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International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 4, Pages 817-820
Research notes

On the convergence of Fourier series

Department of Mathematics, Auburn University, 36849, Alabama, USA

Received 5 March 1984

Copyright © 1984 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We define the space Bp={f:(π,π]R,f(t)=n=0cnbn(t),n=0|cn|<}. Each bn is a special p-atom, that is, a real valued function, defined on (π,π], which is either b(t)=1/2π or b(t)=1|I|1/pXR(t)+1|I|1/pXL(t), where I is an interval in (π,π], L is the left half of I and R is the right half. |I| denotes the length of I and XE the characteristic function of E. Bp is endowed with the norm fBp=Intn=0|cn|, where the infimum is taken over all possible representations of f. Bp is a Banach space for 1/2<p<. Bp is continuously contained in Lp for 1p<, but different. We have THEOREM. Let 1<p<. If fBp then the maximal operator Tf(x)=supn|Sn(f,x)| maps Bp into the Lorentz space L(p,1) boundedly, where Sn(f,x) is the nth-sum of the Fourier Series of f.