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On a certain norm and related applications. (English. Russian original) Zbl 0955.42002

Math. Notes 64, No. 4, 551-554 (1998); translation from Mat. Zametki 64, No. 4, 637-640 (1998).
Let \(QC(\mathbb{T})\) be the closure of trigonometrical polynomials with respect to the norm \[ \|f\|_{QC}:= \int^1_0 \Biggl\|\sum^\infty_{k=0} r_k(\omega) \sum_{2^{k-1}\leq |n|< 2^k}\widehat f(n)e^{inx}\Biggr\|_{L_\infty(\mathbb{T})}d\omega, \] where \(\{r_k(\omega)\}^\infty_{k= 0}\) is the Rademacher system. Using this and norms closed to it, the authors accounce several inequalities for trigonometric polynomials and their applications to in order sharp estimates of entropy numbers and Kolmogorov width.

MSC:

42A05 Trigonometric polynomials, inequalities, extremal problems
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
42A10 Trigonometric approximation
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