Determination of the jumps of a bounded function by its Fourier series
It is well known that the jumps of a function of bounded variation can be determined by means of the partial sums of its differentiated Fourier series as follows: \[ \lim_{n\to\infty} {(S_n(f;\theta))'\over n}= {1\over\pi} (f(\theta+)- f(\theta-)),\tag{\(*\)} \] where \(f\) is a \(2\pi\)-periodic function of bounded variation on \([-\pi,\pi]\), in sign: \(f\in V\). \textit{B. I. Golubov} [Mat. Zametki 12,19-28 (1972; Zbl 0256.42005)] generalized \((*)\) for classes \(V_p\) with \(p\geq 1\) and higher derivatives of the Fourier partial sums. Now, the present author extends \((*)\) to even larger classes of functions, such as \(V_\phi\), \(\Lambda\text{BV}\) (introduced by D. Waterman), and \(V[v]\) (introduced by Z. A. Čanturija), under some additional conditions on the generalized variation. It is also shown that these additional conditions are the best possible in certain sense.
jumps of a function of bounded variation
scientific article; zbMATH DE number 1146507
scientific article