Remarks on some classes of Fourier coefficients
The paper concerns the general problem of finding some conditions on the coefficients of trigonometric series \({a_0\over2}+\sum_{n=1}^\infty a_n\cos(nx)\) under which it is the Fourier series of some \(f\) in \(L^1(0,\pi)\) and converges in \(L^1(0,\pi)\). Several classes were already introduced in previous papers. Roughly speaking, all these classes deal with some perturbation of the monotony. For example, in [Math. Inequal. Appl. 1, No. 4, 527--532 (1998; Zbl 0922.42001)], the author introduced the class \({\mathcal S}_p(\delta)\) (where \(p>1\)) of null sequences \((a_n)\) verifying that there exists a sequence \((A_n)\), which is \(\delta\)-quasi-monotone, such that \(\sum_{n=1}^\infty n\delta_n < \infty\), \(\sum_{n=1}^\infty A_n<\infty\) and \({1\over n}\sum_{k=1}^n{| a_k-a_{k+1}| ^p\over A_k^p}\) is bounded. Recall that, given a sequence \(\delta=(\delta_n)\) of positive numbers, a sequence \((A_n)\) is \(\delta\)-quasi-monotone if \(A_n\rightarrow 0\), \(A_n>0\) ultimately and for every \(n\), \(A_{n+1}-A_n\leq\delta_n\). Here the author shows that this class coincides with several other classes of coefficients introduced by \textit{C. V. Stanojevic} and \textit{V. B. Stanojevic} [Proc. Am. Math. Soc. 101, 679--684 (1987; Zbl 0647.42007)] and \textit{S. Sheng} [Proc. Am. Math. Soc. 110, No. 4, 895--904 (1990; Zbl 0717.42004)], namely the classes \({\mathcal S}_p\) and \({\mathcal S}'_p\).
scientific article; zbMATH DE number 2098990