Correction of step functions and summation of their Fourier series by the Cesàro methods of negative order (Q1099353)

D. E. Men'shov stated the following theorem: Let f(x) be a function measurable and bounded on [0.2\(\pi\) ] a.e. then for each \(\epsilon >0\) there exists a function g(x) identical to f(x) on some set E, \(mE>2\pi - \epsilon\) such that the Fourier series \(\sigma\) (g) converges on [0.2\(\pi\) ] uniformly. [See \textit{N. K. Bari}, Trigonometric series (Moscow, (1961)) p. 448]. In 1971 Men'shov posed the question wheather his theorem can be extended to the Cesaro's summation of negative order \(\alpha >-\). [\textit{D. E. Men'shov}, Mat. Sb. N. Ser. 86(128), 419-445 (1971; Zbl 0223.42006)]. The author's work is closely concerned with the Men'shov's theorem. Let \(-<\alpha \leq 2\). Then for each \(\epsilon >0\) and \(f(t)\in \Lambda_ N:\) \[ \Lambda_ N=\{f(t):f(t)=y_ k,t\in \Delta_{k'}\equiv (\frac{2\pi (k'-1)}{N},\frac{2\pi k'}{N}),\quad k'\cdot k\equiv 0\quad (mod N),\quad k=1,2,...,N\}. \] \(\| f\|_{L^{\infty}}\leq 1\) there exists a function \(g(t^)\in \Lambda_ N\) such that 1. \(m\{\) t:f(t)\(\neq g(t)\}\leq 2\pi \epsilon\); 2. \(_{n}\| \sigma_ n^{\alpha}(g)\|_{L^{\infty}}\leq C_{\alpha}/\epsilon,\) where \(\sigma_ n^{\alpha}(g)\) is the Cesaro's summation of g, \(C_{\alpha}\) is a constant. Obviously, the Men'shov's problem is still open.