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Fourier coefficients of characteristic functions of intervals with respect to a complete orthonormal system bounded in \(L^p([0,1])\), \(2<p<\infty\). (English. Russian original) Zbl 1322.42034
Math. Notes 97, No. 4, 647-651 (2015); translation from Mat. Zametki 97, No. 4, 632-635 (2015).
Main result: for any orthonormal system \(\{\phi_j\}_{j=1}^{\infty}\) which is complete in \(L^2([0,1])\) and uniformly bounded in \(L^p([0,1]),2<p<\infty,\) there exists \(x\in (0,1]\) such that \[ \sum_{j=1}^{\infty}\left|\int_0^x\phi_j(t)dt\right|^{(p-2)/(p-1)}=\infty. \]
MSC:
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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