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 of rational functions for the space of functions analytic on *E*, with the topology of compact convergence; or the space of functions analytic on = the closure of *E*, with an inductive limit topology. It is shown that , the convergence being uniform for *z* and *w* on suitable subsets of the plane. A sequence of elements of (resp. ) is said to be absolutely effective on *E*(resp. ) if it is an absolute basis for (resp. ) and the coefficients arise by matrix multiplication from the expansion of . Conditions for absolute effectivity are derived from W. F. Newns' generalization of work of J. M. Whittaker and B. Cannon. Moreover, if is absolutely effective on a certain simply-connected set associated with *E*, the sequence is extended to an absolutely effective basis for (or ) such that . This last construction applies to a large class of orthogonal polynomials.

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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