converges uniformly (or more generally is uniformly summable (C, k)) to zero as w-+ oc. Moreover, the coefficients am exhibit a regularity in their behavior as i m oo which is connected with that of ?(u). This result enables one to obtain theorems about the behavior of trigonometric integrals from the known theorems about trigonometric series. It is also shown in [5] that integrals of the type fleixudq(u) are equiconvergent with integrals of the form f7ei-xua(u)du, where a(u) is continuous. The purpose of this paper is to extend these results in certain cases to double trigonometric integrals, using restricted and circular convergence. When circular convergence is used extensions to higher dimensions are also possible. 1.1. Definitions and notations. Let 4 be a real-valued set function defined on the class of Borel sets 23 in the plane and satisfying the conditions