zbMATH — the first resource for mathematics

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.

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.
Full Text: DOI