On the degree of approximation of conjugate of a function belonging to weighted \(W(L^p,\xi(t))\) class by matrix summability means of conjugate series of a Fourier series (Q2711244)

Let the Fourier series associated with \(f\in L_p\) \((p\geq 1)\) at a point \(x\) be given by NEWLINE\[NEWLINE{1\over 2} a_0+ \sum^\infty_{n=1} (a_n\cos nx+ b_n\sin nx).\tag{1}NEWLINE\]NEWLINE Then the series conjugate to (1) is given by NEWLINE\[NEWLINE\sum^\infty_{n=1} (a_n\sin nx- b_n\cos nx).\tag{2}NEWLINE\]NEWLINE Also write \(\psi(t)= f(x+ t)- f(x- t)\) and NEWLINE\[NEWLINE\overline f(x)= -{1\over 2\pi} \int^\pi_0 \psi(t)\text{cot }{1\over 2} t dt,NEWLINE\]NEWLINE which exists for almost all values of \(x\) and is called conjugate function of \(f\).NEWLINENEWLINENEWLINELet \(A= (a_{nk})\) \((k,n= 0,1,\dots)\) be a lower-triangular infinite matrix of nonnegative real numbers \(a_{nk}\) and let \(T_n(f)\) denote the \(A\)-transform of \(s_n(f)\), the \(n\)th partial sum of (1). In 1988, the present reviewer [Acta Math. Hung. 52, 199-205 (1988; Zbl 0704.42004)] obtained the degree of approximation of \(f\in L_p\) \((p\geq 1)\) by \(T_n(f)\) whenever \(\{a_{nk}\}^n_{k=0}\) is monotonic and that \(\sum^n_{k=0} a_{nk}= 1\). Since \(f\in L_p\) \((p> 1)\Rightarrow\overline f\in L_p\) \((p> 1)\) and (2) turns out to be the Fourier series of \(\overline f\), therefore it is trivial to obtain analogous results for the degree of approximation of \(\overline f\) by \(T_n(\overline f)\).NEWLINENEWLINENEWLINEUnder some complicated integrability conditions, in the present paper, the author has obtained the degree of approximation of \(\overline f\) by the \(A\)-transform of the Fourier series of \(\overline f\) whenever \(\{a_{nk}\}^n_{k=0}\) is monotonically increasing and \(f\in W(L_p\xi(t))\), a subspace of \(L_p\)-space defined by the present author [Tamkang J. Math. 30, 47-52 (1999; Zbl 1032.42004, preceding review)].NEWLINENEWLINENEWLINERemarks: We observe that the space \(W(L_p,\xi(t))\) is neither well-defined nor significant. Also the proof of the theorem is not correct since its proof breaks down at several places, for example, see page 285; line 3. One cannot get \(\int^{1/n}_1 t^{-q(\beta+ 2)}dt\), which provides the desired estimate. Also in line 12 (p. 285), one cannot get \(\xi({1\over n})\) from the previous line, since \(\xi(1/y)\) is decreasing with \(y\). Consequently, the required estimate cannot be obtained. Further, we observe that all the particular cases of the matrix means, as mentioned on page 280, cannot be deduced from the matrix considered in the theorem.