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Approximations in $$L^1$$ L 1 with convergent Fourier series
Mathematische Zeitschrift ( IF 0.8 ) Pub Date : 2021-04-25 , DOI: 10.1007/s00209-021-02734-6
Zhirayr Avetisyan , Martin Grigoryan , Michael Ruzhansky

For a separable finite diffuse measure space \({\mathcal {M}}\) and an orthonormal basis \(\{\varphi _n\}\) of \(L^2({\mathcal {M}})\) consisting of bounded functions \(\varphi _n\in L^\infty ({\mathcal {M}})\), we find a measurable subset \(E\subset {\mathcal {M}}\) of arbitrarily small complement \(|{\mathcal {M}}{\setminus } E|<\epsilon \), such that every measurable function \(f\in L^1({\mathcal {M}})\) has an approximant \(g\in L^1({\mathcal {M}})\) with \(g=f\) on E and the Fourier series of g converges to g, and a few further properties. The subset E is universal in the sense that it does not depend on the function f to be approximated. Further in the paper this result is adapted to the case of \({\mathcal {M}}=G/H\) being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of n-spheres with spherical harmonics is discussed. The construction of the subset E and approximant g is sketched briefly at the end of the paper.



中文翻译:

收敛傅里叶级数的$$ L ^ 1 $$ L 1逼近

对于一个可分离有限漫测度空间\({\ mathcal {M}} \)和一个正交基\(\ {\ varphi _n \} \)\(L ^ 2({\ mathcal {M}})\)由有界函数\(\ varphi _n \ in L ^ \ infty({\ mathcal {M}})\)组成,我们找到了任意小补码的可测量子集\(E \ subset {\ mathcal {M}} \\)\(| {\ mathcal {M}} {\ setminus} E | <\ epsilon \),这样每个可测量的函数\(f \ in L ^ 1({\ mathcal {M}})\)都有一个近似值\ (g \ in L ^ 1({\ mathcal {M}})\)E上具有\(g = f \)并且g的傅立叶级数收敛到g,以及其他一些属性。在不依赖于要近似的函数f的意义上,子集E是通用的。在本文中,该结果还适用于\({\ mathcal {M}} = G / H \)是无限紧凑的第二可数Hausdorff群的齐次空间的情况。作为有用的说明,讨论了具有球谐函数的n球的情况。在本文的末尾简要概述了子集E和近似值g的构造。

更新日期:2021-04-26
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