On the uniform convergence of sine integrals

The classical theorem of Chaundry and Jolliffe states that the sine series ∑ k = 1 ∞ a k sin k x with coefficients a 1 ⩾ a 2 ⩾ ⋯ ⩾ a k ⩾ ⋯ ⩾ 0 converges uniformly in x if and only if (∗) k a k → 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 ∫ 0 ∞ f ( x ) sin t x d x , where f : R + → C is a measurable function with the property x f ( x ) ∈ L loc 1 ( R + ) .