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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

##### MSC:
 42A20 Convergence and absolute convergence of Fourier and trigonometric series 42A32 Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
##### Keywords:
integrability; Fourier sine series; $$L^ 1$$-convergence
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