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Remark on uniqueness of summable trigonometric series associated with conjugate series. (English) Zbl 0129.28103

The conjugate series of trigonometric series (i) \(\displaystyle\tfrac12 a_0 + \sum_{n=1}^\infty (a_n \cos nx + \sin nx)\) is defined by (ii) \( \sum_{n=1}^\infty (a_n \sin nx - b_n \cos nx)\). D. Menchoff (D. Men’shov) [C. R. Acad. Sci., Paris 163, 433–436 (1916; JFM 46.0457.02)] and A. Zygmund [Trans. Am. Math. Soc. 34, 435–446 (1932; Zbl 0005.06303; JFM 58.0280.01)] have independently constructed examples of trigonometric series not identical to zero but converging almost everywhere to zero. On the other hand N. Lusin and J. Priwaloff [Ann. Sci. Éc. Norm. Supér. (3) 42, 143–191 (1925; JFM 51.0245.01)] have found a trigonometric series not identical to zero, but being Abel-summable together with its conjugate series almost everywhere to the value zero. It is natural to inquire whether there is a trigonometric series not identical to zero but converging together with its conjugate series almost everywhere to zero in \((0, 2\pi)\). The author has shown that the answer to this problem is in the negative by establishing that if (i) and (ii) are both summable \((C, k)\), for some \(k > -1\), to zero in a set \(E\) of positive measure, then both series are identical to zero. He further proves that given any arbitrary set \(E\) of zero measure, there exists a pair of conjugate trigonometric series in the forms (i) and (ii) which are not identical to zero, such that both series are summable \((C, k)\), \(k > 0\), to zero in \(E\).

MSC:

42-XX Harmonic analysis on Euclidean spaces
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References:

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