The notion of cardinal exponential spline interpolant (of order zero) has been introduced and thoroughly investigated by Schoenberg [9, lo]. In a recent paper [I], Greville et al. have extended this notion to the case of order r > 1 (cf. Section 4 below). Moreover, these authors have thoroughly studied their extension in terms of the shift operator. On the other hand, the author of the present paper has used an alternative approach to certain classes of cardinal spline functions, adopting the viewpoint of integral transform theory and complex analysis. Since this method is based on the notion of “discontinuous factor” it produces complex contour integral representations (with non-compact integration paths) of cardinal spline functions in a very natural way [4-61. From these representations all the information that is needed may be obtained by the calculus of residues. It is the purpose of the present paper to establish in the same vein a complex contour integral representation of the cardinal exponential spline interpolants of the first order (Section 4). On the basis of this result we shall determine the pointwise convergence behaviour on R of the cardinal exponential spline interpolants of order r> 0 and successively higher degree (Section 5). For the reader’s convenience, a resume of the complex contour integral representation of cardinal exponential splines (of order zero) and the closely related Euler-Frobenius polynomials will be given in Sections 2 and 3, respectively. Finally, Section 6 summarizes some general principles concerning the application of integral transform techniques to the theory of (univariate) cardinal and periodic spline functions. The object is to emphasize the close connection of these classes of splines with the harmonic analysis.