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A lower bound for the \(L_1\)-norm of a Fourier series of power type. (English. Russian original) Zbl 1054.43003
Math. Notes 73, No. 6, 900-903 (2003); translation from Mat. Zametki 73, No. 6, 951-953 (2003).
The following is known as Littlewood’s conjecture: there exists an absolute constant \(C\) such that \[ \int^\pi_{-\pi}\,\Biggl| \sum^n_{k=1} e^{im_k x}\Biggr|\,dx\geq C\log n \] for any set of positive integers \(\{m_k\}^n_1\) and \(n= 1,2,\dots\) . The conjecture was proved true independently by S. V. Konyagin [Izv. Akad. Nauk SSSR, Ser. Mat. 45, 243–265 (1981; Zbl 0493.42004); Engl. translation in Math. USSR, Izv. 18, 205–225 (1982; Zbl 0511.42003)] and O. C. McGehee, L. Pigno and B. Smith [Ann. Math. (2) 113, 613–618 (1981; Zbl 0473.42001)].
The present paper provides the following extension of the above result: There exists an absolute constant \(C\) such that \[ \int^\pi_{-\pi}\,\Biggl| \sum^\infty_{k=1} c_k e^{im_kx}\Biggr|\, dx\geq C \sum^\infty_{j=1}\,\Biggl[\sum_{2^{j-1}\leq k< 2^j} {| c_k|^2\over k}\Biggr]^{{1\over 2}} \] for any increasing sequence of positive integers \(\{m_k\}^\infty_1\) and any sequence of coefficients \(\{c_k\}^\infty_1\).
To see how exactly Littlewood’s conjecture follows from here the reader should consult the paper itself as well as the above mentioned paper by McGehee at al.

MSC:
43A17 Analysis on ordered groups, \(H^p\)-theory
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
43A20 \(L^1\)-algebras on groups, semigroups, etc.
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