Ray sequences of Laurent-type rational functions (Q1381038)

Let \(A\) be a bounded Jordan region of connectivity \(n+1\) in the complex plane. The complement of \(\overline A\) consists of an unbounded region \(\Omega\) and \(n\) bounded regions \(G_l\). In each \(G_l\) a fixed point \(a_l\) is chosen and Laurent-type rational functions \[ R_N(z)=\sum_{j=0}^k t^N_jz^j +\sum_{l=1}^n\sum_{j=1}^{m_l} s_{l,j}^N(z-a_l)^{-j} \] are considered with multi-index \(N:=(k,m_1,\dots,m_n)\). A sequence of such functions is called a ray sequence when the fractions \(m_l/(k+m_1+\dots +m_n)\) converge to numbers \(\alpha_l\) for \(l=1,\dots,n\). When the ray sequence converges locally uniformly in \(A\) to a function \(f\not\equiv 0\) and satisfies an additional condition, then the normalized counting measure of the zeros of \(R_N\) converges weakly to a measure which is a convex combination of harmonic measures on the boundary components of \(A\) with coefficients \(\alpha_l\). Let \(f\) be a function analytic in \(\overline A\). A ray sequence which approximates \(f\) in the uniform norm on \(\overline A\) converges to \(f\) geometrically. A lower bound for the rate of convergence is derived which depends on the position of the singularities of \(f\) outside \(A\). It is shown that this bound is attained. The bound can be optimized by choosing suitable \(\alpha_l\). Subsequences of the zero distribution of the optimal approximating ray sequence converge weakly to harmonic measures on the boundary components of an extended region \(A_{an}\) which has the property that on each boundary component of \(A_{an}\) there is a singularity of \(f\). Similar results are shown for best approximating ray sequences in \(L_p(A)\) for \(1\leq p \leq \infty\).