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On the Coefficients of Multiple Walsh-Fourier Series with Small Gaps
http://hdl.handle.net/2261/00074368
http://hdl.handle.net/2261/000743688a499b33-a37c-4127-b235-6982e1e74d0b
名前 / ファイル | ライセンス | アクション |
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jms230401.pdf (146.0 kB)
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Item type | 紀要論文 / Departmental Bulletin Paper(1) | |||||
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公開日 | 2018-04-04 | |||||
タイトル | ||||||
タイトル | On the Coefficients of Multiple Walsh-Fourier Series with Small Gaps | |||||
言語 | ||||||
言語 | eng | |||||
キーワード | ||||||
主題 | Multiple Walsh-Fourier coefficient | |||||
主題Scheme | Other | |||||
キーワード | ||||||
主題 | Function of bounded p-variation in several variables | |||||
主題Scheme | Other | |||||
キーワード | ||||||
主題 | order of magnitude | |||||
主題Scheme | Other | |||||
キーワード | ||||||
主題 | Lacunary Fourier series with small gaps | |||||
主題Scheme | Other | |||||
資源タイプ | ||||||
資源 | http://purl.org/coar/resource_type/c_6501 | |||||
タイプ | departmental bulletin paper | |||||
著者 |
Ghodadra, Bhikha Lila
× Ghodadra, Bhikha Lila |
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著者所属 | ||||||
著者所属 | Department of Mathematics, Faculty of Science, The M. S. University of Baroda | |||||
抄録 | ||||||
内容記述タイプ | Abstract | |||||
内容記述 | For a Lebesgue integrable complex-valued function $f$ defined over the $m$-dimensional torus $\mathbb {I}^m:=[0,1)^m$, let $\hat f({\bf n})$ denote the multiple Walsh-Fourier coefficient of $f$, where ${\bf n}=\left(n^{(1)},\dots,n^{(m)}\right)\in (\mathbb {Z}^+)^m$, $\mathbb {Z^+}=\mathbb {N}\cup \{0\}$. The Riemann-Lebesgue lemma shows that $\hat f({\bf n})=o(1)$ as $|{\bf n}|\to \infty$ for any $f\in {\rm L}^1(\mathbb I^m)$. However, it is known that, these Fourier coefficients can tend to zero as slowly as we wish. The definitive result is due to Ghodadra Bhikha Lila for functions of bounded $p$-variation. We shall prove that this is just a matter only of local bounded $p$-variation for functions with multiple Walsh-Fourier series lacunary with small gaps. Our results, as in the case of trigonometric For a Lebesgue integrable complex-valued function $f$ defined over the $m$-dimensional torus $\mathbb {I}^m:=[0,1)^m$, let $\hat f({\bf n})$ denote the multiple Walsh-Fourier coefficient of $f$, where ${\bf n}=\left(n^{(1)},\dots,n^{(m)}\right)\in (\mathbb {Z}^+)^m$, $\mathbb {Z^+}=\mathbb {N}\cup \{0\}$. The Riemann-Lebesgue lemma shows that $\hat f({\bf n})=o(1)$ as $|{\bf n}|\to \infty$ for any $f\in {\rm L}^1(\mathbb I^m)$. However, it is known that, these Fourier coefficients can tend to zero as slowly as we wish. The definitive result is due to Ghodadra Bhikha Lila for functions of bounded $p$-variation. We shall prove that this is just a matter only of local bounded $p$-variation for functions with multiple Walsh-Fourier series lacunary with small gaps. Our results, as in the case of trigonometric Fourier series due to J.R. Patadia and R.G. Vyas, illustrate the interconnection between `localness' of the hypothesis and `type of lacunarity' and allow us to interpolate the results. | |||||
書誌情報 |
Journal of Mathematical Sciences, The University of Tokyo 巻 23, 号 4, p. 727-740, 発行日 2016-10-31 |
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ISSN | ||||||
収録物識別子タイプ | ISSN | |||||
収録物識別子 | 13405705 | |||||
書誌レコードID | ||||||
収録物識別子タイプ | NCID | |||||
収録物識別子 | AA11021653 | |||||
著者版フラグ | ||||||
値 | publisher | |||||
Mathmatical Subject Classification | ||||||
42C10(MSC2010) | ||||||
Mathmatical Subject Classification | ||||||
42B05(MSC2010) | ||||||
Mathmatical Subject Classification | ||||||
26B30(MSC2010) | ||||||
Mathmatical Subject Classification | ||||||
26D15(MSC2010) | ||||||
出版者 | ||||||
出版者 | Graduate School of Mathematical Sciences, The University of Tokyo | |||||
原稿受領日 | ||||||
2014-11-17 |