Approximation of \(W(L^p,\xi(t))\) function by \((N,p,q)C_1\) means of its Fourier series (Q1768190)

Let \((P_n)\) and \((q_n)\) be sequences of nonnegative real numbers such that \(p_0>0\), \(q_0>0\) and \(R_n=\sum^n_{k=0} p_{n-k}q_k\neq 0\) for all \(n\geq 0\). Then the \((N,p,q)C_1\) transform of the \(n\)-th partial sum \(s_n(f;x)\) of the Fourier series of the \(2\pi\)-periodic and Lebesgue integrable function \(f\in(0,2\pi)\) is given by \[ t^{p,q,C_1}_n(f;x)=\frac{1}{R_n}\sum^n_{k=0}\frac{p_{n-k}q_k}{k+1}\sum^k_{\nu=0}s_\nu(f;x). \] In this paper, the author claims to prove the following estimate for \(f\in W(L^p,\xi(t))\): \[ \| t^{p,q,C_1}_n(f)-f\|_p=O\left\{(n+1)^{\beta+\frac1p}\xi\left(\frac{1}{n+1}\right)\right\}, \] where \(\beta\geq 0\), \(p > 1\) and \(0 < \xi(t)\) is increasing whenever \((p_n)\) and \((q_n)\) are non-decreasing such that \[ \sum^n_{k=0}\frac{p_{n-k}q_k}{k+1}=O\left(\frac{R_n}{n+1}\right). \] Remarks. The reviewer observes that there seems to be no reason to assume \(p > 1\) to be an integer (see page 20; line 12). Also the space \(W(L^p;\xi(t))\), as defined by the author on page 20; line 15, is not properly defined. Finally, the reviewer observes that the proof of the theorem breaks down on page 24; lines 2, 3 and 4, also last two lines of the page.