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On the integrability and \(L^ 1\)-convergence of sine series. (English) Zbl 0671.42006
The author studies the integrability and \(L^ 1\)-convergence of Fourier sine series \(f(x)=\sum^{\infty}_{k=1}a_ k \sin kx.\) He introduces the classes \(\tilde C,\) \(\tilde BV\) and \(\tilde V_ p\) and proves among other things: (a) if \(\{a_ k\}\in \tilde C\cap \tilde BV\) then \(f\in L^ 1(0,\pi)\) and (b) if \(\{a_ k\}\in \tilde C\cap \tilde BV\) or \(\{a_ k\}\in \tilde V_ p\) and \(f\in L^ 1(0,\pi)\) then \(a_ k \log k\to 0\) is necessary and sufficient for the \(L^ 1\)-convergence of the partial sums to f.
Reviewer: J.Marschall

42A20 Convergence and absolute convergence of Fourier and trigonometric series
42A32 Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
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