We discuss determination of jumps for functions with generalized bounded variation. The questions are motivated by A. Gelb
and E. Tadmor , F. M�ricz  and  and Q. L. Shi and X. L. Shi . Corollary 1 improves the results proved in B. I.
Golubov  and G. Kvernadze .
We study when sums of trigonometric series belong to given function classes. For this purpose we describe the Nikol’skii class
of functions and, in particular, the generalized Lipschitz class. Results for series with positive and general monotone coefficients
We prove that the conjugate convolution operators can be used to calculate jumps for functions. Our results generalize the theorems established by He and Shi. Furthermore, by using Lukács and Móricz's idea, we solve an open question posed by Shi and Hu.
In 1953 Nash  introduced the class of functions Φ. In this paper the behaviour of generalized Cesàro (C,αn)-means (αn∊(−1,0)) of trigonometric Fourier series of the classes Hω∩Φ in the space of continuous functions is studied. The sharpness of the results obtained is shown.
We describe the functions from Nikol’skii class in terms of behavior of their Fourier coefficients. Results for series with
general monotone coefficients are presented. The problem of strong approximation of Fourier series is also studied.
We generalize some old and new results on the determination of jumps of a periodic, Lebesgue integrable function f at each point of discontinuity of first kind in terms of the partial sums of the conjugate series to the Fourier series of
f in , and in terms of the Abel-Poisson means in , to some more general linear operators which satisfy some certain conditions.
The linear operators in discussion include the Fejér means, de la Vallée-Poussin means, and Bernstein-Rogosinski sums.