On L1-convergence of Walsh-Fourier series
Let G denote the dyadic group, which has as its dual group the Walsh(-Paley) functions. In this paper we formulate a condition for functions in L1(G) which implies that their Walsh-Fourier series converges in L1(G)-norm. As a corollary we obtain a Dini-Lipschitz-type theorem for L1(G) convergence an...
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| Format: | Article |
| Language: | English |
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Wiley
1978-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S016117127800006X |
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| _version_ | 1850232884332331008 |
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| author | C. W. Onneweer |
| author_facet | C. W. Onneweer |
| author_sort | C. W. Onneweer |
| collection | DOAJ |
| description | Let G denote the dyadic group, which has as its dual group the Walsh(-Paley) functions. In this paper we formulate a condition for functions in L1(G) which implies that their Walsh-Fourier series converges in L1(G)-norm. As a corollary we obtain a Dini-Lipschitz-type theorem for L1(G) convergence and we prove that the assumption on the L1(G) modulus of continuity in this theorem cannot be weakened. Similar results also hold for functions on the circle group T and their (trigonometric) Fourier series. |
| format | Article |
| id | doaj-art-0ca86895766147c7a43eb0432dbfb2c6 |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1978-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-0ca86895766147c7a43eb0432dbfb2c62025-08-20T02:03:04ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251978-01-0111475610.1155/S016117127800006XOn L1-convergence of Walsh-Fourier seriesC. W. Onneweer0Department of Mathematics, University of New Mexico, Albuquerque 87131, New Mexico, USALet G denote the dyadic group, which has as its dual group the Walsh(-Paley) functions. In this paper we formulate a condition for functions in L1(G) which implies that their Walsh-Fourier series converges in L1(G)-norm. As a corollary we obtain a Dini-Lipschitz-type theorem for L1(G) convergence and we prove that the assumption on the L1(G) modulus of continuity in this theorem cannot be weakened. Similar results also hold for functions on the circle group T and their (trigonometric) Fourier series.http://dx.doi.org/10.1155/S016117127800006XWalsh-Fourier seriesconvergence in normDini-Lipschitz-type test. |
| spellingShingle | C. W. Onneweer On L1-convergence of Walsh-Fourier series International Journal of Mathematics and Mathematical Sciences Walsh-Fourier series convergence in norm Dini-Lipschitz-type test. |
| title | On L1-convergence of Walsh-Fourier series |
| title_full | On L1-convergence of Walsh-Fourier series |
| title_fullStr | On L1-convergence of Walsh-Fourier series |
| title_full_unstemmed | On L1-convergence of Walsh-Fourier series |
| title_short | On L1-convergence of Walsh-Fourier series |
| title_sort | on l1 convergence of walsh fourier series |
| topic | Walsh-Fourier series convergence in norm Dini-Lipschitz-type test. |
| url | http://dx.doi.org/10.1155/S016117127800006X |
| work_keys_str_mv | AT cwonneweer onl1convergenceofwalshfourierseries |