Strong summability of Fourier series of the periodic functions from \(H^ p(0 <p\leq 1)\) (Q1922534)

Let \(f(x)\) be boundary values of a function from the Hardy class in \(|z|<1\), and \[ \sigma^{\alpha,k}_m(f):= \sum^m_{j=0} \Biggl(1-\Biggl({1\over m}\Biggr)^k\Biggr)^\alpha \widehat f(j)e^{ijx}. \] When \(\alpha= 0\), \(\sigma^{0,k}_m(f)\) become the partial sums \(S_m(f)\) of the Fourier series. The author gives a simple ``harmonic'' proof of the inequality \[ (\log N)^{-1}\sum^N_{m=1}{1\over m} |S_m(f)|_{H^1}\leq C|f|_{H^1}, \] furthermore he proves two theorems: I. For every \(f\in H^1(\mathbb{T})\), \[ \sum^N_{j=1} {1\over j} |f-S_j(f)|_{H^1}\simeq\sum^N_{j=1} {1\over j} E_j(f;H^1). \] II. For every \(f\in H^p\) and \(\alpha={1\over p}-1\), \[ \sum^N_{j=1} {1\over j} |\sigma^{\alpha,2}_j-f|^p_p\simeq \sum^N_{j=1} {1\over j} E^p_j(f;H^p). \] Finally, he presents a new formula for the \(K\)-functional between \(H^p\) and \(H^{p,k}\).