Analytic-function bases for multiply-connected regions

Abstract

Let E be a nonempty (not necessarily bounded) region of finite connectivity, whose boundary consists of a finite number of nonintersecting analytic Jordan curves. Work of J. L. Walsh is utilized to construct an absolute basis (Qn5 n = O, + 1, +?2, ) of rational functions for the space H(E) of functions analytic on E, with the topology of compact convergence; or the space H(Cl (E)) of functions analytic on Cl (E)=the closure of E, with an inductive limit topology. It is shown that o Qn(z)Q-n-l(w)=l/(w-z), the convergence being uniform for z and w on suitable subsets of the plane. A sequence (Pn, n = 0, + 1, +2 . ) of elements of H(E) (resp. H(Cl (E))) is said to be absolutely effective on E (resp. Cl (E)) if it is an absolute basis for H(E) (resp. H(Cl (E))) and the coefficients arise by matrix multiplication from the expansion of (Qn). Conditions for absolute effectivity are derived from W. F. Newns' generalization of work of J. M. Whittaker and B. Cannon. Moreover, if (Pn, n = 0, 1, 2, . . .) is absolutely effective on a certain simply-connected set associated with E, the sequence is extended to an absolutely effective basis (Pn, n = 0, + 1, + 2, . . .) for H(E) (or H(Cl (E))) such that En=o P,(Z)Pn 1(w)= 1/(w-z). This last construction applies to a large class of orthogonal polynomials. 1. Interpolation bases. For any subset S of the extended complex plane, let H(S) be the set of functions analytic on S (that is, f is in H(S) if and only if f is analytic on some open set containing S), and zero at infinity if the point at infinity is in S. The convergence of a sequence of elements (f,) of H(S) will be said to be compact-open on S if and only if the sequence converges uniformly on compact subsets of some open set containing S. Throughout the paper, let E be a nonempty region of finite connectivity, whose boundary consists of a finite number of nonintersecting analytic Jordan curves. If E is bounded, divide the boundary curves into two mutually disjoint sets, let L(O) and L(1) be the respective unions of the curves in each set, and suppose that the curve exterior to all the others is a subset of L(1). Let F be harmonic in E and continuous on the closure of E, and take on the values 0 and 1 on L(O) and L(1) respectively. Then there exist points z1, Z2, Z3,. . ., in the components of the complement of E bounded by L(O), and points W1, W2, W3, .. ., in the components of Received by the editors October 7, 1969 and, in revised form, April 17, 1970. AMS 1969 subject classifications. Primary 3070, 3030.