On properties of the space U(0,2\(\pi\) ) (Q1063195)

Let \(f\in L^ 2(0,2\pi)\) and let \(S_ N(f,x)\) denote the Nth partial sums of its Fourier series. Put \(\| f\|_ U=\sup_{N}\| S_ N(f,x)\|_ C\) and \[ \Omega_ N=\{f:f(x)=\epsilon_ k,x\in (2\pi (k-1)N^{-1},2\pi kN^{-1}),\quad \epsilon_ k=\pm 1,\quad K=1,...,N\}. \] The author proves the following problem posed by B. S. Kashinym: 1) \(2^{-Nj}\sum_{f\in \Omega_ N}\| f\|_ U\asymp \ln \ln N\) (N\(\to \infty):\) 2) There exist \(C>0\) and \(\gamma >0\) such that \(2^{-N}| \{f\in \Omega_ N:\| f\|_ U>C \ln \ln N\}| >\gamma\) for all N. (Here \(| A|\) denotes the number of elements of a finite set A.)