CONVERGENCE OF FOURIER SERIES ALMOST EVERYWHERE AND IN THE $ L$-METRIC

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The following theorems are proved.Theorem 1. There exists a constant such that for any function there is a measurable function for which , and where is a partial sum of the Fourier series of , is a partial sum of the conjugate Fourier series, and is the conjugate function to .Theorem 2. For any function and there exists a measurable function such that , ( is Lebesgue measure), and both the Fourier series of and its conjugate series converge almost everywhere and in the metric of .Bibliography: 11 titles.