On the rate of pointwise \((H, q)\)-summability of Fourier series (Q1812706)

In this paper the authors have given two approximation versions of a slightly modified Tandori's theorem and a fortiori that of Marcinkiewicz's result. For example, they proved the following Theorem: Suppose that \(f\in L(0,2\pi)\), \(-\infty< x<\infty\), \(2\leq q<\infty\) and \(p=q(q-1)^{-1}\). Write \(\varphi_ x(t)=\varphi_ x[f](t)=f(x+t)+f(x-t)-2f(x)\), \[ \omega_ x[f](\alpha,\beta)_ p= \sup_{{0<\lambda\leq\alpha\atop 0<\eta\leq\beta}}\left\{{1\over \lambda} \int^ \lambda_ 0 |\varphi_ x(u)|\Bigl({1\over \eta} \int^{u+\eta}_{u-\eta} |\varphi_ x(v)| dv\Bigr)^{p-1} du\right\}^{1/p} \] and \[ h_ k[f]_ p(x)=\left\{(k+1)^{2(1- p)}\sum^ k_{\mu=0} \sum^ k_{\nu=0}\bigl[(\mu+1)(\nu+1)\bigr]^{p-2}\Bigl(\omega_ x[f]\Bigl({\pi\over \mu+1},{\pi\over \nu+1}\Bigr)_ p\Bigr)^ p\right\}^{1/p}. \] Then \[ H_ n[f]_ q(x)\leq K\left\{{1\over n+1} \sum^ n_{k=0} \bigl(h_ k[f]_ p(x)\bigr)^ q\right\}^{1/q}\quad (n=1,2,3,\ldots). \] {}.