On the uniform convergence of sine integrals

The classical theorem of Chaundry and Jolliffe states that the sine series with coefficients a1⩾a2⩾⋯⩾ak⩾⋯⩾0 converges uniformly in x if and only if (∗) kak→0 as k→∞. Recently the monotonicity condition has been relaxed by a number of authors. An analysis of the proofs of these results reveals that condition (∗) is sufficient for the uniform convergence even in the case of complex coefficients, under appropriately modified conditions. But our main achievement is the extension of these results for the sine integral , where is a measurable function with the property .