On degree of approximation of functions belonging to the weighted \((L^p, \xi(t))\) class by \((C,1)(E,1)\) means (Q2724898)

The author claims the proof of the estimate NEWLINE\[NEWLINE\|f-(CE)_n^1\|_p=O\left(\xi(\frac{1}{n})n^{\beta+\frac{1}{p}}\right)NEWLINE\]NEWLINE under some restrictions on \(\xi\), where NEWLINE\[NEWLINE(CE)_n^1=\frac{1}{n}\sum_{k=1}^n\left(2^{-k}\sum_{i=0}^k\binom ki s_i\right),NEWLINE\]NEWLINE and \(s_i\) are the partial sums of the Fourier series of \(f\), \(f\in W(L^p,\xi(t)),\) i.e., NEWLINE\[NEWLINE\biggl(\int_0^{2\pi}|f(x+t)-f(x)|^p\sin^{\beta}x dx\biggr)^{1/p}=o(\xi(t)).NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEReviewer's remarks: 1.The author uses the representation NEWLINE\[NEWLINEs_n-f(x)=\frac{1}{\pi}\int_0^{\pi}\frac{f(x+t)+f(x-t)-2f(x)}{t}\sin nt dt +O(1)NEWLINE\]NEWLINE and later he does not take into account the \((CE)_n^1\)-means of \(O(1)\)-term! So the claim can not be considered as a correctly proved assertion. 2. As a corollary, the author indicates the estimate \(|(CE)_n^1-f(x)|=O(1/n)\) for \(f\in \operatorname {Lip}(\alpha,\infty),\) \(0<\alpha<1,\), which is the main result of the note [\textit{S. Lal} and \textit{K. N. S. Yadav}, Bull. Calcutta Math. Soc. 93, 191-196 (2001; Zbl 1032.42003), preceding review]. That note was sent to a journal after (!) appearing of the paper under review.