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.
Summary By employing a novel idea and simple techniques, we substantially generalize the Turán type inequality for rational functions with real zeros and prescribed poles established by Min  to include Lp spaces for 1≤ p ≤ ∞ while loosing the restriction ρ > 2 at the same time.
To verify the universal validity of the ``two-sided'' monotonicity condition introduced in , we will apply it to include
more classical examples. The present paper selects the Lp convergence case for this purpose. Furthermore, Theorem 3 shows that our improvements are not trivial.