On the convergence of Fourier series
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...
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| Format: | Article |
| Language: | English |
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Wiley
1984-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171284000843 |
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| _version_ | 1832558751605850112 |
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| author | Geraldo Soares de Souza |
| author_facet | Geraldo Soares de Souza |
| author_sort | Geraldo Soares de Souza |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-bcb93f07e51846cf99d939c97527125f |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1984-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-bcb93f07e51846cf99d939c97527125f2025-02-03T01:31:42ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251984-01-017481782010.1155/S0161171284000843On the convergence of Fourier seriesGeraldo Soares de Souza0Department of Mathematics, Auburn University, 36849, Alabama, USAWe 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.http://dx.doi.org/10.1155/S0161171284000843maximal operatorLorentz spaces and Fourier series. |
| spellingShingle | Geraldo Soares de Souza On the convergence of Fourier series International Journal of Mathematics and Mathematical Sciences maximal operator Lorentz spaces and Fourier series. |
| title | On the convergence of Fourier series |
| title_full | On the convergence of Fourier series |
| title_fullStr | On the convergence of Fourier series |
| title_full_unstemmed | On the convergence of Fourier series |
| title_short | On the convergence of Fourier series |
| title_sort | on the convergence of fourier series |
| topic | maximal operator Lorentz spaces and Fourier series. |
| url | http://dx.doi.org/10.1155/S0161171284000843 |
| work_keys_str_mv | AT geraldosoaresdesouza ontheconvergenceoffourierseries |