International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 4, Pages 817-820
doi:10.1155/S0161171284000843
Research notes
On the convergence of Fourier series
Department of Mathematics, Auburn University, 36849, Alabama, USA
Received 5 March 1984
Copyright © 1984 Geraldo Soares de Souza. 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.
Abstract
We define the space Bp={f:(−π,π]→R, f(t)=∑n=0∞cnbn(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 ‖f‖Bp=Int∑n=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 1≤p<∞, but different. We have THEOREM. Let 1<p<∞. If f∈Bp 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.