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On trigonometric series with positive partial sums. (English. Russian original) Zbl 0913.42004

Lupanov, O. B. (ed.), Analytic number theory and applications. Collected papers in honor of the sixtieth birthday of Professor Anatolii Alexeevich Karatsuba. Moscow: MAIK Nauka/Interperiodica Publishing, Proc. Steklov Inst. Math. 218, 444-447 (1997); translation from Tr. Mat. Inst. Steklova 218, 444-447 (1997).
From the note: “We study trigonometric series (TS) with nonnegative partial sums whose spectrum is \(A\cup (-A)\), where \(A\) is the set of natural numbers free of sums, which is well known in number theory, i.e., the equation \(x+ y=z\) has no solutions in \(A\). We state that TS of this type, \[ 1+ \sum_{\substack{ \nu\in Z\\ |\nu|\in A}} c_\nu e^{i\nu x},\quad c_\nu= \overline{c_{-\nu}}, \] are Fourier series of functions from \(L_3\). It was stated earlier that these series belong to \(L_2\)”.
For the entire collection see [Zbl 0907.00013].

MSC:

42A24 Summability and absolute summability of Fourier and trigonometric series
11L03 Trigonometric and exponential sums (general theory)
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