A continuous function space with a Faber basis |
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Authors: | Josef Obermaier |
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Affiliation: | Institute of Biomathematics and Biometry, GSF—National Research Center for Environment and Health, Ingolstädter Landstr. 1, Neuherberg D-85764, Germany |
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Abstract: | Let be compact with #S=∞ and let C(S) be the set of all real continuous functions on S. We ask for an algebraic polynomial sequence (Pn)n=0∞ with deg Pn=n such that every fC(S) has a unique representation f=∑i=0∞ αiPi and call such a basis Faber basis. In the special case of , 0<q<1, we prove the existence of such a basis. A special orthonormal Faber basis is given by the so-called little q-Legendre polynomials. Moreover, these polynomials state an example with A(Sq)≠U(Sq)=C(Sq), where A(Sq) is the so-called Wiener algebra and U(Sq) is the set of all fC(Sq) which are uniquely represented by its Fourier series. |
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Keywords: | Continuous function Polynomial Basis Little q-Legendre polynomials Fourier series Wiener algebra |
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