Degree of convergence for a class of linear operators (Q2708933)

Let \(\mathcal B\) be the set of functions \(f\) that are bounded measurable and periodic of period \(2\pi\), and denote by \(\widetilde f\) the conjugate of \(f\). Denote \({\mathcal K}_0:=\{f\in{\mathcal B}\mid\widetilde f\in \text{Lip } 1\}\). If \(S_k(f,x)\) are the partial sums of the Fourier series of \(f\), and \({\mathcal D}=(d_{nk})\) is a double infinite matrix, then the \(\mathcal D\) transform of the Fourier series of \(f\) is \(L_n(f,x)=\sum_{k=0}^\infty d_{nk}S_k(f,x)\). The authors take special Nörlund means which they call the generalized harmonic means, \(H_n^\delta(f)\), which are defined for \(\delta=1,2,\dots\) by the sequence \(p_n=b_n^\delta\), where \(\sum_{n=0}^\infty b_n^\delta z^n:=(-\log{(1-z)}/z)^\delta\). If \(\widetilde f\) is the indefinite integral of \(\widetilde f'\), assume that for a positive nondecreasing function \(\Lambda(t)\) in \((0,\pi)\), we have \(|\widetilde f'(x+t)-\widetilde f'(x)|\leq M\Lambda(t)\), uniformly in \(x,t\in(0,\pi)\), and denote by \({\mathcal F}^\delta\) the class of functions \(f\) such that \(\int_0^\pi[\Lambda(t)(\log{K/t})^\delta/t] dt<\infty\), \(K>\pi\) being a constant. Then the paper presumably proves that if \(f\in{\mathcal F}^\delta\) (the paper demands \(f\in{\mathcal F}^1\)), then \(\|f-H_n^\delta(f)\|=O(1/(n\log{(n+1)})\). Moreover, the saturation class corresponding to this order is a proper subclass of \({\mathcal K}_0\) and contains \({\mathcal F}^1\).NEWLINENEWLINEFor the entire collection see [Zbl 0947.00027].