)\) or \(A(X,x_1,\ldots,x_n)\) -- class functions \((R(X)\) are rational functions with poles outside \(X\) and \(A(X,x_1,\ldots,x_n)\) are continuous within \(X\) and analytical on a set \(X^0\cup\{x_1,\ldots,x_n\}\).