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A problem of Littlewood. (English. Russian original) Zbl 0776.42001

Math. Notes 49, No. 2, 217-218 (1991); translation from Mat. Zametki 49, No. 2, 143-144 (1991).
From the text: Denote by \({\mathcal L}\) the set of real trigonometric polynomials of the form \(f(x)=\tfrac12 a_0+\sum^\infty_{n=1} a_n \cos nx\), where \(a_0\geq | a_n|\) for all \(n\). For \(f\in {\mathcal L}\) we put \(f^*(x)= \tfrac12 a_0+\sum^ \infty_{n=1} a^*_n \cos nx\), where \(\{a^*_0,a^*_1,\ldots\}\) is a nonincreasing rearrangement of the sequence \(\{| a_0|,| a_1|,\ldots\}\) (in particular, \(a^*_0 = a_0)\)...
J. E. Littlewood [J. Lond. Math. Soc. 35, 325–365 (1960; Zbl 0099.05403)] conjectured that
\[ \sup_{\substack{f\in\mathcal L \\ f\neq 0}} \int^\pi_{- \pi} | f^*(x)| \,dx\Bigl/\int^\pi_{-\pi} | f(x)| \,dx = +\infty. \]
... Here we prove Littlewood’s conjecture.

MSC:

42A05 Trigonometric polynomials, inequalities, extremal problems

Citations:

Zbl 0099.05403
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References:

[1] G. H. Hardy and J. E. Littlewood, J. London Math. Soc.,23, No. 91, 163-168 (1948). · Zbl 0034.04301
[2] D. H. Lehmer, J. London Math. Soc.,34, No. 136, 395-396, Addendum, 485 (1959). · Zbl 0087.06204
[3] J. E. Littlewood, J. London Math. Soc.,35, No. 139, 352-365 (1960). · Zbl 0099.05403
[4] O. C. McGehee, L. Pigno, and B. Smith, Ann. Math.,113, No. 3, 613-618 (1981). · Zbl 0473.42001
[5] S. V. Konyagin, Izv. Akad. Nauk SSSR, Ser. Mat.,45, No. 2, 243-265 (1981).
[6] N. K. Bari, Trigonometric Series [in Russian], Fizmatgiz, Moscow (1961).
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