L\({}^ 1\)-convergence of Fourier series with O-regularly varying quasimonotonic coefficients (Q921395)

The main result of this paper is the following Theorem: Let \(\{a_ n\}\) be an O-regularly varying quasi-monotonic sequence, and for some even \(f\in L^ 1(0,\Pi)\) let \(\hat f(n)=a_ n\) for \(n=0,1,2..\). Then the necessary and sufficient condition for \(\| S_ n(f)-f\| =o(1)\), \(n\to \infty\) is \(a_ nlg n=o(1)\), \(n\to \infty\), where \(S_ n(f)=a_{o/2}+\sum^{n}_{k=1}a_ k \cos kx\) and \(\| \cdot \|\) denotes \(L^ 1(0,\Pi)\) norm. This theorem includes as corollary a result due to \textit{W. O. Bray} and \textit{Č. V. Stanojevič} [Proc. Am. Math. Soc. 96, 53-61 (1986; Zbl 0607.42008)].