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U. Luther; G. Mastroianni : Fourier Projections in Weighted L_infty-Spaces

U. Luther; G. Mastroianni : Fourier Projections in Weighted L_infty-Spaces

Author(s) :
U. Luther; G. Mastroianni
Title :
Fourier Projections in Weighted L_infty-Spaces
Preprint series
Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 99-10, 1999
Mathematics Subject Classification :
42C10 [ Fourier series in special orthogonal functions ]
41A10 [ Approximation by polynomials ]
Abstract :
It is well-known, that the norms of the classical Fourier projections the space of all 2pi-periodic L_infty-functions can be estimated by ||S_n||< const ln(n+1). In other words, the L_infty(-1,1)-operator norms of the Fourier projections with respect to the normalized Chebyshev polynomials of first kind are bounded by const ln(n+1). In this paper we show that this remains true for Fourier projections with respect to normalized Jacobi polynomials, if we consider them in a weighted L_infty-space, where the weight is a Jacobi weight, which has to fulfil certain conditions. Moreover, we prove that these conditions are necessary, and we also consider the case of pairs of L_infty-spaces with different Jacobi weights. As corollaries we obtain, among others, corresponding results in weighted L_1-spaces and norm estimates of the type O(ln²n) or O(ln³n) for modified Fourier projections in cases where the unmodified Fourier projections can not have a logarithmic norm behaviour.
Keywords :
Fourier projections, weighted function spaces, orthogonal polynomials
Language :
english
Publication time :
12/1999
Notes :
To appear in: Proceedings of the 11th TMP (Chemnitz, March 1999), Operator Theory: Advances and Applications, Birkhäuser Verlag.