On the degree of approximation of continuous functions
A sequence \(c=(c_n)\) of non-negative numbers, tending to zero, is said to be of rest bounded variation (\(c\in \text{RBVS}\)) if there exists a constant \(K\) such that for every integer \(m\), \[ \sum_{n=m}^\infty| c_n-c_{n+1}| \leq Kc_m. \] A sequence \(c=(c_n)\) of non-negative numbers is said to be of head bounded variation (\(c\in \text{HBVS}\)) if there exists a constant \(K\) such that for every integer \(m\), \[ \sum_{n=0}^{m-1}| c_n-c_{n+1}| \leq Kc_m. \] These notions clearly (strictly) extend the usual notions of monotone sequences. The author already used this weakening of the monotony condition to extend miscellaneous results (see for instance [\textit{L. Leindler}, Anal. Math. 27, 279--285 (2001; Zbl 1002.42002) and Anal. Math. 28, No. 4, 279--286 (2002; Zbl 1026.42007)]). Here the author is interested in the generalization of four theorems of \textit{P. Chandra} [Acta Math. Hung. 52, No. 3/4, 199--205 (1988; Zbl 0704.42004); ibid. 62, No.~1, 21--23 (1993; Zbl 0794.42002)] by replacing a hypothesis of monotony by the previous weaker type. Roughly speaking, this deals with the majorization of \(\| T_n(f)-f\| _\infty\), where \(f\) is a \(2\pi\)-periodic continuous function and \(T_n(f)=\sum_{k=0}^na_{n,k}S_n(f)\), with \(S_n(f)\) being the \(n^{\text{th}}\) Fourier-partial sum and where \(A=(a_{n,k})\) is a lower triangular infinite matrix of real numbers. The hypothesis of quasi-monotony concerns the rows of \(A\). We point out a minor mistake (rather a misprint) in the proof of Lemma 3. In (3.6), the \({1\over2}\) has to be replaced by \({\tau+1\over2}\). Then it is easy to conclude, since the RBVS hypothesis easily implies that \(\tau a_{n,\tau}=O(A_{n,\tau})\), that (3.3) (and then the Lemma) remains true.
scientific article; zbMATH DE number 2186343