Everywhere divergent \(\Phi\)-means of Fourier series
It was proved by \textit{L. D. Gogoladze} [Mat. Sb., N. Ser. 135(177), No. 2, 158--168 (1988; Zbl 0647.42008)] and \textit{V. A. Rodin} [Mat. Sb. 182, No. 10, 1463--1478 (1991; Zbl 0741.42017)] that for any \(A>0\) the means \[ n^{-1}\sum^n_{k=1}(\exp\{A| S_k(x)-f(x)| \}-1) \] tend to \(0\) a.e., for every \(f\in L^1(\mathbb{T})\). Here \(S_k(x)\) is the \(k\)-partial sum of the Fourier series of \(f\). This is the ultimate refinement of the strong summability theorem of Zygmund and Marcinkiewicz: Let \(\Phi(t):[0,\infty)\to [0,\infty), \Phi(0)=0,\) be a continuous function satisfying \(\limsup_{t\to\infty}t^{-1}\log\Phi(t)=\infty.\) In [J. Contemp. Math. Anal., Armen. Acad. Sci. 26, No. 2, 66--69 (1991); translation from Izv. Akad. Nauk Armen., Mat. 26, No. 2, 159--162 (1991; Zbl 0849.42006)] and his dissertation the author has shown that a Fourier series \(f\) exists that, together with its conjugate, has an a.e. unboundedly divergent \(\Phi\)-means. In the present paper this result is extended to divergence everywhere.
scientific article; zbMATH DE number 5155495
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