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Transactions of the American Mathematical Society

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Analytic-function bases for multiply-connected regions


Author: Victor Manjarrez
Journal: Trans. Amer. Math. Soc. 154 (1971), 93-103
MSC: Primary 30.70
DOI: https://doi.org/10.1090/S0002-9947-1971-0273029-0
MathSciNet review: 0273029
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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.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0273029-0
Keywords: Absolute basis, multiply-connected region, compact convergence, dual space, inductive limit topology, rational interpolation functions, absolute effectivity, orthogonal polynomials
Article copyright: © Copyright 1971 American Mathematical Society