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 $({Q_n},n = 0, \pm 1, \pm 2, \ldots )$ of rational functions for the space $H(E)$ of functions analytic on E, with the topology of compact convergence; or the space $H({\text {Cl}}\;(E))$ of functions analytic on ${\text {Cl}}\;(E)$ = the closure of E, with an inductive limit topology. It is shown that $\Sigma _{n = 0}^\infty {Q_n}(z){Q_{ - n - 1}}(w) = 1/(w - z)$, the convergence being uniform for z and w on suitable subsets of the plane. A sequence $({P_n},n = 0, \pm 1, \pm 2, \ldots )$ of elements of $H(E)$ (resp. $H({\text {Cl}}\;(E))$) is said to be absolutely effective on E(resp. ${\text {Cl}}\;(E)$) if it is an absolute basis for $H(E)$ (resp. $H({\text {Cl}}\;(E))$) and the coefficients arise by matrix multiplication from the expansion of $({Q_n})$. Conditions for absolute effectivity are derived from W. F. Newns’ generalization of work of J. M. Whittaker and B. Cannon. Moreover, if $({P_n},n = 0,1,2, \ldots )$ is absolutely effective on a certain simply-connected set associated with E, the sequence is extended to an absolutely effective basis $({P_n},n = 0, \pm 1, \pm 2, \ldots )$ for $H(E)$ (or $H({\text {Cl}}\;(E))$) such that $\Sigma _{n = 0}^\infty {P_n}(z){P_{ - n - 1}}(w) = 1/(w - z)$. This last construction applies to a large class of orthogonal polynomials.