Tambetabi, Henry Tabe; (1996) Closure and completeness problems in approximation theory. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

Much of the work in the theory of approximation has crystallised around two classical problems; the first is the Tchebyshev problem of finding a polynomial of degree n which gives the best uniform approximation to a given continuous function on an interval. The second is that of Weierstrass, to show that every continuous function on a closed bounded interval can be uniformly approximated with arbitrarily small error by a polynomial. Closure and completeness problems fall in the second category. In this thesis, we look at some sequences of rational functions in certain Hilbert spaces of bounded analytic functions, and ask when every function in these Hilbert spaces can be approximated arbitrarily closely (in the Hilbert space norm) by finite linear combinations of the rational functions. Furthermore, we provide a characterisation of the closed subspaces that the rational functions generate in the case when this is not the whole space. Finally, we obtain necessary and sufficient conditions in order that the rational functions will constitute Schauder bases in the closed subspaces which they generate.

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