Embedding results pertaining to strong approximation of Fourier series. III (Q1382413)

Let \(\omega\) be a nondecreasing, continuous and subadditive function on \([0,2\pi]\) such that \(\omega(0)=0\). By \(W^r H^\omega_s\), where \(r\) is a nonnegative integer, we denote the class of odd, continuous functions \(f\) with Fourier series \(\sum^\infty_{n=1} b_n\) \(\sin nt \) with partial sums \(s_n(t)\), satisfying the condition \(\sum^n_{k=m}| b_k-b_{k+1}| \leq K(b) | b_m|\) for all \(n\geq m\), and with modulus of continuity \(\omega(f^{(r)},\delta)=O(\omega(\delta))\). It is proved that the embedding \(W^rH^\omega_s \subset H(\beta,p,r,\omega)\) holds, if the sequence \(\{n^{\beta-pr}(\omega(1/n))^p\}\) is quasi monotone increasing, \(\beta,p>0\), and \(H(\beta,p,r,\omega)\) is the class of functions whose strong means \(h_n(f,\beta,p)\) in the \(C_{2\pi}\)-norm of the sequence \(\{| s_k(x)-f(x)|\}\) are \(O(n^{-r}\omega(1/n))\). For Part I and Part II see the preceding two reviews.