On the rate of rational approximation of functions on tangent continua (Q1921743)

Let \(K\) be a union of two bounded continua \(K_1\) and \(K_2\) with connected complements in the complex plane such that \(K_1 \cap K_2 = \{z_0\}\) is a single point \(z_0\). Let \(f\) be a function defined on \(K\) such that \(f(z) = f_j (z)\), \(z \in K_j\), where \(f_j\) is a holomorphic function on \(K_j (j = 1,2)\). Let \(R_n (f,K) : = \inf \{|f - r_n |_K\}\) be the best uniform approximation by rational functions \(r_n\) of order \(\leq n\). Under suitable assumptions on \(K\), the author finds lower and upper estimates of \(R_n (f,K)\). In particular, as a corollary, he shows that if \(\mathbb{C} \backslash K\) is a John domain then \(R_n (f,K) \leq C \exp (- c \sqrt {n \nu_f})\), \(n \geq 1\), where \(C\) and \(c\) are positive constants and \(\nu_f\) denotes the minimum of the orders of the functions \(f_j (z) - f_j (z_0)\) \((j = 1,2)\) at \(z_0\).