Additional results on the strong approximation of Fourier series (Q1059246)

The following two theorems are proved: I. If \(p>0\), \(\omega\) (\(\delta)\) is a modulus of continuity, and r is a nonnegative integer then \[ \sum^{\infty}_{m=1}\| \{\sum^{2^{m+1}}_{n=2^ m+1}n^{rp- 1}\omega^{-p}(1/n)| s_ n-f|^ p\}^{1/p}\| <\infty \] implies \(f\in \omega^ rH^{\omega}\), where \(s_ n\) and \(\| \cdot \|\) denote the n-th partial sum of Fourier series of \(f(\in C_{2\pi})\) and the maximum norm, respectively. II. If \(\omega =\omega_{\alpha}(\delta)\) satisfies the conditions: (i) for any \(\alpha '>\alpha\) there exists a natural number \(\mu =\mu (\alpha ')\) so that \(2^{\mu \alpha '}\omega_{\alpha}(2^{-n- \mu})>2\omega_{\alpha}(2^{-n})\) holds for all n(\(\geq 1)\); ii) for every natural number \(\nu\) there exists a natural number N(\(\nu)\) so that \(2^{\nu \alpha}\omega_{\alpha}(2^{-n-\nu})\leq 2\omega_{\alpha}(2^{-n})\) \((n>N(\nu))\) holds; and \(p>0\), \(0<\alpha <1\), r is a nonnegative integer, then \[ \| \sum^{\infty}_{n=1}n^{rp-1}\omega_{\alpha}^{-p}(1/n)| s_ n-f|^ p\| <\infty \] implies \(f\in W^ rH^{\omega_{\alpha}}\).