Bounding partial sums of Fourier series in weighted L2-norms, with applications to matrix analysis

Authors:
N. Borovykh;M. N. Spijker
Affiliations:
Mathematical Institute, University of Leiden, P.O.Box 9512, 2300 RA Leiden, The Netherlands;Mathematical Institute, University of Leiden, P.O.Box 9512, 2300 RA Leiden, The Netherlands
Venue:
Journal of Computational and Applied Mathematics
Year:
2002

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Abstract

For integrable functions f defined on the interval [ -π, π], we denote the partial sums of the corresponding Fourier series by Sn(f)(n=0,1,2,...). In this paper, we deal with the problem of bounding supn ||Sn||, where ||.|| denotes an operator norm induced by a weighted L2-norm for functions f on [-π,π]. For weight functions w belonging to a class introduced by Helson and Szegö (Ann. Mat. Pura Appl. 51 (1960) 107), we present explicit upper bounds for supn||Sn|| in terms of w.The authors were led to the problem of deriving explicit upper bounds for supn||sn||, by the need for such bounds in an analysis related to the Kreiss matrix theorem-a famous result in the fields of linear algebra and numerical analysis. Accordingly, the present paper highlights the relevance of bounds on supn,||Sn|| to these fields.