On the uniform convergence of the Fourier series of functions of bounded variation
The author proves three interesting theorems regarding the subject given by the title of the paper. Here we recall only one of the theorems:NEWLINENEWLINENEWLINELet \(f\) be a \(2\pi\)-periodic function of bounded variation with the Fourier series NEWLINE\[NEWLINE{a_0\over 2}+ \sum^\infty_{k=1} (a_k\cos kx+ b_k\sin kx).NEWLINE\]NEWLINE Then, for every sequence \(\{n_j\}\) with \(\sum^\infty_{j=m}{1\over n_j}\leq {A\over n_m}\), the series NEWLINE\[NEWLINE\sum^\infty_{j=1} \Biggl|\sum^{n_{j+1}-1}_{k= n_j} (a_k\cos kx+ b_k\sin kx)\Biggr|NEWLINE\]NEWLINE is uniformly convergent at the points of continuity of \(f\) and is uniformly convergent on an interval \([a,b]\) if \(f\) is continuous at each point of this interval.NEWLINENEWLINENEWLINEThe other two theorems present estimates for the coefficients, and the convergence rate of the Fourier series of functions whose derivatives of prescribed order have bounded variation.NEWLINENEWLINEFor the entire collection see [Zbl 0981.00017].
scientific article; zbMATH DE number 1729334