Bispectral and (glN, glM) dualities, discrete versus differential

We define integral transforms which establish a correspondence between spaces of quasi-exponentials and spaces of quasi-polynomials under certain conditions. As a corollary of the properties of our transforms we obtain a correspondence between solutions to the Bethe ansatz equations of two (gl(N), gl(M)) dual quantum integrable models: one is the special trigonometric Gaudin model and the other is the special XXX model. (c) 2007 Elsevier Inc. All rights reserved.